From be28607531a9093bbce827fff7b2efd89b093e99 Mon Sep 17 00:00:00 2001 From: Pedro Maciel Xavier Date: Wed, 15 Feb 2023 23:39:49 -0300 Subject: [PATCH 01/38] Organize manual & booklet chapters in separate files --- docs/make.jl | 16 ++- docs/src/booklet.md | 121 ---------------------- docs/src/booklet/1-intro.md | 11 ++ docs/src/booklet/2-qubo.md | 29 ++++++ docs/src/booklet/3-pbo.md | 23 ++++ docs/src/booklet/4-encoding.md | 25 +++++ docs/src/booklet/5-virtual.md | 21 ++++ docs/src/booklet/6-solvers.md | 21 ++++ docs/src/examples/prime_factorization.md | 2 +- docs/src/{manual.md => manual/1-start.md} | 23 +--- docs/src/manual/2-model.md | 1 + docs/src/manual/3-results.md | 1 + docs/src/manual/4-settings.md | 25 +++++ src/attributes/compiler.jl | 14 ++- src/model/virtual.jl | 24 ++++- 15 files changed, 210 insertions(+), 147 deletions(-) delete mode 100644 docs/src/booklet.md create mode 100644 docs/src/booklet/1-intro.md create mode 100644 docs/src/booklet/2-qubo.md create mode 100644 docs/src/booklet/3-pbo.md create mode 100644 docs/src/booklet/4-encoding.md create mode 100644 docs/src/booklet/5-virtual.md create mode 100644 docs/src/booklet/6-solvers.md rename docs/src/{manual.md => manual/1-start.md} (51%) create mode 100644 docs/src/manual/2-model.md create mode 100644 docs/src/manual/3-results.md create mode 100644 docs/src/manual/4-settings.md diff --git a/docs/make.jl b/docs/make.jl index c9832a7d..86d568e3 100644 --- a/docs/make.jl +++ b/docs/make.jl @@ -19,12 +19,24 @@ makedocs(; authors="Pedro Xavier and Tiago Andrade and Joaquim Garcia and David Bernal", pages=[ "Home" => "index.md", - "Manual" => "manual.md", + "Manual" => [ + "Getting Started" => "manual/1-start.md", + "Running a Model" => "manual/2-model.md", + "Gathering Results" => "manual/3-results.md", + "Compiler Settings" => "manual/4-settings.md", + ], "Examples" => [ "Knapsack" =>"examples/knapsack.md", "Prime Factorization" => "examples/prime_factorization.md", ], - "Booklet" => "booklet.md" + "Booklet" => [ + "Introduction" => "booklet/1-intro.md", + "QUBO" => "booklet/2-qubo.md", + "PBO" => "booklet/3-pbo.md", + "Encoding" => "booklet/4-encoding.md", + "Virtual Mapping" => "booklet/5-virtual.md", + "Solvers" => "booklet/6-solvers.md", + ] ], workdir="." ) diff --git a/docs/src/booklet.md b/docs/src/booklet.md deleted file mode 100644 index 5eec2071..00000000 --- a/docs/src/booklet.md +++ /dev/null @@ -1,121 +0,0 @@ -# ToQUBO.jl Booklet -This booklet aims to gather the theoretical and practical details behind `ToQUBO` and provide documentation for project internals. The target audience includes, among others, advanced users and those willing to contribute to the project. The latter are advised to read the following sections, as they give a glimpse of the ideas employed up to now. - -## QUBO -```math -\begin{array}{rl} - \min & \mathbf{x}^{\intercal} Q\,\mathbf{x} \\ - \text{s.t.} & \mathbf{x} \in \mathbb{B}^{n} -\end{array} -``` - -```@docs -ToQUBO.isqubo -ToQUBO.toqubo -ToQUBO.toqubo! -``` - -```@docs -ToQUBO.toqubo_sense! -ToQUBO.toqubo_variables! -ToQUBO.toqubo_constraint -ToQUBO.toqubo_constraints! -ToQUBO.toqubo_objective! -ToQUBO.toqubo_penalties! -ToQUBO.toqubo_parse! -ToQUBO.toqubo_build! -``` - -## Pseudo-Boolean Optimization -Internally, problems are represented through a Pseudo-Boolean Optimization (PBO) framework. The main goal is to represent a given problem using a Pseudo-Boolean Function (PBF) since there is an immediate correspondence between quadratic PBFs and QUBO forms. - -```@docs -ToQUBO.PBO.PseudoBooleanFunction -ToQUBO.PBO.residual -ToQUBO.PBO.derivative -ToQUBO.PBO.gradient -ToQUBO.PBO.gap -ToQUBO.PBO.sharpness -ToQUBO.PBO.relaxed_gcd -``` - -### A Primer on Submodularity -A set function ``f : 2^{S} \to \mathbb{R}`` is said to be submodular if - -```math -f(X \cup Y) + f(X \cap Y) \le f(X) + f(Y) \forall X, Y \subset S -``` - -holds. - -## Quadratization -In order to successfully achieve a QUBO formulation, sometimes it is needed to quadratize the resulting PBF, i.e., reduce its degree until reaching the quadratic case. There are many quadratization methods available, and `ToQUBO` implements a few of them. - -```@docs -ToQUBO.PBO.quadratize! -``` - -## Virtual Mapping -During reformulation, `ToQUBO` holds two distinct models, namely the *Source Model* and the *Target Model*. The source model is a generic `MOI` model restricted to the supported constraints. The target one is on the QUBO form used during the solving process. Both lie within a *Virtual Model*, which provides the necessary API integration and keeps all variable and constraint mapping tied together. - -This is done in a transparent fashion for both agents since the user will mostly interact with the presented model, and the solvers will only access the generated one. - -## Virtual Variables -Every virtual model stores a collection of virtual variables, intended to provide a link between those in the source and those to be created in the target model. Each virtual variable stores enconding information for later expansion and evaluation. - -```@docs -ToQUBO.VirtualVariable -ToQUBO.encode! -``` - -## Variable Encoding - -### Linear Encoding Methods -```@docs -ToQUBO.LinearEncoding -ToQUBO.Binary -ToQUBO.Unary -ToQUBO.Arithmetic -ToQUBO.OneHot -``` - -### Sequential Encoding Methods -```@docs -ToQUBO.SequentialEncoding -ToQUBO.DomainWall -``` - -### Bounded Coefficients -```@docs -ToQUBO.Bounded -``` - -## Virtual Models -```@docs -ToQUBO.VirtualModel -``` - -### Annealing & Sampling -`ToQUBO`'s main goal is to benefit from non-deterministic samplers, especially *Quantum Adiabatic* devices and other *Annealing* machines. A few `MOI`-compliant interfaces for annealers and samplers are bundled within `ToQUBO` via the `Anneal.jl` submodule and package prototype. Some of them are presented below. - -### Quantum Annealing -Interfacing with [D-Wave](https://www.dwavesys.com/)'s quantum computers is one of the milestones we expect to achieve with this package. Like other proprietary optimization resources such as [Gurobi](https://gurobi.com) and [FICO® Xpress](https://www.fico.com/en/products/fico-xpress-solver), this requires licensing and extra steps are needed to get access to it. In a first moment, for those willing to get started, the *Simulated Annealing* optimizer might be enough. - -While in `JuMP`, run `using Anneal` and look for `QuantumAnnealer.Optimizer`. - -### Simulated Annealing -Provided by D-Wave's open-source code libraries, this [Simulated Annealing](https://en.wikipedia.org/wiki/Simulated_annealing) engine implements some of the features and configuration you would find using the Quantum API. Its adoption is recommended for basic usage, tests, and during early research steps due to its simplicity and ease of use. It does not implement the most advanced Simulated Annealing algorithm available but performs fairly well on small instances. `Anneal.jl` exports this interface as `SimulatedAnnealer.Optimizer`. - -### Random Sampling -This sampler is implemented for test purposes and simply assigns 0 or 1 to each variable according to a given probability bias ``0 \le p \le 1``, which defaults to ``p = 0.5``. After running the `using Anneal` command, `RandomSampler.Optimizer` will be available. - -### Exact Solver (Exaustive Enumeration) -Also made to be used in tests, the `ExactSolver.Optimizer` interface runs through all possible state configurations, which implies in an exponential time complexity on the number of variables. Thus, only problems with no more than 20 variables should be provided. - -## MIQP Solvers -The most accessible alternative to Annealers and Samplers are Mixed-Integer Quadratic Programming (MIQP) Solvers such as [Gurobi](https://github.com/jump-dev/Gurobi.jl) and [CPLEX](https://github.com/jump-dev/CPLEX.jl). These are not intended to be of regular use alongside `ToQUBO` since reformulation usually makes things harder for these folks. Yet, there are still cases where they may be suitable for tests on small instances. - -### Custom Error Types -```@docs -ToQUBO.QUBOError -``` \ No newline at end of file diff --git a/docs/src/booklet/1-intro.md b/docs/src/booklet/1-intro.md new file mode 100644 index 00000000..8696524f --- /dev/null +++ b/docs/src/booklet/1-intro.md @@ -0,0 +1,11 @@ +# ToQUBO.jl Booklet + +This booklet aims to gather the theoretical and practical details behind `ToQUBO` and provide documentation for project internals. +The target audience includes, among others, advanced users and those willing to contribute to the project. +The latter are advised to read the following sections, as they give a glimpse of the ideas employed up to now. + +## Table of Contents +```@contents +Pages = ["2-qubo.md", "3-pbo.md", "4-encoding.md", "5-virtual.md", "6-solvers.md"] +Depth = 2 +``` \ No newline at end of file diff --git a/docs/src/booklet/2-qubo.md b/docs/src/booklet/2-qubo.md new file mode 100644 index 00000000..85d1cd7c --- /dev/null +++ b/docs/src/booklet/2-qubo.md @@ -0,0 +1,29 @@ +# QUBO + +## Definition + +```math +\begin{array}{rl} + \min & \mathbf{x}^{\intercal} Q\,\mathbf{x} \\ + \text{s.t.} & \mathbf{x} \in \mathbb{B}^{n} +\end{array} +``` + +## OK, but why QUBO? + +```@docs +ToQUBO.isqubo +ToQUBO.toqubo +ToQUBO.toqubo! +``` + +```@docs +ToQUBO.toqubo_sense! +ToQUBO.toqubo_variables! +ToQUBO.toqubo_constraint +ToQUBO.toqubo_constraints! +ToQUBO.toqubo_objective! +ToQUBO.toqubo_penalties! +ToQUBO.toqubo_parse! +ToQUBO.toqubo_build! +``` \ No newline at end of file diff --git a/docs/src/booklet/3-pbo.md b/docs/src/booklet/3-pbo.md new file mode 100644 index 00000000..9cbc0b5b --- /dev/null +++ b/docs/src/booklet/3-pbo.md @@ -0,0 +1,23 @@ +# Pseudo-Boolean Optimization +Internally, problems are represented through a Pseudo-Boolean Optimization (PBO) framework. +The main goal is to represent a given problem using a Pseudo-Boolean Function (PBF) since there is an immediate correspondence between quadratic PBFs and QUBO forms. + +```@docs +ToQUBO.PBO.PseudoBooleanFunction +``` + +## Quadratization +In order to successfully achieve a QUBO formulation, sometimes it is needed to quadratize the resulting PBF, i.e., reduce its degree until reaching the quadratic case. There are many quadratization methods available, and `ToQUBO` implements a few of them. + +```@docs +ToQUBO.PBO.quadratize! +``` + +### A Primer on Submodularity +A set function ``f : 2^{S} \to \mathbb{R}`` is said to be submodular if + +```math +f(X \cup Y) + f(X \cap Y) \le f(X) + f(Y) \forall X, Y \subset S +``` + +holds. \ No newline at end of file diff --git a/docs/src/booklet/4-encoding.md b/docs/src/booklet/4-encoding.md new file mode 100644 index 00000000..26e7b651 --- /dev/null +++ b/docs/src/booklet/4-encoding.md @@ -0,0 +1,25 @@ +# Encoding Methods + +## Variable Encoding + +### Linear Encoding Methods +```@docs +ToQUBO.LinearEncoding +ToQUBO.Binary +ToQUBO.Unary +ToQUBO.Arithmetic +ToQUBO.OneHot +``` + +### Sequential Encoding Methods +```@docs +ToQUBO.SequentialEncoding +ToQUBO.DomainWall +``` + +### Bounded Coefficients +```@docs +ToQUBO.Bounded +``` + +## Constraint Encoding \ No newline at end of file diff --git a/docs/src/booklet/5-virtual.md b/docs/src/booklet/5-virtual.md new file mode 100644 index 00000000..ff61ed04 --- /dev/null +++ b/docs/src/booklet/5-virtual.md @@ -0,0 +1,21 @@ +# Virtual Mapping +During reformulation, `ToQUBO` holds two distinct models, namely the *Source Model* and the *Target Model*. +The source model is a generic `MOI` model restricted to the supported constraints. +The target one is on the QUBO form used during the solving process. +Both lie within a *Virtual Model*, which provides the necessary API integration and keeps all variable and constraint mapping tied together. + +This is done in a transparent fashion for both agents since the user will mostly interact with the presented model, and the solvers will only access the generated one. + +## Virtual Model +```@docs +ToQUBO.VirtualModel +``` + +## Virtual Variables +Every virtual model stores a collection of virtual variables, intended to provide a link between those in the source and those to be created in the target model. +Each virtual variable stores enconding information for later expansion and evaluation. + +```@docs +ToQUBO.VirtualVariable +ToQUBO.encode! +``` \ No newline at end of file diff --git a/docs/src/booklet/6-solvers.md b/docs/src/booklet/6-solvers.md new file mode 100644 index 00000000..475b52ab --- /dev/null +++ b/docs/src/booklet/6-solvers.md @@ -0,0 +1,21 @@ +# QUBO Solvers + +## Solvers, Annealers & Samplers +`ToQUBO`'s main goal is to benefit from non-deterministic samplers, especially *Quantum Adiabatic* devices and other *Annealing* machines. A few `MOI`-compliant interfaces for annealers and samplers are bundled within `ToQUBO` via the `Anneal.jl` submodule and package prototype. Some of them are presented below. + +### Simulated Annealing +Provided by D-Wave's open-source code libraries, this [Simulated Annealing](https://en.wikipedia.org/wiki/Simulated_annealing) engine implements some of the features and configuration you would find using the Quantum API. Its adoption is recommended for basic usage, tests, and during early research steps due to its simplicity and ease of use. It does not implement the most advanced Simulated Annealing algorithm available but performs fairly well on small instances. `Anneal.jl` exports this interface as `SimulatedAnnealer.Optimizer`. + +### Quantum Annealing +Interfacing with [D-Wave](https://www.dwavesys.com/)'s quantum computers is one of the milestones we expect to achieve with this package. Like other proprietary optimization resources such as [Gurobi](https://gurobi.com) and [FICO® Xpress](https://www.fico.com/en/products/fico-xpress-solver), this requires licensing and extra steps are needed to get access to it. In a first moment, for those willing to get started, the *Simulated Annealing* optimizer might be enough. + +While in `JuMP`, run `using Anneal` and look for `QuantumAnnealer.Optimizer`. + +### Random Sampling +This sampler is implemented for test purposes and simply assigns 0 or 1 to each variable according to a given probability bias ``0 \le p \le 1``, which defaults to ``p = 0.5``. After running the `using Anneal` command, `RandomSampler.Optimizer` will be available. + +### Exact Solver (Exaustive Enumeration) +Also made to be used in tests, the `ExactSolver.Optimizer` interface runs through all possible state configurations, which implies in an exponential time complexity on the number of variables. Thus, only problems with no more than 20 variables should be provided. + +## MIQP Solvers +The most accessible alternative to Annealers and Samplers are Mixed-Integer Quadratic Programming (MIQP) Solvers such as [Gurobi](https://github.com/jump-dev/Gurobi.jl) and [CPLEX](https://github.com/jump-dev/CPLEX.jl). These are not intended to be of regular use alongside `ToQUBO` since reformulation usually makes things harder for these folks. Yet, there are still cases where they may be suitable for tests on small instances. diff --git a/docs/src/examples/prime_factorization.md b/docs/src/examples/prime_factorization.md index 86391dce..670cfc84 100644 --- a/docs/src/examples/prime_factorization.md +++ b/docs/src/examples/prime_factorization.md @@ -25,7 +25,7 @@ using ToQUBO using DWaveNeal function factor(R::Integer; optimizer = DWaveNeal.Optimizer) - return factor(identity, R; optimizer = optimizer) + return factor(identity, R; optimizer) end function factor(config!::Function, R::Integer; optimizer = DWaveNeal.Optimizer) diff --git a/docs/src/manual.md b/docs/src/manual/1-start.md similarity index 51% rename from docs/src/manual.md rename to docs/src/manual/1-start.md index bc4c311a..5deae6c3 100644 --- a/docs/src/manual.md +++ b/docs/src/manual/1-start.md @@ -19,23 +19,8 @@ optimize!(model) solution_summary(model) ``` -## Compiler Flags - -### Architecture -```@docs -ToQUBO.Attributes.Architecture -``` - -### Quadratization -```@docs -ToQUBO.Attributes.Quadratize -ToQUBO.Attributes.QuadratizationMethod -ToQUBO.Attributes.StableQuadratization -``` - -### Variable & Constraint Encoding -```@docs -ToQUBO.Attributes.VariableEncodingMethod -ToQUBO.Attributes.VariableEncodingPenalty -ToQUBO.Attributes.ConstraintEncodingPenalty +## Table of Contents +```@contents +Pages = ["2-model.md", "3-results.md", "4-settings.md"] +Depth = 2 ``` \ No newline at end of file diff --git a/docs/src/manual/2-model.md b/docs/src/manual/2-model.md new file mode 100644 index 00000000..b0e734d3 --- /dev/null +++ b/docs/src/manual/2-model.md @@ -0,0 +1 @@ +# Running a Model \ No newline at end of file diff --git a/docs/src/manual/3-results.md b/docs/src/manual/3-results.md new file mode 100644 index 00000000..70a7b2cc --- /dev/null +++ b/docs/src/manual/3-results.md @@ -0,0 +1 @@ +# Gathering Results \ No newline at end of file diff --git a/docs/src/manual/4-settings.md b/docs/src/manual/4-settings.md new file mode 100644 index 00000000..09e18235 --- /dev/null +++ b/docs/src/manual/4-settings.md @@ -0,0 +1,25 @@ +# Compiler Settings + +## Architecture +```@docs +ToQUBO.Attributes.Architecture +``` + +## Quadratization +```@docs +ToQUBO.Attributes.Quadratize +ToQUBO.Attributes.QuadratizationMethod +ToQUBO.Attributes.StableQuadratization +``` + +## Variable & Constraint Encoding +```@docs +ToQUBO.Attributes.VariableEncodingBits +ToQUBO.Attributes.DefaultVariableEncodingBits +ToQUBO.Attributes.VariableEncodingATol +ToQUBO.Attributes.DefaultVariableEncodingATol +ToQUBO.Attributes.VariableEncodingMethod +ToQUBO.Attributes.DefaultVariableEncodingMethod +ToQUBO.Attributes.VariableEncodingPenalty +ToQUBO.Attributes.ConstraintEncodingPenalty +``` \ No newline at end of file diff --git a/src/attributes/compiler.jl b/src/attributes/compiler.jl index b0396078..c3bc34ea 100644 --- a/src/attributes/compiler.jl +++ b/src/attributes/compiler.jl @@ -1,6 +1,14 @@ module Attributes -import ..ToQUBO +import ..ToQUBO: + ToQUBO, + Unary, + Binary, + Arithmetic, + OneHot, + DomainWall, + Bounded + import MathOptInterface as MOI const MOIU = MOI.Utilities const VI = MOI.VariableIndex @@ -75,7 +83,7 @@ end QuadratizationMethod() Defines which quadratization method to use. -Available options are defined in the [`PBO`](@ref) submodule. +Available options are defined in the `PBO` submodule. """ struct QuadratizationMethod <: CompilerAttribute end function MOI.get(model::ToQUBO.VirtualModel, ::QuadratizationMethod) @@ -293,7 +301,7 @@ end MOI.is_set_by_optimize(::ConstraintEncodingPenalty) = true @doc raw""" - QUBONormalForm + QUBONormalForm() """ struct QUBONormalForm <: CompilerAttribute end function MOI.get( diff --git a/src/model/virtual.jl b/src/model/virtual.jl index 3d938512..26690ba0 100644 --- a/src/model/virtual.jl +++ b/src/model/virtual.jl @@ -470,7 +470,17 @@ function encode!( return encode!(model, e, x, a, b, n) end -abstract type SequentialEncoding <: Encoding end +@doc raw""" + SequentialEncoding + +A *sequential encoding* is one of the form + +```math +\xi(\mathbf{y}) = \sum_{i = 1}^{n} \gamma_{i} \left({y_{i + 1} \ast y_{i}}\right) +``` + +where ``\mathbf{y} \in \mathbb{B}^{n + 1}``. +""" abstract type SequentialEncoding <: Encoding end function encode!( model::VirtualModel{T}, @@ -551,8 +561,17 @@ function encode!( end @doc raw""" + Bounded{E,T}(μ::T) where {E<:Encoding,T} + +The bounded-coefficient encoding method[^Karimi2019] consists in limiting the magnitude of the +coefficients in the encoding expansion to a parameter ``\mu``. + +[^Karimi2019]: + Karimi, S. & Ronagh, P. **Practical integer-to-binary mapping for quantum annealers**. *Quantum Inf Process 18, 94* (2019). [{doi}](https://doi.org/10.1007/s11128-019-2213-x) + Bounded{Binary,T}(μ::T) where {T} +## Rationale Let ``x \in [a, b] \subset \mathbb{Z}`` and ``n = b - a``. First, @@ -582,6 +601,7 @@ and \epsilon = n - 2^{k} + 1 - r \times \mu ``` +```math \gamma_{j} = \left\lbrace\begin{array}{cl} 2^{j} & \text{if } 1 \le j \le k \\ \mu & \text{if } k < j < r + k \\ @@ -591,6 +611,8 @@ and Bounded{Unary,T}(μ::T) where {T} +Let ``x \in [a, b] \subset \mathbb{Z}`` and ``n = b - a``. + """ struct Bounded{E<:LinearEncoding,T} <: LinearEncoding μ::T From 604620b4cd66ab980c21b99f53ad6ccc13bf9b37 Mon Sep 17 00:00:00 2001 From: Pedro Maciel Xavier Date: Thu, 16 Feb 2023 10:18:33 -0300 Subject: [PATCH 02/38] Fix authors' names --- README.md | 4 ++-- docs/make.jl | 22 +++++++++++----------- 2 files changed, 13 insertions(+), 13 deletions(-) diff --git a/README.md b/README.md index 26982553..862027b9 100644 --- a/README.md +++ b/README.md @@ -136,12 +136,12 @@ If you think this list is incomplete, consider creating an [Issue](https://githu If you use `ToQUBO.jl` in your work, we kindly ask you to include the following citation: ```tex @software{toqubo:2022, - author = {Pedro Xavier and Tiago Andrade and Joaquim Garcia and David Bernal}, + author = {Pedro Maciel Xavier and Pedro Ripper and Tiago Andrade and Joaquim Dias Garcia and David E. Bernal Neira}, title = {{ToQUBO.jl}}, month = {mar}, year = {2022}, publisher = {Zenodo}, - version = {v0.1.0}, + version = {v0.1.5}, doi = {10.5281/zenodo.6387592}, url = {https://doi.org/10.5281/zenodo.6387592} } diff --git a/docs/make.jl b/docs/make.jl index 86d568e3..97125efc 100644 --- a/docs/make.jl +++ b/docs/make.jl @@ -7,17 +7,17 @@ DocMeta.setdocmeta!( ) makedocs(; - modules=[ToQUBO], - doctest=true, - clean=true, - format=Documenter.HTML( - assets = ["assets/extra_styles.css", "assets/favicon.ico"], - mathengine=Documenter.MathJax2(), - sidebar_sitename=false, + modules = [ToQUBO], + doctest = true, + clean = true, + format = Documenter.HTML( + assets = ["assets/extra_styles.css", "assets/favicon.ico"], + mathengine = Documenter.KaTeX(), + sidebar_sitename = false, ), - sitename="ToQUBO.jl", - authors="Pedro Xavier and Tiago Andrade and Joaquim Garcia and David Bernal", - pages=[ + sitename = "ToQUBO.jl", + authors = "Pedro Maciel Xavier and Pedro Ripper and Tiago Andrade and Joaquim Dias Garcia and David E. Bernal Neira", + pages = [ "Home" => "index.md", "Manual" => [ "Getting Started" => "manual/1-start.md", @@ -26,7 +26,7 @@ makedocs(; "Compiler Settings" => "manual/4-settings.md", ], "Examples" => [ - "Knapsack" =>"examples/knapsack.md", + "Knapsack" => "examples/knapsack.md", "Prime Factorization" => "examples/prime_factorization.md", ], "Booklet" => [ From aff1f3b11d4481d0666c200251a8b087a96a92b4 Mon Sep 17 00:00:00 2001 From: Pedro Maciel Xavier Date: Thu, 16 Feb 2023 10:54:43 -0300 Subject: [PATCH 03/38] Update text + Fix typos --- docs/src/booklet/5-virtual.md | 2 +- docs/src/booklet/6-solvers.md | 34 +++++++++++++++++++++------------- 2 files changed, 22 insertions(+), 14 deletions(-) diff --git a/docs/src/booklet/5-virtual.md b/docs/src/booklet/5-virtual.md index ff61ed04..6c64d332 100644 --- a/docs/src/booklet/5-virtual.md +++ b/docs/src/booklet/5-virtual.md @@ -13,7 +13,7 @@ ToQUBO.VirtualModel ## Virtual Variables Every virtual model stores a collection of virtual variables, intended to provide a link between those in the source and those to be created in the target model. -Each virtual variable stores enconding information for later expansion and evaluation. +Each virtual variable stores encoding information for later expansion and evaluation. ```@docs ToQUBO.VirtualVariable diff --git a/docs/src/booklet/6-solvers.md b/docs/src/booklet/6-solvers.md index 475b52ab..7cdfc31c 100644 --- a/docs/src/booklet/6-solvers.md +++ b/docs/src/booklet/6-solvers.md @@ -1,21 +1,29 @@ # QUBO Solvers ## Solvers, Annealers & Samplers -`ToQUBO`'s main goal is to benefit from non-deterministic samplers, especially *Quantum Adiabatic* devices and other *Annealing* machines. A few `MOI`-compliant interfaces for annealers and samplers are bundled within `ToQUBO` via the `Anneal.jl` submodule and package prototype. Some of them are presented below. +[`ToQUBO.jl`](https://github.com/psrenergy/ToQUBO.jl)'s main goal is to make use of parameterized stochastic optimization solvers, particularly those relying on non-conventional hardware such as *Quantum Annealing* and other *Ising Machines*. +A few `MOI`-compliant interfaces for annealers and samplers are bundled within [`ToQUBO.jl`](https://github.com/psrenergy/ToQUBO.jl) via the [`Anneal.jl`](https://github.com/psrenergy/Anneal.jl) companion package. +Some of them are presented below. -### Simulated Annealing -Provided by D-Wave's open-source code libraries, this [Simulated Annealing](https://en.wikipedia.org/wiki/Simulated_annealing) engine implements some of the features and configuration you would find using the Quantum API. Its adoption is recommended for basic usage, tests, and during early research steps due to its simplicity and ease of use. It does not implement the most advanced Simulated Annealing algorithm available but performs fairly well on small instances. `Anneal.jl` exports this interface as `SimulatedAnnealer.Optimizer`. +## Simulated Annealing +Provided by D-Wave's open-source code libraries, this [Simulated Annealing](https://en.wikipedia.org/wiki/Simulated_annealing) engine implements some of the features and configuration you would find using the Quantum API. +Its adoption is recommended for basic usage, tests, and research due to its robustness, simplicity and ease of use. +The [`DWaveNeal.jl`](https://github.com/psrenergy/DWaveNeal.jl) package uses [`Anneal.jl`](https://github.com/psrenergy/Anneal.jl) to deliver an interface to this sampler. -### Quantum Annealing -Interfacing with [D-Wave](https://www.dwavesys.com/)'s quantum computers is one of the milestones we expect to achieve with this package. Like other proprietary optimization resources such as [Gurobi](https://gurobi.com) and [FICO® Xpress](https://www.fico.com/en/products/fico-xpress-solver), this requires licensing and extra steps are needed to get access to it. In a first moment, for those willing to get started, the *Simulated Annealing* optimizer might be enough. +## Quantum Annealing +Interfacing with [D-Wave](https://www.dwavesys.com/)'s quantum computers is one of the milestones we expect to achieve with this package. +Like other proprietary optimization resources such as [Gurobi](https://gurobi.com) and [FICO® Xpress](https://www.fico.com/en/products/fico-xpress-solver), this requires licensing and extra steps are needed to get access to it. +In a first moment, for those willing to get started, the [`DWaveNeal.jl`](https://github.com/psrenergy/DWaveNeal.jl) optimizer might be enough. -While in `JuMP`, run `using Anneal` and look for `QuantumAnnealer.Optimizer`. +## Random Sampling +This sampler is implemented for test purposes and simply assigns 0 or 1 to each variable according to a given probability bias ``0 \le p \le 1``, which defaults to ``p = 0.5``. +After running the `using Anneal` command, `RandomSampler.Optimizer` will be available. -### Random Sampling -This sampler is implemented for test purposes and simply assigns 0 or 1 to each variable according to a given probability bias ``0 \le p \le 1``, which defaults to ``p = 0.5``. After running the `using Anneal` command, `RandomSampler.Optimizer` will be available. +## Exact Solver (Exaustive Enumeration) +Also made to be used in tests, the `ExactSolver.Optimizer` interface runs through all possible state configurations, which implies in an exponential time complexity on the number of variables. +Thus, only problems with no more than 20 variables should be provided. -### Exact Solver (Exaustive Enumeration) -Also made to be used in tests, the `ExactSolver.Optimizer` interface runs through all possible state configurations, which implies in an exponential time complexity on the number of variables. Thus, only problems with no more than 20 variables should be provided. - -## MIQP Solvers -The most accessible alternative to Annealers and Samplers are Mixed-Integer Quadratic Programming (MIQP) Solvers such as [Gurobi](https://github.com/jump-dev/Gurobi.jl) and [CPLEX](https://github.com/jump-dev/CPLEX.jl). These are not intended to be of regular use alongside `ToQUBO` since reformulation usually makes things harder for these folks. Yet, there are still cases where they may be suitable for tests on small instances. +## Mixed-Integer Quadratic Programming +The most accessible alternative to the forementioned methods are Mixed-Integer Quadratic Programming (MIQP) solvers such as [Gurobi](https://github.com/jump-dev/Gurobi.jl), [CPLEX](https://github.com/jump-dev/CPLEX.jl), [SCIP](https://github.com/scipopt/SCIP.jl) and [BARON](https://github.com/jump-dev/BARON.jl). +These are not intended to be of regular use alongside [`ToQUBO.jl`](https://github.com/psrenergy/ToQUBO.jl) since providing a QUBO reformulation will usually make things harder for non-specialized solvers. +Yet, there are still a few cases where they may be suitable, such as tests, benchmarks, or any other situation where global optimality is a must. From 2b29b2962656d4a76ebdbc6dedebf87da5110b1a Mon Sep 17 00:00:00 2001 From: Pedro Maciel Xavier Date: Thu, 16 Feb 2023 11:13:13 -0300 Subject: [PATCH 04/38] Update text [no ci] --- docs/src/booklet/6-solvers.md | 12 ++++++------ docs/src/manual/2-model.md | 6 +++++- docs/src/manual/3-results.md | 6 +++++- 3 files changed, 16 insertions(+), 8 deletions(-) diff --git a/docs/src/booklet/6-solvers.md b/docs/src/booklet/6-solvers.md index 7cdfc31c..5348f149 100644 --- a/docs/src/booklet/6-solvers.md +++ b/docs/src/booklet/6-solvers.md @@ -6,22 +6,22 @@ A few `MOI`-compliant interfaces for annealers and samplers are bundled within [ Some of them are presented below. ## Simulated Annealing -Provided by D-Wave's open-source code libraries, this [Simulated Annealing](https://en.wikipedia.org/wiki/Simulated_annealing) engine implements some of the features and configuration you would find using the Quantum API. +Provided by D-Wave's open-source code libraries, this [Simulated Annealing](https://en.wikipedia.org/wiki/Simulated_annealing) engine implements some of the features and configurations you would find using the Quantum API. Its adoption is recommended for basic usage, tests, and research due to its robustness, simplicity and ease of use. The [`DWaveNeal.jl`](https://github.com/psrenergy/DWaveNeal.jl) package uses [`Anneal.jl`](https://github.com/psrenergy/Anneal.jl) to deliver an interface to this sampler. ## Quantum Annealing -Interfacing with [D-Wave](https://www.dwavesys.com/)'s quantum computers is one of the milestones we expect to achieve with this package. -Like other proprietary optimization resources such as [Gurobi](https://gurobi.com) and [FICO® Xpress](https://www.fico.com/en/products/fico-xpress-solver), this requires licensing and extra steps are needed to get access to it. -In a first moment, for those willing to get started, the [`DWaveNeal.jl`](https://github.com/psrenergy/DWaveNeal.jl) optimizer might be enough. +Interfacing with [D-Wave](https://www.dwavesys.com/)'s quantum annealer is one of the milestones we expect to achieve with this package. +Like other proprietary optimization resources such as [Gurobi](https://gurobi.com), [FICO® Xpress](https://www.fico.com/en/products/fico-xpress-solver) and [IBM® CPLEX®](https://www.ibm.com/products/ilog-cplex-optimization-studio/cplex-optimizer), this requires licensing and extra steps are needed to get an access token. +In a first moment, for those willing to get started, the [`DWaveNeal.jl`](https://github.com/psrenergy/DWaveNeal.jl) optimizer might be enough to learn the ropes. ## Random Sampling This sampler is implemented for test purposes and simply assigns 0 or 1 to each variable according to a given probability bias ``0 \le p \le 1``, which defaults to ``p = 0.5``. After running the `using Anneal` command, `RandomSampler.Optimizer` will be available. -## Exact Solver (Exaustive Enumeration) +## Exact Solver (Exhaustive Enumeration) Also made to be used in tests, the `ExactSolver.Optimizer` interface runs through all possible state configurations, which implies in an exponential time complexity on the number of variables. -Thus, only problems with no more than 20 variables should be provided. +Thus, only problems with at most ``\sym 20`` variables should be provided since visiting ``2^{20} \sym 10^{6}`` can already take up to a few seconds. ## Mixed-Integer Quadratic Programming The most accessible alternative to the forementioned methods are Mixed-Integer Quadratic Programming (MIQP) solvers such as [Gurobi](https://github.com/jump-dev/Gurobi.jl), [CPLEX](https://github.com/jump-dev/CPLEX.jl), [SCIP](https://github.com/scipopt/SCIP.jl) and [BARON](https://github.com/jump-dev/BARON.jl). diff --git a/docs/src/manual/2-model.md b/docs/src/manual/2-model.md index b0e734d3..41a52791 100644 --- a/docs/src/manual/2-model.md +++ b/docs/src/manual/2-model.md @@ -1 +1,5 @@ -# Running a Model \ No newline at end of file +# Running a Model + +!!! warning "Work in progress" + We hope to write this part of the documentation soon. + Please come back later! \ No newline at end of file diff --git a/docs/src/manual/3-results.md b/docs/src/manual/3-results.md index 70a7b2cc..b3e497d7 100644 --- a/docs/src/manual/3-results.md +++ b/docs/src/manual/3-results.md @@ -1 +1,5 @@ -# Gathering Results \ No newline at end of file +# Gathering Results + +!!! warning "Work in progress" + We hope to write this part of the documentation soon. + Please come back later! \ No newline at end of file From 433488c3ab3690a539ea1c64e062f555d7208459 Mon Sep 17 00:00:00 2001 From: Pedro Maciel Xavier Date: Thu, 16 Feb 2023 11:19:06 -0300 Subject: [PATCH 05/38] Update text --- docs/make.jl | 3 ++- docs/src/booklet/1-intro.md | 2 +- docs/src/booklet/2-qubo.md | 15 ++++----------- docs/src/booklet/6-compiler.md | 18 ++++++++++++++++++ .../src/booklet/{6-solvers.md => 7-solvers.md} | 0 5 files changed, 25 insertions(+), 13 deletions(-) create mode 100644 docs/src/booklet/6-compiler.md rename docs/src/booklet/{6-solvers.md => 7-solvers.md} (100%) diff --git a/docs/make.jl b/docs/make.jl index 97125efc..d6c5a600 100644 --- a/docs/make.jl +++ b/docs/make.jl @@ -35,7 +35,8 @@ makedocs(; "PBO" => "booklet/3-pbo.md", "Encoding" => "booklet/4-encoding.md", "Virtual Mapping" => "booklet/5-virtual.md", - "Solvers" => "booklet/6-solvers.md", + "The Compiler" => "booklet/6-compiler.md" + "Solvers" => "booklet/7-solvers.md", ] ], workdir="." diff --git a/docs/src/booklet/1-intro.md b/docs/src/booklet/1-intro.md index 8696524f..7507364e 100644 --- a/docs/src/booklet/1-intro.md +++ b/docs/src/booklet/1-intro.md @@ -6,6 +6,6 @@ The latter are advised to read the following sections, as they give a glimpse of ## Table of Contents ```@contents -Pages = ["2-qubo.md", "3-pbo.md", "4-encoding.md", "5-virtual.md", "6-solvers.md"] +Pages = ["2-qubo.md", "3-pbo.md", "4-encoding.md", "5-virtual.md", "6-compiler.md", "7-solvers.md"] Depth = 2 ``` \ No newline at end of file diff --git a/docs/src/booklet/2-qubo.md b/docs/src/booklet/2-qubo.md index 85d1cd7c..5559bb41 100644 --- a/docs/src/booklet/2-qubo.md +++ b/docs/src/booklet/2-qubo.md @@ -11,19 +11,12 @@ ## OK, but why QUBO? +!!! warning "Work in progress" + We hope to write this part of the documentation soon. + Please come back later! + ```@docs ToQUBO.isqubo ToQUBO.toqubo ToQUBO.toqubo! ``` - -```@docs -ToQUBO.toqubo_sense! -ToQUBO.toqubo_variables! -ToQUBO.toqubo_constraint -ToQUBO.toqubo_constraints! -ToQUBO.toqubo_objective! -ToQUBO.toqubo_penalties! -ToQUBO.toqubo_parse! -ToQUBO.toqubo_build! -``` \ No newline at end of file diff --git a/docs/src/booklet/6-compiler.md b/docs/src/booklet/6-compiler.md new file mode 100644 index 00000000..2828fd72 --- /dev/null +++ b/docs/src/booklet/6-compiler.md @@ -0,0 +1,18 @@ +# The Compiler + +!!! warning "Work in progress" + We hope to write this part of the documentation soon. + Please come back later! + +## Compilation Steps + +```@docs +ToQUBO.toqubo_sense! +ToQUBO.toqubo_variables! +ToQUBO.toqubo_constraint +ToQUBO.toqubo_constraints! +ToQUBO.toqubo_objective! +ToQUBO.toqubo_penalties! +ToQUBO.toqubo_parse! +ToQUBO.toqubo_build! +``` \ No newline at end of file diff --git a/docs/src/booklet/6-solvers.md b/docs/src/booklet/7-solvers.md similarity index 100% rename from docs/src/booklet/6-solvers.md rename to docs/src/booklet/7-solvers.md From 20958a32bbd2ce9a6d017382dce9fbff73132e31 Mon Sep 17 00:00:00 2001 From: Pedro Maciel Xavier Date: Thu, 16 Feb 2023 12:41:27 -0300 Subject: [PATCH 06/38] Fix citation --- README.md | 12 ++++++------ docs/src/index.md | 16 ++++++++-------- 2 files changed, 14 insertions(+), 14 deletions(-) diff --git a/README.md b/README.md index 862027b9..2669ce6b 100644 --- a/README.md +++ b/README.md @@ -135,15 +135,15 @@ If you think this list is incomplete, consider creating an [Issue](https://githu ## Citing ToQUBO.jl If you use `ToQUBO.jl` in your work, we kindly ask you to include the following citation: ```tex -@software{toqubo:2022, +@software{toqubo:2023, author = {Pedro Maciel Xavier and Pedro Ripper and Tiago Andrade and Joaquim Dias Garcia and David E. Bernal Neira}, - title = {{ToQUBO.jl}}, - month = {mar}, - year = {2022}, + title = {ToQUBO.jl}, + month = {feb}, + year = {2023}, publisher = {Zenodo}, version = {v0.1.5}, - doi = {10.5281/zenodo.6387592}, - url = {https://doi.org/10.5281/zenodo.6387592} + doi = {10.5281/zenodo.7644291}, + url = {https://doi.org/10.5281/zenodo.7644291} } ``` diff --git a/docs/src/index.md b/docs/src/index.md index e7938866..a8ea841d 100644 --- a/docs/src/index.md +++ b/docs/src/index.md @@ -37,14 +37,14 @@ end ## Citing ToQUBO.jl If you use `ToQUBO.jl` in your work, we kindly ask you to include the following citation: ```tex -@software{toqubo:2022, - author = {Pedro Xavier and Tiago Andrade and Joaquim Garcia and David Bernal}, - title = {{ToQUBO.jl}}, - month = mar, - year = 2022, +@software{toqubo:2023, + author = {Pedro Maciel Xavier and Pedro Ripper and Tiago Andrade and Joaquim Dias Garcia and David E. Bernal Neira}, + title = {ToQUBO.jl}, + month = {feb}, + year = {2023}, publisher = {Zenodo}, - version = {v0.1.0}, - doi = {10.5281/zenodo.6387592}, - url = {https://doi.org/10.5281/zenodo.6387592} + version = {v0.1.5}, + doi = {10.5281/zenodo.7644291}, + url = {https://doi.org/10.5281/zenodo.7644291} } ``` \ No newline at end of file From 6706d58f61b3910909244580803fdf945b8b1168 Mon Sep 17 00:00:00 2001 From: Pedro Maciel Xavier Date: Thu, 16 Feb 2023 12:42:03 -0300 Subject: [PATCH 07/38] Fix `make.jl` --- docs/make.jl | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/docs/make.jl b/docs/make.jl index d6c5a600..27f3c310 100644 --- a/docs/make.jl +++ b/docs/make.jl @@ -35,7 +35,7 @@ makedocs(; "PBO" => "booklet/3-pbo.md", "Encoding" => "booklet/4-encoding.md", "Virtual Mapping" => "booklet/5-virtual.md", - "The Compiler" => "booklet/6-compiler.md" + "The Compiler" => "booklet/6-compiler.md", "Solvers" => "booklet/7-solvers.md", ] ], From b61557270f75cba9b9248e68a5b55806214facee Mon Sep 17 00:00:00 2001 From: Pedro Maciel Xavier Date: Thu, 16 Feb 2023 15:55:17 -0300 Subject: [PATCH 08/38] Add example to Quick Start --- docs/src/index.md | 18 ++++++------------ 1 file changed, 6 insertions(+), 12 deletions(-) diff --git a/docs/src/index.md b/docs/src/index.md index a8ea841d..46a212a9 100644 --- a/docs/src/index.md +++ b/docs/src/index.md @@ -2,7 +2,7 @@ `ToQUBO.jl` is a Julia Package intended to automatically translate models written in [JuMP](https://github.com/jump-dev/JuMP.jl), into the [QUBO](https://en.wikipedia.org/wiki/Quadratic_unconstrained_binary_optimization) mathematical optimization framework. -## Getting Started +## Quick Start ### Installation ```julia @@ -11,27 +11,21 @@ julia> import Pkg julia> Pkg.add("ToQUBO") ``` -### Running -```julia +### Example +```@example using JuMP using ToQUBO -using Anneal +using DWaveNeal -model = Model(() -> ToQUBO.Optimizer(ExactSampler.Optimizer)) +model = Model(() -> ToQUBO.Optimizer(DWaveNeal.Optimizer)) @variable(model, x[1:3], Bin) - @objective(model, Max, 1.0*x[1] + 2.0*x[2] + 3.0*x[3]) - @constraint(model, 0.3*x[1] + 0.5*x[2] + 1.0*x[3] <= 1.6) optimize!(model) -for i = 1:result_count(model) - xᵢ = value.(x, result = i) - yᵢ = objective_value(model, result = i) - println("f($xᵢ) = $yᵢ") -end +solution_summary(model) ``` ## Citing ToQUBO.jl From 2b596f0fb8bcd33ca27c6e92b4a334af02614ebd Mon Sep 17 00:00:00 2001 From: pedroripper Date: Thu, 16 Feb 2023 17:38:00 -0300 Subject: [PATCH 09/38] Update docs(qubo and quadratization) --- docs/src/assets/figs/quantum-chip.png | Bin 0 -> 603115 bytes docs/src/booklet/2-qubo.md | 19 +++++++++++++++---- docs/src/booklet/3-pbo.md | 19 +++++++++++++++++-- src/lib/pbo/quadratization.jl | 18 +++++++++++++++--- 4 files changed, 47 insertions(+), 9 deletions(-) create mode 100644 docs/src/assets/figs/quantum-chip.png diff --git a/docs/src/assets/figs/quantum-chip.png b/docs/src/assets/figs/quantum-chip.png new 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ziCwZxMs(Y@ZCRN)#)Jwv=V@9fxYKzEDRAIg>pZRNyj;d>@ILlktA{O@D-%-;)qu&c zY%_UxJ`XWQVpcT!w(o0RE|HOOZVv z>tzBdWL*dpIoK9%2xO?B0xD<#4cv=$pbBWLLW@1nT10~YM$IqaBXc)$c*_f_-e23` zHv^^}Vg}TzK?B}7PgG5h4 zRK-1)`8c+EEG2hU6adZ5Z>yzTuh+{sKL7bvOGzm_Jf2gR0P#AnFR!ovdcMBAjQdd> z1&T4G6jCCN)}klR&fWX3pjIiRZpVIN9Y$M{LL6_Go9x{S>LxC*?Rk!8(&$p#4Q`9Zppz*a&{*pY1oQFYTC2u1$yVtUp zyzM{pSGd{wwHgM}6bp(KRHRj@tcWh#v5Yg$8#CvJQ`TYlz8z`Um4{D16Ee_r1E|AAaft_7TP&WQmu@Hup=VPd`;-M^=y+U3-=U8s%|L#YBlYh(JE7srYkVmaqGlB@-8ba)IY7_|X$LV=2$1WN;c8nY% zd5hO^yv%dg#jX#Wm?NpGyo_Z()_LQ^X=e%s0~2$bIP7u|kp0-UeVyhjvGVNbgAi91K99=uglg*W$cAkV};LTKp2b=k&`OW;97j75X!lZ$;PngIby{doPjUY5%Rnd8u>M#Vax z&u?#U>$(yVk#inHB5LjS9v9~Rtzezu=MRZad4CdV$OZ9;+I^HV};A27GJ<2I_Kyl zz|1PffVFBF-BGGF*mV({*Xvv_>%8Wdv8YyNwo(~vYoMR!*}@k` Date: Fri, 17 Feb 2023 15:30:43 -0300 Subject: [PATCH 10/38] Add docs for quadratization and fix typos --- docs/src/booklet/2-qubo.md | 4 ++-- docs/src/booklet/3-pbo.md | 17 ++++++++++++++--- src/compiler/interface.jl | 7 +++++++ src/lib/pbo/quadratization.jl | 7 ++++++- 4 files changed, 29 insertions(+), 6 deletions(-) diff --git a/docs/src/booklet/2-qubo.md b/docs/src/booklet/2-qubo.md index ce206eac..864f45db 100644 --- a/docs/src/booklet/2-qubo.md +++ b/docs/src/booklet/2-qubo.md @@ -20,9 +20,9 @@ We can represent a QUBO as follows. ## OK, but why QUBO? -QUBOs are suited for representing non-convex global optimization problems. With that said, the significant advances in computing systems and algorithms specialized for sampling QUBOs has been contributing to their popularity. +QUBOs are suited for representing non-convex global optimization problems. With that said, the significant advances in computing systems and algorithms specialized for sampling QUBOs have contributed to their popularity. -Some of the paradigms that stand out for running QUBOs are quantum gate-based optimization algorithms (QAOA and VQE), quantum annealers, hardware-accelerated platforms (Coherent Ising Machines and Simulated Bifurcation Machines). +Some of the paradigms that stand out for running QUBOs are quantum gate-based optimization algorithms (QAOA and VQE), quantum annealers and hardware-accelerated platforms (Coherent Ising Machines and Simulated Bifurcation Machines). ![D-Wave Washington 1000Q](../assets/figs/quantum-chip.png) diff --git a/docs/src/booklet/3-pbo.md b/docs/src/booklet/3-pbo.md index 1044a4b1..e5a87eb3 100644 --- a/docs/src/booklet/3-pbo.md +++ b/docs/src/booklet/3-pbo.md @@ -7,7 +7,16 @@ ToQUBO.PBO.PseudoBooleanFunction ``` ## Quadratization -In order to successfully achieve a QUBO formulation, sometimes it is needed to quadratize the resulting PBF, i.e., reduce its degree until reaching the quadratic case. There are many quadratization methods available[^Dattani2019], and `ToQUBO` implements a two of them for now. However, using Julia's multiple dispatch paradigm, it's possible to extend the quadratization method coverage with your own algorithms. +In order to successfully achieve a QUBO formulation, sometimes it is needed to quadratize the resulting PBF, i.e., reduce its degree until reaching the quadratic case. + +A quadratization is a mapping ``\mathcal{Q}: \mathscr{F} \to \mathscr{F}^{2}`` such that + +```math +\forall f \in \mathscr{F}, \forall x \in \{0, 1\}^{n}, \min_{y} \mathcal{Q}\left\lbrace{}f\right\rbrace{}(x; y) = f(x) + +``` + +There are many quadratization methods available[^Dattani2019], and `ToQUBO` implements two of them for now. However, using Julia's multiple dispatch paradigm, it's possible to extend the quadratization method coverage with your own algorithms. ```@docs ToQUBO.PBO.quadratize! @@ -32,7 +41,9 @@ f(X \cup Y) + f(X \cap Y) \le f(X) + f(Y) \forall X, Y \subset S holds. -### + + + [^Dattani2019]: - Dattani, Nikesh S.., **Quadratization in discrete optimization and quantum mechanics**, *ArXiv*, 2019 [{link}](https://arxiv.org/abs/1901.04405) + Dattani, Nikesh S.., **Quadratization in discrete optimization and quantum mechanics**, *ArXiv*, 2019 [{doi}](https://doi.org/10.48550/arXiv.1901.04405) diff --git a/src/compiler/interface.jl b/src/compiler/interface.jl index 6a1d09df..45bbc3ea 100644 --- a/src/compiler/interface.jl +++ b/src/compiler/interface.jl @@ -82,6 +82,13 @@ Parses the given MOI function `f` into PBF `g`. toqubo_build!(model::VirtualModel, ::AbstractArchitecture) """ function toqubo_build! end +@doc raw""" + toqubo_quadratize!(model::VirtualModel, arch::AbstractArchitecture) + +Quadratizes the objective function from a model + +""" function toqubo_quadratize! end + @doc raw""" toqubo_empty!(model::VirtualModel, ::AbstractArchitecture) """ function toqubo_empty! end \ No newline at end of file diff --git a/src/lib/pbo/quadratization.jl b/src/lib/pbo/quadratization.jl index 0b678e15..e637ed96 100644 --- a/src/lib/pbo/quadratization.jl +++ b/src/lib/pbo/quadratization.jl @@ -12,7 +12,7 @@ end @doc raw""" quadratize!(aux::Function, f::PBF{S, T}, ::Quadratization{Q}) where {S,T,Q} -Quadratizes a given PBF in-place, i.e. applies a mapping ``Q : \mathscr{F}^{k} \to \mathscr{F}^{2}``, where Q is the quadratization method. +Quadratizes a given PBF in-place, i.e. applies a mapping ``\mathcal{Q} : \mathscr{F}^{k} \to \mathscr{F}^{2}``, where ``\mathcal{Q}`` is the quadratization method. ```julia aux(::Nothing)::S @@ -139,6 +139,11 @@ function quadratize!( return nothing end +@doc raw""" + quadratize!(aux::Function, f::PBF{S,T}, quad::Quadratization{TERM_BY_TERM}) where {S,T} + + Receives a higher-degree pseudo-Boolean function +""" function quadratize!( aux::Function, f::PBF{S,T}, From e8eee66023fec3afd5f60e786358cb722891891a Mon Sep 17 00:00:00 2001 From: Pedro Maciel Xavier Date: Thu, 23 Feb 2023 08:55:48 -0300 Subject: [PATCH 11/38] Add double brackets to title in TeX citation --- CITATION.bib | 2 +- README.md | 2 +- docs/src/index.md | 2 +- 3 files changed, 3 insertions(+), 3 deletions(-) diff --git a/CITATION.bib b/CITATION.bib index 3d2bda4d..3a69a9de 100644 --- a/CITATION.bib +++ b/CITATION.bib @@ -1,7 +1,7 @@ % If you use this software, please cite it as below. @software{toqubo:2023, author = {Pedro Maciel Xavier and Pedro Ripper and Tiago Andrade and Joaquim Dias Garcia and David E. Bernal Neira}, - title = {ToQUBO.jl}, + title = {{ToQUBO.jl}}, month = {feb}, year = {2023}, publisher = {Zenodo}, diff --git a/README.md b/README.md index 2669ce6b..c5aef2f0 100644 --- a/README.md +++ b/README.md @@ -137,7 +137,7 @@ If you use `ToQUBO.jl` in your work, we kindly ask you to include the following ```tex @software{toqubo:2023, author = {Pedro Maciel Xavier and Pedro Ripper and Tiago Andrade and Joaquim Dias Garcia and David E. Bernal Neira}, - title = {ToQUBO.jl}, + title = {{ToQUBO.jl}}, month = {feb}, year = {2023}, publisher = {Zenodo}, diff --git a/docs/src/index.md b/docs/src/index.md index 46a212a9..1dd37f16 100644 --- a/docs/src/index.md +++ b/docs/src/index.md @@ -33,7 +33,7 @@ If you use `ToQUBO.jl` in your work, we kindly ask you to include the following ```tex @software{toqubo:2023, author = {Pedro Maciel Xavier and Pedro Ripper and Tiago Andrade and Joaquim Dias Garcia and David E. Bernal Neira}, - title = {ToQUBO.jl}, + title = {{ToQUBO.jl}}, month = {feb}, year = {2023}, publisher = {Zenodo}, From afc69be9c4ced0ce739a7821a8676ff4da9ab9b7 Mon Sep 17 00:00:00 2001 From: Pedro Maciel Xavier Date: Thu, 23 Feb 2023 09:11:55 -0300 Subject: [PATCH 12/38] Enhance quadatization methods' docs --- src/lib/pbo/quadratization.jl | 14 ++++++++------ 1 file changed, 8 insertions(+), 6 deletions(-) diff --git a/src/lib/pbo/quadratization.jl b/src/lib/pbo/quadratization.jl index e637ed96..f5d99bdb 100644 --- a/src/lib/pbo/quadratization.jl +++ b/src/lib/pbo/quadratization.jl @@ -26,11 +26,12 @@ aux(::Integer)::Vector{S} Negative Term Reduction NTR-KZFD (Kolmogorov & Zabih, 2004; Freedman & Drineas, 2005) ```math --x_1 x_2 \dots x_k \to (k-1) x_{aux} - \sum_i x_i x_{aux} +f(x_{1}, \dots, x_{k}) = - x_{1} \cdots x_{k} \to +\mathcal{Q}\left\lbrace{f}\right\rbrace(x_{1}, \dots, x_{k}; z) = (k - 1) z - \sum_{i = 1}^{k} x_{i} z ``` !!! info - Introduces one new variable and no non-submodular terms. + Introduces a new variable ``z`` and no non-submodular terms. """ struct NTR_KZFD <: QuadratizationMethod end function quadratize!( @@ -70,13 +71,14 @@ end Positive Term Reduction PTR-BG (Boros & Gruber, 2014) ```math -x_1 x_2 \dots x_k \to \left[{ - \sum^{k-2}_{i=1}x_{a_i} \left({k - i - 1 + x_i + \sum^k_{j=i+1} x_j}\right) - }\right] + x_{k-1}x_k +f(x_{1}, \dots, x_{k}) = x_{1} x_{2} \dots x_{k} \to +\mathcal{Q}\left\lbrace{f}\right\rbrace(x_{1}, \dots, x_{k}; z_{1}, \dots, z_{k-2} ) = \left[{ + \sum_{i = 1}^{k-2} z_{i} \left({ k - i - 1 + x_{i} + \sum_{j = i+1}^{k} x_{j} }\right) +}\right] + x_{k-1} x_{k} ``` !!! info - Introduces ``k - 2`` new variables and ``k - 1`` non-submodular terms. + Introduces ``k - 2`` new variables ``z_{1}, \dots, z_{k-2}`` and ``k - 1`` non-submodular terms. """ struct PTR_BG <: QuadratizationMethod end function quadratize!( From 43cc8098ef508fcabc0db061d6162809f2fab462 Mon Sep 17 00:00:00 2001 From: Pedro Maciel Xavier Date: Thu, 23 Feb 2023 09:16:27 -0300 Subject: [PATCH 13/38] Minor Fix --- docs/src/booklet/2-qubo.md | 3 ++- 1 file changed, 2 insertions(+), 1 deletion(-) diff --git a/docs/src/booklet/2-qubo.md b/docs/src/booklet/2-qubo.md index 864f45db..1d7e807b 100644 --- a/docs/src/booklet/2-qubo.md +++ b/docs/src/booklet/2-qubo.md @@ -20,7 +20,8 @@ We can represent a QUBO as follows. ## OK, but why QUBO? -QUBOs are suited for representing non-convex global optimization problems. With that said, the significant advances in computing systems and algorithms specialized for sampling QUBOs have contributed to their popularity. +QUBOs are suited for representing non-convex global optimization problems. +With that said, the significant advances in computing systems and algorithms specialized for sampling QUBOs have contributed to their popularity. Some of the paradigms that stand out for running QUBOs are quantum gate-based optimization algorithms (QAOA and VQE), quantum annealers and hardware-accelerated platforms (Coherent Ising Machines and Simulated Bifurcation Machines). From 3df0a055dca22c661b8b63f32d6af8a6f195b4c2 Mon Sep 17 00:00:00 2001 From: Pedro Maciel Xavier Date: Fri, 24 Feb 2023 02:41:13 -0300 Subject: [PATCH 14/38] Update docs --- docs/make.jl | 9 ++++- docs/src/assets/README.md | 23 ------------ docs/src/assets/extra_styles.css | 4 ++ .../assets/{figs => figures}/quantum-chip.png | Bin ...ower LICENSE.txt => Sunflower-LICENSE.txt} | 0 docs/src/assets/init.js | 24 ++++++++++++ docs/src/booklet/1-intro.md | 2 +- docs/src/booklet/2-qubo.md | 8 ++-- docs/src/booklet/3-pbo.md | 5 ++- docs/src/booklet/4-encoding.md | 2 + docs/src/booklet/5-virtual.md | 16 ++++++++ docs/src/booklet/7-solvers.md | 2 +- docs/src/booklet/8-appendix.md | 35 ++++++++++++++++++ docs/src/manual/4-settings.md | 14 ++++++- src/attributes/compiler.jl | 10 ++++- src/compiler/architectures.jl | 7 +++- 16 files changed, 124 insertions(+), 37 deletions(-) delete mode 100644 docs/src/assets/README.md rename docs/src/assets/{figs => figures}/quantum-chip.png (100%) rename docs/src/assets/fonts/{Sunflower LICENSE.txt => Sunflower-LICENSE.txt} (100%) create mode 100644 docs/src/assets/init.js create mode 100644 docs/src/booklet/8-appendix.md diff --git a/docs/make.jl b/docs/make.jl index 27f3c310..567eb02c 100644 --- a/docs/make.jl +++ b/docs/make.jl @@ -11,7 +11,11 @@ makedocs(; doctest = true, clean = true, format = Documenter.HTML( - assets = ["assets/extra_styles.css", "assets/favicon.ico"], + assets = [ + "assets/extra_styles.css", + "assets/favicon.ico", + "assets/init.js", + ], mathengine = Documenter.KaTeX(), sidebar_sitename = false, ), @@ -37,7 +41,8 @@ makedocs(; "Virtual Mapping" => "booklet/5-virtual.md", "The Compiler" => "booklet/6-compiler.md", "Solvers" => "booklet/7-solvers.md", - ] + "Appendix" => "booklet/8-appendix.md", + ], ], workdir="." ) diff --git a/docs/src/assets/README.md b/docs/src/assets/README.md deleted file mode 100644 index 900ceebf..00000000 --- a/docs/src/assets/README.md +++ /dev/null @@ -1,23 +0,0 @@ -# Logo - -The ideia behind [ToQUBO](/)'s logo comes from a wordplay in Portuguese and Spanish. The package's main purpose is to assemble QUBO Models, which sounds like *cubo*[¹](#1), the translation for *cube*. - -[![ToQUBO.jl](logo.svg)](/docs/src/assets) - -## Colors - -The colors were chosen according to *Julia's Reference for logo graphics*[²](#2). Text color matches the innermost shape and renders fairly well in both light and dark background themes. - -## Typography -*MADETYPE Sunflower*[³](#3) Font was used. It was converted to a SVG path through the *Google Font to Svg Path*[⁴](#4) online tool. - -¹ IPA: \[ˈkubʊ\] - -² [github.com/JuliaLang/julia-logo-graphics](https://github.com/JuliaLang/julia-logo-graphics/) - -³ [Licensed](/docs/src/assets/fonts/Sunflower%20LICENSE.txt) by the authors for use in this project - -⁴ [danmarshall.github.io/google-font-to-svg-path](https://danmarshall.github.io/google-font-to-svg-path/) - -# Web Icon [![ToQUBO.jl](favicon.ico)](/docs/src/assets) -The icon used to decorate the documentation resembles an assembled version of the cube with its blue face making up the background. \ No newline at end of file diff --git a/docs/src/assets/extra_styles.css b/docs/src/assets/extra_styles.css index 760d2ee1..37ee7a0b 100644 --- a/docs/src/assets/extra_styles.css +++ b/docs/src/assets/extra_styles.css @@ -1,4 +1,8 @@ .docs-sidebar .docs-logo > img { max-height: 16rem !important; margin: auto; +} + +.mermaid { + } \ No newline at end of file diff --git a/docs/src/assets/figs/quantum-chip.png b/docs/src/assets/figures/quantum-chip.png similarity index 100% rename from docs/src/assets/figs/quantum-chip.png rename to docs/src/assets/figures/quantum-chip.png diff --git a/docs/src/assets/fonts/Sunflower LICENSE.txt b/docs/src/assets/fonts/Sunflower-LICENSE.txt similarity index 100% rename from docs/src/assets/fonts/Sunflower LICENSE.txt rename to docs/src/assets/fonts/Sunflower-LICENSE.txt diff --git a/docs/src/assets/init.js b/docs/src/assets/init.js new file mode 100644 index 00000000..f3ead049 --- /dev/null +++ b/docs/src/assets/init.js @@ -0,0 +1,24 @@ +/* Ref: +** https://discourse.julialang.org/t/use-javascript-library-with-documenter-jl/34984/8 +*/ +require.config({ + paths: { + mermaid: "https://cdnjs.cloudflare.com/ajax/libs/mermaid/9.4.0/mermaid.min" + } +}); + +require(['mermaid'], function (mermaid) { + mermaid.initialize({ + startOnLoad : true, + theme : 'neutral', + // themeVariables: { + // primaryColor: '#BB2528', + // primaryTextColor: '#fff', + // primaryBorderColor: '#7C0000', + // lineColor: '#F8B229', + // secondaryColor: '#006100', + // tertiaryColor: '#aaa', + // noteBorderColor: '#ff0', + // } + }) +}); \ No newline at end of file diff --git a/docs/src/booklet/1-intro.md b/docs/src/booklet/1-intro.md index 7507364e..4f15124b 100644 --- a/docs/src/booklet/1-intro.md +++ b/docs/src/booklet/1-intro.md @@ -6,6 +6,6 @@ The latter are advised to read the following sections, as they give a glimpse of ## Table of Contents ```@contents -Pages = ["2-qubo.md", "3-pbo.md", "4-encoding.md", "5-virtual.md", "6-compiler.md", "7-solvers.md"] +Pages = ["2-qubo.md", "3-pbo.md", "4-encoding.md", "5-virtual.md", "6-compiler.md", "7-solvers.md", "8-appendix.md"] Depth = 2 ``` \ No newline at end of file diff --git a/docs/src/booklet/2-qubo.md b/docs/src/booklet/2-qubo.md index 1d7e807b..196a855f 100644 --- a/docs/src/booklet/2-qubo.md +++ b/docs/src/booklet/2-qubo.md @@ -1,7 +1,7 @@ # QUBO ## Definition -QUBO stands for Quadratic Unconstrained Binary Optimization. +QUBO stands for _**Q**uadratic **U**nconstrained **B**inary **O**ptimization_. So every QUBO model has the following characteristics: - Its objective function is at most quadratic @@ -12,8 +12,8 @@ We can represent a QUBO as follows. ```math \begin{array}{rl} - \min & \mathbf{x}' Q\,\mathbf{x} \\ - \text{s.t.} & \mathbf{x} \in \mathbb{B}^{n} + \min & \mathbf{x}' Q\,\mathbf{x} \\ + \mathrm{s.t.} & \mathbf{x} \in \mathbb{B}^{n} \end{array} ``` @@ -25,7 +25,7 @@ With that said, the significant advances in computing systems and algorithms spe Some of the paradigms that stand out for running QUBOs are quantum gate-based optimization algorithms (QAOA and VQE), quantum annealers and hardware-accelerated platforms (Coherent Ising Machines and Simulated Bifurcation Machines). -![D-Wave Washington 1000Q](../assets/figs/quantum-chip.png) +![D-Wave Washington 1000Q](../assets/figures/quantum-chip.png) ```@docs ToQUBO.isqubo diff --git a/docs/src/booklet/3-pbo.md b/docs/src/booklet/3-pbo.md index e5a87eb3..97c5c57f 100644 --- a/docs/src/booklet/3-pbo.md +++ b/docs/src/booklet/3-pbo.md @@ -1,6 +1,6 @@ # Pseudo-Boolean Optimization Internally, problems are represented through a Pseudo-Boolean Optimization (PBO) framework. -The main goal is to represent a given problem using a Pseudo-Boolean Function (PBF) since there is an immediate correspondence between quadratic PBFs and QUBO forms. +The main goal is to represent a given problem using a Pseudo-Boolean Function (PBF) since there is an immediate correspondence between optimization over quadratic PBFs and the QUBO formalism. ```@docs ToQUBO.PBO.PseudoBooleanFunction @@ -16,7 +16,8 @@ A quadratization is a mapping ``\mathcal{Q}: \mathscr{F} \to \mathscr{F}^{2}`` s ``` -There are many quadratization methods available[^Dattani2019], and `ToQUBO` implements two of them for now. However, using Julia's multiple dispatch paradigm, it's possible to extend the quadratization method coverage with your own algorithms. +There are many quadratization methods available[^Dattani2019], and `ToQUBO` implements two of them for now. +However, using Julia's multiple dispatch paradigm, it's possible to extend the quadratization method coverage with your own algorithms. ```@docs ToQUBO.PBO.quadratize! diff --git a/docs/src/booklet/4-encoding.md b/docs/src/booklet/4-encoding.md index 26e7b651..c323b23e 100644 --- a/docs/src/booklet/4-encoding.md +++ b/docs/src/booklet/4-encoding.md @@ -4,7 +4,9 @@ ### Linear Encoding Methods ```@docs +ToQUBO.Mirror ToQUBO.LinearEncoding +ToQUBO.Linear ToQUBO.Binary ToQUBO.Unary ToQUBO.Arithmetic diff --git a/docs/src/booklet/5-virtual.md b/docs/src/booklet/5-virtual.md index 6c64d332..dce6a7eb 100644 --- a/docs/src/booklet/5-virtual.md +++ b/docs/src/booklet/5-virtual.md @@ -4,6 +4,22 @@ The source model is a generic `MOI` model restricted to the supported constraint The target one is on the QUBO form used during the solving process. Both lie within a *Virtual Model*, which provides the necessary API integration and keeps all variable and constraint mapping tied together. +```@raw html +

+flowchart LR; + JUMP_MODEL["JuMP Model"]; + + subgraph VIRTUAL_MODEL["Virtual Model"]; + SOURCE_MODEL["Source Model"]; + TARGET_MODEL["Target Model"]; + + SOURCE_MODEL -->|ToQUBO.jl| TARGET_MODEL; + end + + JUMP_MODEL -->|MOI Bridges| SOURCE_MODEL; +
+``` + This is done in a transparent fashion for both agents since the user will mostly interact with the presented model, and the solvers will only access the generated one. ## Virtual Model diff --git a/docs/src/booklet/7-solvers.md b/docs/src/booklet/7-solvers.md index 5348f149..237764a2 100644 --- a/docs/src/booklet/7-solvers.md +++ b/docs/src/booklet/7-solvers.md @@ -21,7 +21,7 @@ After running the `using Anneal` command, `RandomSampler.Optimizer` will be avai ## Exact Solver (Exhaustive Enumeration) Also made to be used in tests, the `ExactSolver.Optimizer` interface runs through all possible state configurations, which implies in an exponential time complexity on the number of variables. -Thus, only problems with at most ``\sym 20`` variables should be provided since visiting ``2^{20} \sym 10^{6}`` can already take up to a few seconds. +Thus, only problems with at most ``\approxeq 20`` variables should be provided since visiting ``2^{20} \approxeq 10^{6}`` states can already take up to a few seconds. ## Mixed-Integer Quadratic Programming The most accessible alternative to the forementioned methods are Mixed-Integer Quadratic Programming (MIQP) solvers such as [Gurobi](https://github.com/jump-dev/Gurobi.jl), [CPLEX](https://github.com/jump-dev/CPLEX.jl), [SCIP](https://github.com/scipopt/SCIP.jl) and [BARON](https://github.com/jump-dev/BARON.jl). diff --git a/docs/src/booklet/8-appendix.md b/docs/src/booklet/8-appendix.md new file mode 100644 index 00000000..c1d5707a --- /dev/null +++ b/docs/src/booklet/8-appendix.md @@ -0,0 +1,35 @@ +# Appendix + +## ToQUBO.jl's Assets + +### Logo + +The ideia behind [ToQUBO.jl](https://github.com/psrenergy/ToQUBO.jl)'s logo comes from a wordplay in Portuguese and Spanish. +The package's main purpose is to assemble QUBO Models, which sounds like *cubo*[^1], the translation for *cube*. + +[![ToQUBO.jl Logo](../assets/logo.svg)]() + +### Colors + +The colors were picked according to *Julia's Reference for logo graphics*[^2]. +Text color matches the innermost shape and renders fairly well in both light and dark background themes. + +### Typography + +The *MADETYPE Sunflower*[^3] font was chosen. +It was converted to a SVG path using the *Google Font to Svg Path*[^4] online tool. + +[^1]: + IPA: \[ˈkubʊ\] + +[^2]: + [github.com/JuliaLang/julia-logo-graphics](https://github.com/JuliaLang/julia-logo-graphics/) + +[^3]: + [Licensed](../../assets/fonts/Sunflower-LICENSE.txt) by the authors for use in this project + +[^4]: + [danmarshall.github.io/google-font-to-svg-path](https://danmarshall.github.io/google-font-to-svg-path/) + +### Web Icon [![ToQUBO.jl Icon](../../assets/favicon.ico)]() +The icon used to decorate the documentation resembles an assembled version of the cube with its blue face making up the background. \ No newline at end of file diff --git a/docs/src/manual/4-settings.md b/docs/src/manual/4-settings.md index 09e18235..65cba01e 100644 --- a/docs/src/manual/4-settings.md +++ b/docs/src/manual/4-settings.md @@ -1,10 +1,15 @@ # Compiler Settings -## Architecture +## Working with solver architectures ```@docs ToQUBO.Attributes.Architecture ``` +```@docs +ToQUBO.AbstractArchitecture +ToQUBO.GenericArchitecture +``` + ## Quadratization ```@docs ToQUBO.Attributes.Quadratize @@ -22,4 +27,9 @@ ToQUBO.Attributes.VariableEncodingMethod ToQUBO.Attributes.DefaultVariableEncodingMethod ToQUBO.Attributes.VariableEncodingPenalty ToQUBO.Attributes.ConstraintEncodingPenalty -``` \ No newline at end of file +``` + +## Discretization +```@docs +ToQUBO.Attributes.Discretize +``` diff --git a/src/attributes/compiler.jl b/src/attributes/compiler.jl index c3bc34ea..e7ad3a6a 100644 --- a/src/attributes/compiler.jl +++ b/src/attributes/compiler.jl @@ -7,7 +7,8 @@ import ..ToQUBO: Arithmetic, OneHot, DomainWall, - Bounded + Bounded, + GenericArchitecture import MathOptInterface as MOI const MOIU = MOI.Utilities @@ -34,6 +35,9 @@ export @doc raw""" Architecture() + +Selects which solver architecture to use. +Defaults to [`GenericArchitecture`](@ref). """ struct Architecture <: CompilerAttribute end function MOI.get(model::ToQUBO.VirtualModel, ::Architecture)::ToQUBO.AbstractArchitecture @@ -120,6 +124,8 @@ end @doc raw""" DefaultVariableEncodingMethod() + +Fallback value for [`VariableEncodingMethod`](@ref). """ struct DefaultVariableEncodingMethod <: CompilerAttribute end function MOI.get(model::ToQUBO.VirtualModel, ::DefaultVariableEncodingMethod)::ToQUBO.Encoding @@ -134,6 +140,8 @@ end @doc raw""" DefaultVariableEncodingATol() + +Fallback value for [`VariableEncodingATol`](@ref). """ struct DefaultVariableEncodingATol <: CompilerAttribute end function MOI.get(model::ToQUBO.VirtualModel{T}, ::DefaultVariableEncodingATol)::T where {T} diff --git a/src/compiler/architectures.jl b/src/compiler/architectures.jl index 52c0cefe..d1f1e998 100644 --- a/src/compiler/architectures.jl +++ b/src/compiler/architectures.jl @@ -2,7 +2,12 @@ AbstractArchitecture """ abstract type AbstractArchitecture end -struct GenericArchitecture <: AbstractArchitecture end +@doc raw""" + GenericArchitecture() + +This type is used to reach fallback implementations for [`AbstractArchitecture`](@ref) and, therefore, +should not have any methods directely related to it. +""" struct GenericArchitecture <: AbstractArchitecture end @doc raw""" """ function infer_architecture end From 1152620e2d537f582c61296136b50643ffcad5aa Mon Sep 17 00:00:00 2001 From: Pedro Maciel Xavier Date: Fri, 24 Feb 2023 09:47:59 -0300 Subject: [PATCH 15/38] Center images & diagrams --- docs/src/assets/extra_styles.css | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/docs/src/assets/extra_styles.css b/docs/src/assets/extra_styles.css index 37ee7a0b..4e0f31fb 100644 --- a/docs/src/assets/extra_styles.css +++ b/docs/src/assets/extra_styles.css @@ -4,5 +4,5 @@ } .mermaid { - + margin: 0 auto; } \ No newline at end of file From 80ee186a93994a83474cfb4d1ffa18a9a82a9558 Mon Sep 17 00:00:00 2001 From: Pedro Maciel Xavier Date: Fri, 24 Feb 2023 09:49:02 -0300 Subject: [PATCH 16/38] Fix `extra_styles.css` --- docs/src/assets/extra_styles.css | 11 +++++++++-- 1 file changed, 9 insertions(+), 2 deletions(-) diff --git a/docs/src/assets/extra_styles.css b/docs/src/assets/extra_styles.css index 4e0f31fb..2d249309 100644 --- a/docs/src/assets/extra_styles.css +++ b/docs/src/assets/extra_styles.css @@ -3,6 +3,13 @@ margin: auto; } -.mermaid { +/* Center images & diagrams */ +p:has(img) { + text-align: center; margin: 0 auto; -} \ No newline at end of file +} + +div.mermaid { + text-align: center; + margin: 0 auto; +} From 7eb1ae4d1a48f5fba59ff312786cb5b96a6293e9 Mon Sep 17 00:00:00 2001 From: pedroripper Date: Fri, 24 Feb 2023 14:19:30 -0300 Subject: [PATCH 17/38] Add encoding intro --- docs/src/booklet/4-encoding.md | 14 ++++++++++++++ 1 file changed, 14 insertions(+) diff --git a/docs/src/booklet/4-encoding.md b/docs/src/booklet/4-encoding.md index c323b23e..5f78ce98 100644 --- a/docs/src/booklet/4-encoding.md +++ b/docs/src/booklet/4-encoding.md @@ -2,6 +2,20 @@ ## Variable Encoding +As you should already know, QUBO models are comprised only of binary variables. So when we are reformulating general optimization problems, one important step is to encode variables into binary ones. + +`ToQUBO` currently implements 6 encoding techniques. Each method introduces a different number of variables, quadratic terms and linear terms. Also, they differ in the range of their coefficients($\Delta$). + +| Encoding | Binary Variables | # Linear terms | # Quadratic terms | $\Delta$ | +|:---------------------:|:------------------:|:----------------:|:-------------------:|:----------:| +| Binary | $O(log \ n)$ | $O(log \ n)$ | - | $O(n)$ | +| Unary | $O(n)$ | $O(n)$ | - | $O(1)$ | +| One-Hot | $O(n)$ | $O(n)$ | $O(n^2)$ | $O(n)$ | +| Domain-Wall | $O(n)$ | $O(n)$ | $O(n)$ | $O(n)$ | +| Bounded-Coefficient | $O(n)$ | $O(n)$ | - | $O(1)$ | +| Arithmetic Prog | $O(\sqrt{n})$ | $O(\sqrt{n})$ | - | $O(\sqrt{n})$ | + + ### Linear Encoding Methods ```@docs ToQUBO.Mirror From e56193c93d10ad18d8b191dc157f577d9d8de63f Mon Sep 17 00:00:00 2001 From: pedroripper Date: Tue, 28 Feb 2023 16:45:12 -0300 Subject: [PATCH 18/38] Update docs for constraint encoding --- docs/src/booklet/4-encoding.md | 15 +++++- src/compiler/constraints.jl | 96 +++++++++++++++++++++++++++++++--- src/compiler/interface.jl | 9 ++-- 3 files changed, 109 insertions(+), 11 deletions(-) diff --git a/docs/src/booklet/4-encoding.md b/docs/src/booklet/4-encoding.md index 5f78ce98..296ba290 100644 --- a/docs/src/booklet/4-encoding.md +++ b/docs/src/booklet/4-encoding.md @@ -38,4 +38,17 @@ ToQUBO.DomainWall ToQUBO.Bounded ``` -## Constraint Encoding \ No newline at end of file +## Constraint Encoding + +As you should already know, a QUBO model is unconstrained. So when `ToQUBO` is reformulating a problem, it needs to encode all constraints into the objective function loosing as little information as possible. + +As constraints are introduced into the objective function, we need to make sure that they won't be violated. In order to do that, `ToQUBO` multiplies the encoded constraint by a large penalty $\rho$, so that any violation would result in an infeasible solution to the problem. + +Sometimes, moving a constraint to the objective fuction might introduce higher-order terms (degree > 2). If that is the case, `ToQUBO` needs to reduce it back to a quadratic function. + +As of today, `ToQUBO` provides encoding for the following constraints: + +```@docs +ToQUBO.toqubo_constraint +``` + diff --git a/src/compiler/constraints.jl b/src/compiler/constraints.jl index e2c435ac..16cc4ae5 100644 --- a/src/compiler/constraints.jl +++ b/src/compiler/constraints.jl @@ -40,20 +40,29 @@ function toqubo_constraint( end @doc raw""" - toqubo_constraint(model::VirtualModel{T}, f::SAF{T}, s::EQ{T}, ::AbstractArchitecture) where {T} + toqubo_constraint( + model::VirtualModel{T}, + f::SAF{T}, + s::EQ{T}, + ::AbstractArchitecture + ) where {T} Turns constraints of the form ```math + \begin{array}{rl} \text{s.t} & \mathbf{a}'\mathbf{x} - b = 0 \end{array} + ``` into ```math -\left\Vertg(\mathbf{x})\right\Vert_{\left\lbrace{0}\right\rbrace} = \left(\mathbf{a}'\mathbf{x} - b\right)^{2} + +\left\Vert(\mathbf{x})\right\Vert_{\left\lbrace{0}\right\rbrace} = \left(\mathbf{a}'\mathbf{x} - b\right)^{2} + ``` """ function toqubo_constraint( @@ -81,7 +90,12 @@ function toqubo_constraint( end @doc raw""" - toqubo_constraint(model::VirtualModel{T}, f::SAF{T}, s::LT{T}, ::AbstractArchitecture) where {T} + toqubo_constraint( + model::VirtualModel{T}, + f::SAF{T}, + s::LT{T}, + ::AbstractArchitecture + ) where {T} Turns constraints of the form @@ -94,7 +108,8 @@ Turns constraints of the form into ```math -\left\Vertg(\mathbf{x})\right\Vert_{\left\lbrace{0}\right\rbrace} = \left(\mathbf{a}'\mathbf{x} - b\right + z)^{2} +\left\Vert(\mathbf{x})\right\Vert_{\left\lbrace{0}\right\rbrace} = (\mathbf{a}'\mathbf{x} - b + z)^{2} + ``` by adding a slack variable ``z``. @@ -131,13 +146,37 @@ function toqubo_constraint( return g^2 end +@doc raw""" + toqubo_constraint( + model::VirtualModel{T}, + f::SQF{T}, + s::EQ{T}, + arch::AbstractArchitecture, + ) where {T} + +Turns constraints of the form + +```math +\begin{array}{rl} +\text{s.t} & \mathbf{x}'\mathbf{Q}\mathbf{x} + \mathbf{a}'\mathbf{x} - b = 0 +\end{array} +``` + +into + +```math +\left\Vert(\mathbf{x})\right\Vert_{\left\lbrace{0}\right\rbrace} = (\mathbf{x}'\mathbf{Q}\mathbf{x} + \mathbf{a}'\mathbf{x} - b)^{2} + +``` + +""" function toqubo_constraint( model::VirtualModel{T}, f::SQF{T}, s::EQ{T}, arch::AbstractArchitecture, ) where {T} - # Scalar Quadratic Equality Constraint: g(x) = x Q x + a x - b = 0 + # Scalar Quadratic Equality Constraint: g(x) = x' Q x + a' x - b = 0 g = toqubo_parse(model, f, s, arch) PBO.discretize!(g) @@ -158,13 +197,39 @@ function toqubo_constraint( return g^2 end + +@doc raw""" + toqubo_constraint( + model::VirtualModel{T}, + f::SQF{T}, + s::LT{T}, + arch::AbstractArchitecture, + ) where {T} + +Turns constraints of the form + +```math +\begin{array}{rl} +\text{s.t} & \mathbf{x}'\mathbf{Q}\mathbf{x} + \mathbf{a}'\mathbf{x} - b \leq 0 +\end{array} +``` + +into + +```math +\left\Vert(\mathbf{x})\right\Vert_{\left\lbrace{0}\right\rbrace} = (\mathbf{x}'\mathbf{Q}\mathbf{x} + \mathbf{a}'\mathbf{x} - b + z)^{2} + +``` + +by adding a slack variable ``z``. +""" function toqubo_constraint( model::VirtualModel{T}, f::SQF{T}, s::LT{T}, arch::AbstractArchitecture, ) where {T} - # Scalar Quadratic Inequality Constraint: g(x) = x Q x + a x - b ≤ 0 + # Scalar Quadratic Inequality Constraint: g(x) = x' Q x + a' x - b ≤ 0 g = toqubo_parse(model, f, s, arch) PBO.discretize!(g) @@ -193,6 +258,25 @@ function toqubo_constraint( return g^2 end +@doc raw""" + toqubo_constraint( + model::VirtualModel{T}, + x::MOI.VectorOfVariables, + ::MOI.SOS1{T}, + ::AbstractArchitecture, + ) where {T} + +Turns constraints of the form + +```math +\begin{array}{rl} +\text{s.t} & \sum{\mathbf{x}} \leq \min \mathbf{x} +\end{array} +``` + +into + +""" function toqubo_constraint( model::VirtualModel{T}, x::MOI.VectorOfVariables, diff --git a/src/compiler/interface.jl b/src/compiler/interface.jl index 45bbc3ea..434fef92 100644 --- a/src/compiler/interface.jl +++ b/src/compiler/interface.jl @@ -54,11 +54,12 @@ Copies `MOI.ObjectiveSense` from `model.source_model` to `model.target_model`. toqubo_constraints!(model::VirtualModel, ::AbstractArchitecture) """ function toqubo_constraints! end -@doc raw""" - toqubo_constraint +# """ +# toqubo_constraint -Returns the pseudo-boolean function associated to a given constraint from the source model. -""" function toqubo_constraint end +# Returns the pseudo-boolean function associated to a given constraint from the source model. +# """ +function toqubo_constraint end @doc raw""" toqubo_parse!( From a7786ccf4d85e6a0c2faab71aafaf1811fb4e401 Mon Sep 17 00:00:00 2001 From: Pedro Maciel Xavier Date: Fri, 17 Mar 2023 15:38:44 -0300 Subject: [PATCH 19/38] Update docs --- docs/make.jl | 3 ++- docs/src/booklet/3-pbo.md | 11 +++-------- docs/src/booklet/4-encoding.md | 9 ++++++++- docs/src/booklet/5-virtual.md | 18 +++++------------- src/model/virtual.jl | 16 +++++++++------- 5 files changed, 27 insertions(+), 30 deletions(-) diff --git a/docs/make.jl b/docs/make.jl index 567eb02c..7f6dbfe5 100644 --- a/docs/make.jl +++ b/docs/make.jl @@ -14,7 +14,8 @@ makedocs(; assets = [ "assets/extra_styles.css", "assets/favicon.ico", - "assets/init.js", + asset("https://tikzjax.com/v1/fonts.css"; class = :css), + asset("https://tikzjax.com/v1/tikzjax.js"; class = :js), ], mathengine = Documenter.KaTeX(), sidebar_sitename = false, diff --git a/docs/src/booklet/3-pbo.md b/docs/src/booklet/3-pbo.md index 97c5c57f..77617401 100644 --- a/docs/src/booklet/3-pbo.md +++ b/docs/src/booklet/3-pbo.md @@ -23,6 +23,9 @@ However, using Julia's multiple dispatch paradigm, it's possible to extend the q ToQUBO.PBO.quadratize! ``` +[^Dattani2019]: + Nikesh S. Dattani, **Quadratization in discrete optimization and quantum mechanics**, *ArXiv*, 2019 [{doi}](https://doi.org/10.48550/arXiv.1901.04405) + ### Implemented Quadratization Techniques Currently, `ToQUBO` has two reduction algorithms, one for negative and another for positive terms. @@ -32,7 +35,6 @@ ToQUBO.PBO.NTR_KZFD ToQUBO.PBO.PTR_BG ``` - ### A Primer on Submodularity A set function ``f : 2^{S} \to \mathbb{R}`` is said to be submodular if @@ -41,10 +43,3 @@ f(X \cup Y) + f(X \cap Y) \le f(X) + f(Y) \forall X, Y \subset S ``` holds. - - - - - -[^Dattani2019]: - Dattani, Nikesh S.., **Quadratization in discrete optimization and quantum mechanics**, *ArXiv*, 2019 [{doi}](https://doi.org/10.48550/arXiv.1901.04405) diff --git a/docs/src/booklet/4-encoding.md b/docs/src/booklet/4-encoding.md index c323b23e..1d10a29b 100644 --- a/docs/src/booklet/4-encoding.md +++ b/docs/src/booklet/4-encoding.md @@ -5,6 +5,9 @@ ### Linear Encoding Methods ```@docs ToQUBO.Mirror +``` + +```@docs ToQUBO.LinearEncoding ToQUBO.Linear ToQUBO.Binary @@ -24,4 +27,8 @@ ToQUBO.DomainWall ToQUBO.Bounded ``` -## Constraint Encoding \ No newline at end of file +### Constraint Encoding + +!!! warning "Work in progress" + We hope to write this part of the documentation soon. + Please come back later! diff --git a/docs/src/booklet/5-virtual.md b/docs/src/booklet/5-virtual.md index dce6a7eb..6ef0b68f 100644 --- a/docs/src/booklet/5-virtual.md +++ b/docs/src/booklet/5-virtual.md @@ -5,19 +5,11 @@ The target one is on the QUBO form used during the solving process. Both lie within a *Virtual Model*, which provides the necessary API integration and keeps all variable and constraint mapping tied together. ```@raw html -
-flowchart LR; - JUMP_MODEL["JuMP Model"]; - - subgraph VIRTUAL_MODEL["Virtual Model"]; - SOURCE_MODEL["Source Model"]; - TARGET_MODEL["Target Model"]; - - SOURCE_MODEL -->|ToQUBO.jl| TARGET_MODEL; - end - - JUMP_MODEL -->|MOI Bridges| SOURCE_MODEL; -
+ ``` This is done in a transparent fashion for both agents since the user will mostly interact with the presented model, and the solvers will only access the generated one. diff --git a/src/model/virtual.jl b/src/model/virtual.jl index 26690ba0..01fdecfc 100644 --- a/src/model/virtual.jl +++ b/src/model/virtual.jl @@ -188,7 +188,7 @@ function encode!( end @doc raw""" - Mirror + Mirror() Mirrors binary variable ``x \in \mathbb{B}`` with a twin variable ``y \in \mathbb{B}``. """ struct Mirror <: LinearEncoding end @@ -198,7 +198,7 @@ function encode!(model::VirtualModel{T}, e::Mirror, x::Union{VI,Nothing}) where end @doc raw""" - Linear + Linear() """ struct Linear <: LinearEncoding end function encode!( @@ -219,7 +219,7 @@ end Let ``x \in [a, b] \subset \mathbb{Z}, n = b - a, \mathbf{y} \in \mathbb{B}^{n}``. ```math -x \approx \xi(\mathbf{y}) = a + \sum_{j = 1}^{b - a} y_{j} +x = \xi(\mathbf{y}) = a + \sum_{j = 1}^{b - a} y_{j} ``` """ struct Unary <: LinearEncoding end @@ -271,7 +271,7 @@ function encode!( end @doc raw""" - Binary + Binary() Binary Expansion within the closed interval ``[\alpha, \beta]``. @@ -338,7 +338,9 @@ function encode!( end @doc raw""" - Arithmetic + Arithmetic() + + """ struct Arithmetic <: LinearEncoding end function encode!( @@ -476,10 +478,10 @@ end A *sequential encoding* is one of the form ```math -\xi(\mathbf{y}) = \sum_{i = 1}^{n} \gamma_{i} \left({y_{i + 1} \ast y_{i}}\right) +\xi(\mathbf{y}) = \sum_{i = 1}^{n} \gamma_{i} \left({y_{i + 1} \circast y_{i}}\right) ``` -where ``\mathbf{y} \in \mathbb{B}^{n + 1}``. +where ``\mathbf{y} \in \mathbb{B}^{n + 1}`` and ``\circast`` is a binary operator. """ abstract type SequentialEncoding <: Encoding end function encode!( From 11383fbec21d62829e8987c260db62a6764fec2e Mon Sep 17 00:00:00 2001 From: pedroripper Date: Mon, 3 Apr 2023 16:07:34 -0300 Subject: [PATCH 20/38] Update docs [no ci] --- docs/src/booklet/4-encoding.md | 2 +- src/model/virtual.jl | 2 +- 2 files changed, 2 insertions(+), 2 deletions(-) diff --git a/docs/src/booklet/4-encoding.md b/docs/src/booklet/4-encoding.md index e8b6b625..62accdf4 100644 --- a/docs/src/booklet/4-encoding.md +++ b/docs/src/booklet/4-encoding.md @@ -43,7 +43,7 @@ ToQUBO.Bounded ## Constraint Encoding -As you should already know, a QUBO model is unconstrained. So when `ToQUBO` is reformulating a problem, it needs to encode all constraints into the objective function loosing as little information as possible. +A QUBO model is unconstrained. So when `ToQUBO` is reformulating a problem, it needs to encode all constraints into the objective function loosing as little information as possible. As constraints are introduced into the objective function, we need to make sure that they won't be violated. In order to do that, `ToQUBO` multiplies the encoded constraint by a large penalty $\rho$, so that any violation would result in an infeasible solution to the problem. diff --git a/src/model/virtual.jl b/src/model/virtual.jl index 01fdecfc..f0c384d1 100644 --- a/src/model/virtual.jl +++ b/src/model/virtual.jl @@ -396,7 +396,7 @@ end OneHot() The one-hot encoding is a linear technique used to represent a variable -``x \in \left\lbrace{\gamma_{j}}}\right\rbrace_{j \in \left[n\right]``. +``x \in \{ \gamma_{j} \}_{j \in [n]}``. The encoding function is combined with a constraint assuring that only one and exactly one of the expansion's variables ``y_{j}`` is activated From 44b3b9ba5c8ea2a0b943454f0808a9597c42de07 Mon Sep 17 00:00:00 2001 From: Pedro Maciel Xavier Date: Wed, 5 Apr 2023 14:08:22 -0300 Subject: [PATCH 21/38] Update README [no ci] --- README.md | 20 +++++++++----------- 1 file changed, 9 insertions(+), 11 deletions(-) diff --git a/README.md b/README.md index c5aef2f0..82d3c6c7 100644 --- a/README.md +++ b/README.md @@ -20,7 +20,7 @@ ## Introduction -ToQUBO.jl is a Julia package to reformulate general optimization problems into [QUBO](https://en.wikipedia.org/wiki/Quadratic_unconstrained_binary_optimization) (Quadratic Unconstrained Binary Optimization) instances. This tool aims to convert a broad range of [JuMP](https://github.com/jump-dev/JuMP.jl) problems for straightforward application in many physics and physics-inspired solution methods whose normal optimization form is equivalent to the QUBO. These methods include quantum annealing, quantum gate-circuit optimization algorithms (Quantum Optimization Alternating Ansatz, Variational Quantum Eigensolver), other hardware-accelerated platforms, such as Coherent Ising Machines and Simulated Bifurcation Machines, and more traditional methods such as simulated annealing. During execution, ToQUBO.jl encodes both discrete and continuous variables, maps constraints, and computes their penalties, performing a few model optimization steps along the process. A simple interface to connect various annealers and samplers as QUBO solvers is defined in [Anneal.jl](https://github.com/psrenergy/Anneal.jl). +ToQUBO.jl is a Julia package to reformulate general optimization problems into [QUBO](https://en.wikipedia.org/wiki/Quadratic_unconstrained_binary_optimization) (Quadratic Unconstrained Binary Optimization) instances. This tool aims to convert a broad range of [JuMP](https://github.com/jump-dev/JuMP.jl) problems for straightforward application in many physics and physics-inspired solution methods whose normal optimization form is equivalent to the QUBO. These methods include quantum QUBODriversing, quantum gate-circuit optimization algorithms (Quantum Optimization Alternating Ansatz, Variational Quantum Eigensolver), other hardware-accelerated platforms, such as Coherent Ising Machines and Simulated Bifurcation Machines, and more traditional methods such as simulated QUBODriversing. During execution, ToQUBO.jl encodes both discrete and continuous variables, maps constraints, and computes their penalties, performing a few model optimization steps along the process. A simple interface to connect various QUBODriversers and samplers as QUBO solvers is defined in [QUBODrivers.jl](https://github.com/psrenergy/QUBODrivers.jl). ToQUBO.jl was written as a [MathOptInterface](https://github.com/jump-dev/MathOptInterface.jl) (MOI) layer that automatically maps between input and output models, thus providing a smooth JuMP modeling experience. @@ -28,10 +28,7 @@ ToQUBO.jl was written as a [MathOptInterface](https://github.com/jump-dev/MathOp ### Installation ToQUBO is available via Julia's Pkg: -```julia -julia> ]add ToQUBO -``` -or + ```julia julia> using Pkg @@ -42,7 +39,7 @@ julia> Pkg.add("ToQUBO") ```julia using JuMP using ToQUBO -using Anneal +using QUBODrivers model = Model(() -> ToQUBO.Optimizer(ExactSampler.Optimizer)) @@ -53,9 +50,10 @@ model = Model(() -> ToQUBO.Optimizer(ExactSampler.Optimizer)) optimize!(model) for i = 1:result_count(model) - xᵢ = value.(x, result = i) - yᵢ = objective_value(model, result = i) - println("f($xᵢ) = $yᵢ") + xi = value.(x, result = i) + yi = objective_value(model, result = i) + + println("f($xi) = $yi") end ``` @@ -152,8 +150,8 @@ If you use `ToQUBO.jl` in your work, we kindly ask you to include the following ToQUBO.jl - - Anneal.jl + + QUBODrivers.jl QUBOTools.jl From ea009d9587951c7b651e944b45012883a52bf632 Mon Sep 17 00:00:00 2001 From: Pedro Maciel Xavier Date: Wed, 5 Apr 2023 14:23:36 -0300 Subject: [PATCH 22/38] Replace `Anneal -> QUBODrivers` --- docs/Project.toml | 4 ++-- docs/src/booklet/7-solvers.md | 6 +++--- 2 files changed, 5 insertions(+), 5 deletions(-) diff --git a/docs/Project.toml b/docs/Project.toml index 96713b1e..32c9e357 100644 --- a/docs/Project.toml +++ b/docs/Project.toml @@ -1,5 +1,5 @@ [deps] -Anneal = "e4d9eb7f-b088-426e-aeb5-1c0dae3d8abb" +QUBODrivers = "e4d9eb7f-b088-426e-aeb5-1c0dae3d8abb" CSV = "336ed68f-0bac-5ca0-87d4-7b16caf5d00b" DWaveNeal = "870cdf72-5502-4b10-839c-127ceab78f22" DataFrames = "a93c6f00-e57d-5684-b7b6-d8193f3e46c0" @@ -11,7 +11,7 @@ Random = "9a3f8284-a2c9-5f02-9a11-845980a1fd5c" ToQUBO = "9a412ddf-83fa-43b6-9748-7843c851aa65" [compat] -Anneal = "0.6" +QUBODrivers = "0.1" CSV = "0.10" DataFrames = "1.3" Documenter = "0.27" diff --git a/docs/src/booklet/7-solvers.md b/docs/src/booklet/7-solvers.md index 237764a2..2f504a24 100644 --- a/docs/src/booklet/7-solvers.md +++ b/docs/src/booklet/7-solvers.md @@ -2,13 +2,13 @@ ## Solvers, Annealers & Samplers [`ToQUBO.jl`](https://github.com/psrenergy/ToQUBO.jl)'s main goal is to make use of parameterized stochastic optimization solvers, particularly those relying on non-conventional hardware such as *Quantum Annealing* and other *Ising Machines*. -A few `MOI`-compliant interfaces for annealers and samplers are bundled within [`ToQUBO.jl`](https://github.com/psrenergy/ToQUBO.jl) via the [`Anneal.jl`](https://github.com/psrenergy/Anneal.jl) companion package. +A few `MOI`-compliant interfaces for annealers and samplers are bundled within [`ToQUBO.jl`](https://github.com/psrenergy/ToQUBO.jl) via the [`QUBODrivers.jl`](https://github.com/psrenergy/QUBODrivers.jl) companion package. Some of them are presented below. ## Simulated Annealing Provided by D-Wave's open-source code libraries, this [Simulated Annealing](https://en.wikipedia.org/wiki/Simulated_annealing) engine implements some of the features and configurations you would find using the Quantum API. Its adoption is recommended for basic usage, tests, and research due to its robustness, simplicity and ease of use. -The [`DWaveNeal.jl`](https://github.com/psrenergy/DWaveNeal.jl) package uses [`Anneal.jl`](https://github.com/psrenergy/Anneal.jl) to deliver an interface to this sampler. +The [`DWaveNeal.jl`](https://github.com/psrenergy/DWaveNeal.jl) package uses [`QUBODrivers.jl`](https://github.com/psrenergy/QUBODrivers.jl) to deliver an interface to this sampler. ## Quantum Annealing Interfacing with [D-Wave](https://www.dwavesys.com/)'s quantum annealer is one of the milestones we expect to achieve with this package. @@ -17,7 +17,7 @@ In a first moment, for those willing to get started, the [`DWaveNeal.jl`](https: ## Random Sampling This sampler is implemented for test purposes and simply assigns 0 or 1 to each variable according to a given probability bias ``0 \le p \le 1``, which defaults to ``p = 0.5``. -After running the `using Anneal` command, `RandomSampler.Optimizer` will be available. +After running the `using QUBODrivers` command, `RandomSampler.Optimizer` will be available. ## Exact Solver (Exhaustive Enumeration) Also made to be used in tests, the `ExactSolver.Optimizer` interface runs through all possible state configurations, which implies in an exponential time complexity on the number of variables. From b99166e9dd8fbfe93ea69acb1d54523cb1c6d7ee Mon Sep 17 00:00:00 2001 From: pedroripper Date: Fri, 14 Apr 2023 08:58:53 -0300 Subject: [PATCH 23/38] Add Portfolio Optimization tutorial --- docs/Project.toml | 5 +- docs/make.jl | 1 + docs/src/examples/portfolio_optimization.md | 85 +++++++++++++++++++++ 3 files changed, 89 insertions(+), 2 deletions(-) create mode 100644 docs/src/examples/portfolio_optimization.md diff --git a/docs/Project.toml b/docs/Project.toml index 32c9e357..87bf3fbb 100644 --- a/docs/Project.toml +++ b/docs/Project.toml @@ -1,5 +1,4 @@ [deps] -QUBODrivers = "e4d9eb7f-b088-426e-aeb5-1c0dae3d8abb" CSV = "336ed68f-0bac-5ca0-87d4-7b16caf5d00b" DWaveNeal = "870cdf72-5502-4b10-839c-127ceab78f22" DataFrames = "a93c6f00-e57d-5684-b7b6-d8193f3e46c0" @@ -7,14 +6,16 @@ Documenter = "e30172f5-a6a5-5a46-863b-614d45cd2de4" JuMP = "4076af6c-e467-56ae-b986-b466b2749572" MQLib = "16f11440-1623-44c9-850c-358a6c72f3c9" MathOptInterface = "b8f27783-ece8-5eb3-8dc8-9495eed66fee" +QUBODrivers = "e4d9eb7f-b088-426e-aeb5-1c0dae3d8abb" Random = "9a3f8284-a2c9-5f02-9a11-845980a1fd5c" +Statistics = "10745b16-79ce-11e8-11f9-7d13ad32a3b2" ToQUBO = "9a412ddf-83fa-43b6-9748-7843c851aa65" [compat] -QUBODrivers = "0.1" CSV = "0.10" DataFrames = "1.3" Documenter = "0.27" JuMP = "1" MathOptInterface = "1" +QUBODrivers = "0.1" ToQUBO = "0.1" diff --git a/docs/make.jl b/docs/make.jl index 7f6dbfe5..a0852c66 100644 --- a/docs/make.jl +++ b/docs/make.jl @@ -33,6 +33,7 @@ makedocs(; "Examples" => [ "Knapsack" => "examples/knapsack.md", "Prime Factorization" => "examples/prime_factorization.md", + "Portfolio Optimization" => "examples/portfolio_optimization.md", ], "Booklet" => [ "Introduction" => "booklet/1-intro.md", diff --git a/docs/src/examples/portfolio_optimization.md b/docs/src/examples/portfolio_optimization.md new file mode 100644 index 00000000..f6f142c7 --- /dev/null +++ b/docs/src/examples/portfolio_optimization.md @@ -0,0 +1,85 @@ +# Portfolio Optimization + +For this example, we will be using the data provided for a tutorial on Portolio Optimization in the [JuMP documentation](https://jump.dev/JuMP.jl/stable/tutorials/nonlinear/portfolio/), where they provide the stock prices for three assets $\texttt{IBM}$, $\texttt{WMT}$ and $\texttt{SEHI}$. + +Portfolio Optimization aims to sort out the best asset distribution that maximizes the return and minimizes the financial risk. We will be using the following approach to model this problem: +```math +\begin{array}{r l} + \max & \mu'x - \lambda x'Q x\\ + \text{s.t.} & 0.0 \leq x \leq 1.0 \\ + & \sum{x_i} = 1.0 +\end{array} +``` +where, +- ``\mu`` is the expected return value for each investment +- ``Q`` is the covariance matrix +- ``\lambda`` is a penalization + +## Importing the required packages +```@example portfolio-optimization +using JuMP +using DataFrames +using Statistics +using ToQUBO +using DWaveNeal +``` + +## Stock prices +```@example portfolio-optimization +df = DataFrames.DataFrame( + [ + 93.043 51.826 1.063 + 84.585 52.823 0.938 + 111.453 56.477 1.000 + 99.525 49.805 0.938 + 95.819 50.287 1.438 + 114.708 51.521 1.700 + 111.515 51.531 2.540 + 113.211 48.664 2.390 + 104.942 55.744 3.120 + 99.827 47.916 2.980 + 91.607 49.438 1.900 + 107.937 51.336 1.750 + 115.590 55.081 1.800 + ], + [:IBM, :WMT, :SEHI], +) +``` + +## Solving + + +```@example portfolio-optimization +function solve(df::DataFrame, λ::Float64 = 10.; optimizer = DWaveNeal.Optimizer) + return solve(identity, df, λ; optimizer) +end + +function solve(config!::Function, df::DataFrame, λ::Float64 = 10.; optimizer = DWaveNeal.Optimizer) + r = diff(Matrix(df); dims = 1) ./ Matrix(df[1:end-1, :]) + + # Expected montly return value for each stock + μ = vec(Statistics.mean(r; dims = 1)) + + # Covariance matrix + Σ = Statistics.cov(r) + + model = Model(() -> ToQUBO.Optimizer(optimizer)) + + @variable(model, 0.0 <= x[1:3] <= 1.0) + @objective(model, Max, μ'x - λ*x'*Σ*x) + @constraint(model, sum(x) == 1) + + config!(model) + + optimize!(model) + + return value.(x) +end +``` + + +```@example portfolio-optimization +solve(df) do model + JuMP.set_silent(model) +end +``` \ No newline at end of file From 95239a863923510820742efa0eec5edf72378955 Mon Sep 17 00:00:00 2001 From: Pedro Maciel Xavier Date: Mon, 17 Apr 2023 13:13:02 -0300 Subject: [PATCH 24/38] Update docs dependencies --- docs/Project.toml | 4 +--- 1 file changed, 1 insertion(+), 3 deletions(-) diff --git a/docs/Project.toml b/docs/Project.toml index 8c1fba37..d545a0b4 100644 --- a/docs/Project.toml +++ b/docs/Project.toml @@ -4,9 +4,8 @@ DWaveNeal = "870cdf72-5502-4b10-839c-127ceab78f22" DataFrames = "a93c6f00-e57d-5684-b7b6-d8193f3e46c0" Documenter = "e30172f5-a6a5-5a46-863b-614d45cd2de4" JuMP = "4076af6c-e467-56ae-b986-b466b2749572" -MQLib = "16f11440-1623-44c9-850c-358a6c72f3c9" MathOptInterface = "b8f27783-ece8-5eb3-8dc8-9495eed66fee" -QUBODrivers = "a3f166f7-2cd3-47b6-9e1e-6fbfe0449eb0" +PythonPlot = "274fc56d-3b97-40fa-a1cd-1b4a50311bf9" Random = "9a3f8284-a2c9-5f02-9a11-845980a1fd5c" Statistics = "10745b16-79ce-11e8-11f9-7d13ad32a3b2" ToQUBO = "9a412ddf-83fa-43b6-9748-7843c851aa65" @@ -17,5 +16,4 @@ DataFrames = "1.3" Documenter = "0.27" JuMP = "1" MathOptInterface = "1" -QUBODrivers = "0.1" ToQUBO = "0.1" From e783c574f46d8e800adb2743d73a36d3793100b3 Mon Sep 17 00:00:00 2001 From: Pedro Maciel Xavier Date: Mon, 17 Apr 2023 13:13:16 -0300 Subject: [PATCH 25/38] Polish portfolio optimization example --- docs/make.jl | 4 +- docs/src/examples/portfolio_optimization.md | 107 +++++++++++++------- 2 files changed, 74 insertions(+), 37 deletions(-) diff --git a/docs/make.jl b/docs/make.jl index a0852c66..4a289630 100644 --- a/docs/make.jl +++ b/docs/make.jl @@ -31,8 +31,8 @@ makedocs(; "Compiler Settings" => "manual/4-settings.md", ], "Examples" => [ - "Knapsack" => "examples/knapsack.md", - "Prime Factorization" => "examples/prime_factorization.md", + "Knapsack" => "examples/knapsack.md", + "Prime Factorization" => "examples/prime_factorization.md", "Portfolio Optimization" => "examples/portfolio_optimization.md", ], "Booklet" => [ diff --git a/docs/src/examples/portfolio_optimization.md b/docs/src/examples/portfolio_optimization.md index f6f142c7..1f718175 100644 --- a/docs/src/examples/portfolio_optimization.md +++ b/docs/src/examples/portfolio_optimization.md @@ -1,72 +1,75 @@ # Portfolio Optimization -For this example, we will be using the data provided for a tutorial on Portolio Optimization in the [JuMP documentation](https://jump.dev/JuMP.jl/stable/tutorials/nonlinear/portfolio/), where they provide the stock prices for three assets $\texttt{IBM}$, $\texttt{WMT}$ and $\texttt{SEHI}$. +In this example, we will be exploring an optimization model for asset distribution where the expected return is maximized while mitigating the financial risk. +The following approach was inspired by a [JuMP tutorial](https://jump.dev/JuMP.jl/stable/tutorials/nonlinear/portfolio/), where monthly stock prices for three assets are provided, namely `IBM`, `WMT` and `SEHI`. + -Portfolio Optimization aims to sort out the best asset distribution that maximizes the return and minimizes the financial risk. We will be using the following approach to model this problem: ```math -\begin{array}{r l} - \max & \mu'x - \lambda x'Q x\\ - \text{s.t.} & 0.0 \leq x \leq 1.0 \\ - & \sum{x_i} = 1.0 +\begin{array}{rll} + \max_{\mathbf{x}} & \mathbf{\mu}'\mathbf{x} - \lambda\, \mathbf{x}' \Sigma \mathbf{x} \\ + \textrm{s.t.} & 0 \leq {x}_{i} \leq 1 & \forall i \\ + & \sum_{i} {x}_{i} = 1 \end{array} ``` -where, -- ``\mu`` is the expected return value for each investment -- ``Q`` is the covariance matrix -- ``\lambda`` is a penalization -## Importing the required packages +where ``\mu_{i}`` is the expected return value for each investment ``i``; ``\Sigma`` is the covariance matrix and ``\lambda`` is a penalization factor. + +## Stock prices ```@example portfolio-optimization -using JuMP using DataFrames using Statistics -using ToQUBO -using DWaveNeal -``` -## Stock prices -```@example portfolio-optimization +assets = [:IBM, :WMT, :SEHI] + df = DataFrames.DataFrame( [ - 93.043 51.826 1.063 - 84.585 52.823 0.938 + 93.043 51.826 1.063 + 84.585 52.823 0.938 111.453 56.477 1.000 - 99.525 49.805 0.938 - 95.819 50.287 1.438 + 99.525 49.805 0.938 + 95.819 50.287 1.438 114.708 51.521 1.700 111.515 51.531 2.540 113.211 48.664 2.390 104.942 55.744 3.120 - 99.827 47.916 2.980 - 91.607 49.438 1.900 + 99.827 47.916 2.980 + 91.607 49.438 1.900 107.937 51.336 1.750 115.590 55.081 1.800 ], - [:IBM, :WMT, :SEHI], + assets, ) ``` ## Solving - - ```@example portfolio-optimization -function solve(df::DataFrame, λ::Float64 = 10.; optimizer = DWaveNeal.Optimizer) - return solve(identity, df, λ; optimizer) -end +using JuMP +using ToQUBO +using DWaveNeal + +function solve( + config!::Function, + df::DataFrame, + λ::Float64 = 10.; + optimizer = DWaveNeal.Optimizer +) + # Number of assets + n = size(df, 2) -function solve(config!::Function, df::DataFrame, λ::Float64 = 10.; optimizer = DWaveNeal.Optimizer) + # Relative monthly return r = diff(Matrix(df); dims = 1) ./ Matrix(df[1:end-1, :]) - # Expected montly return value for each stock + # Expected monthly return value for each stock μ = vec(Statistics.mean(r; dims = 1)) # Covariance matrix Σ = Statistics.cov(r) + # Build model model = Model(() -> ToQUBO.Optimizer(optimizer)) - @variable(model, 0.0 <= x[1:3] <= 1.0) - @objective(model, Max, μ'x - λ*x'*Σ*x) + @variable(model, 0 <= x[1:n] <= 1) + @objective(model, Max, μ'x - λ * x' * Σ * x) @constraint(model, sum(x) == 1) config!(model) @@ -75,11 +78,45 @@ function solve(config!::Function, df::DataFrame, λ::Float64 = 10.; optimizer = return value.(x) end -``` +function solve(df::DataFrame, λ::Float64 = 10.; optimizer = DWaveNeal.Optimizer) + return solve(identity, df, λ; optimizer) +end +``` ```@example portfolio-optimization -solve(df) do model +solve(df, λ) do model JuMP.set_silent(model) + JuMP.set_optimizer_attribute(model, "num_reads", 2_000) end +``` + +## Penalty Analysis +To finish our discussion, we are going to sketch some graphics to help our reasoning on how the penalty factor ``\lambda`` affects our investments. + +```@example portfolio-optimization +using Plots; pythonplot() + +Λ = collect(0.:10.:100.) +X = Dict{Symbol,Vector{Float64}}(tag => [] for tag in assets) + +for λ = Λ + x = solve(df, λ) + + for (i, tag) in enumerate(assets) + push!(X[tag], x[i]) + end +end + +plt = plot(; + title="Portfolio Optimization", + xlabel=raw"penalty factor ($\lambda$)", + ylabel=raw"investment share ($x$)", +) + +for tag in assets + plot!(plt, Λ, X[tag]; label=string(tag)) +end + +plt ``` \ No newline at end of file From e0d68b1ef7fd7adc4ccc318d63831341131cb32a Mon Sep 17 00:00:00 2001 From: Pedro Maciel Xavier Date: Mon, 17 Apr 2023 13:15:58 -0300 Subject: [PATCH 26/38] Enhance plots --- docs/src/examples/portfolio_optimization.md | 7 ++++--- 1 file changed, 4 insertions(+), 3 deletions(-) diff --git a/docs/src/examples/portfolio_optimization.md b/docs/src/examples/portfolio_optimization.md index 1f718175..51b0859a 100644 --- a/docs/src/examples/portfolio_optimization.md +++ b/docs/src/examples/portfolio_optimization.md @@ -4,10 +4,11 @@ In this example, we will be exploring an optimization model for asset distributi The following approach was inspired by a [JuMP tutorial](https://jump.dev/JuMP.jl/stable/tutorials/nonlinear/portfolio/), where monthly stock prices for three assets are provided, namely `IBM`, `WMT` and `SEHI`. + ```math \begin{array}{rll} \max_{\mathbf{x}} & \mathbf{\mu}'\mathbf{x} - \lambda\, \mathbf{x}' \Sigma \mathbf{x} \\ - \textrm{s.t.} & 0 \leq {x}_{i} \leq 1 & \forall i \\ + \textrm{s.t.} & 0 \le {x}_{i} \le 1 & \forall i \\ & \sum_{i} {x}_{i} = 1 \end{array} ``` @@ -85,7 +86,7 @@ end ``` ```@example portfolio-optimization -solve(df, λ) do model +solve(df) do model JuMP.set_silent(model) JuMP.set_optimizer_attribute(model, "num_reads", 2_000) end @@ -97,7 +98,7 @@ To finish our discussion, we are going to sketch some graphics to help our reaso ```@example portfolio-optimization using Plots; pythonplot() -Λ = collect(0.:10.:100.) +Λ = collect(0.:5.:50.) X = Dict{Symbol,Vector{Float64}}(tag => [] for tag in assets) for λ = Λ From 38d240f46d33af2d587d745189ca9612762d1829 Mon Sep 17 00:00:00 2001 From: Pedro Maciel Xavier Date: Mon, 17 Apr 2023 13:31:38 -0300 Subject: [PATCH 27/38] Add text --- docs/src/examples/portfolio_optimization.md | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/docs/src/examples/portfolio_optimization.md b/docs/src/examples/portfolio_optimization.md index 51b0859a..11132a25 100644 --- a/docs/src/examples/portfolio_optimization.md +++ b/docs/src/examples/portfolio_optimization.md @@ -3,7 +3,7 @@ In this example, we will be exploring an optimization model for asset distribution where the expected return is maximized while mitigating the financial risk. The following approach was inspired by a [JuMP tutorial](https://jump.dev/JuMP.jl/stable/tutorials/nonlinear/portfolio/), where monthly stock prices for three assets are provided, namely `IBM`, `WMT` and `SEHI`. - +The modelling presented below aggregates the risk measurement ``\mathbf{x}' \Sigma \mathbf{x}`` as a penalty term to the objective function, thus yielding ```math \begin{array}{rll} @@ -13,7 +13,7 @@ The following approach was inspired by a [JuMP tutorial](https://jump.dev/JuMP.j \end{array} ``` -where ``\mu_{i}`` is the expected return value for each investment ``i``; ``\Sigma`` is the covariance matrix and ``\lambda`` is a penalization factor. +where ``\mu_{i} = \mathbb{E}[r_{i}]`` is the expected return value for each investment ``i``; ``\Sigma`` is the covariance matrix and ``\lambda`` is the risk-aversion penalty factor. ## Stock prices ```@example portfolio-optimization From a0618c4c67916ffb6f65d472423da2de607b234b Mon Sep 17 00:00:00 2001 From: Pedro Maciel Xavier Date: Mon, 17 Apr 2023 14:02:42 -0300 Subject: [PATCH 28/38] Update dependencies --- docs/Project.toml | 2 ++ 1 file changed, 2 insertions(+) diff --git a/docs/Project.toml b/docs/Project.toml index d545a0b4..8b1a1e9f 100644 --- a/docs/Project.toml +++ b/docs/Project.toml @@ -5,6 +5,7 @@ DataFrames = "a93c6f00-e57d-5684-b7b6-d8193f3e46c0" Documenter = "e30172f5-a6a5-5a46-863b-614d45cd2de4" JuMP = "4076af6c-e467-56ae-b986-b466b2749572" MathOptInterface = "b8f27783-ece8-5eb3-8dc8-9495eed66fee" +Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80" PythonPlot = "274fc56d-3b97-40fa-a1cd-1b4a50311bf9" Random = "9a3f8284-a2c9-5f02-9a11-845980a1fd5c" Statistics = "10745b16-79ce-11e8-11f9-7d13ad32a3b2" @@ -12,6 +13,7 @@ ToQUBO = "9a412ddf-83fa-43b6-9748-7843c851aa65" [compat] CSV = "0.10" +DWaveNeal = "0.4" DataFrames = "1.3" Documenter = "0.27" JuMP = "1" From 263b5b8783fe82472f31dd2eb7b0f87294fa9d7b Mon Sep 17 00:00:00 2001 From: Pedro Maciel Xavier Date: Mon, 17 Apr 2023 14:27:15 -0300 Subject: [PATCH 29/38] Update docs action --- .github/workflows/documentation.yml | 6 ++++-- 1 file changed, 4 insertions(+), 2 deletions(-) diff --git a/.github/workflows/documentation.yml b/.github/workflows/documentation.yml index 1d9f992c..3fe05a41 100644 --- a/.github/workflows/documentation.yml +++ b/.github/workflows/documentation.yml @@ -14,10 +14,12 @@ jobs: - uses: julia-actions/setup-julia@latest with: version: '1.6' + - name: Develop ToQUBO.jl + run: julia --project=docs -e 'using Pkg; Pkg.develop(path=pwd())' - uses: julia-actions/julia-buildpkg@latest with: - project: "docs/" + project: "docs" - name: Build and deploy env: GITHUB_TOKEN: ${{ secrets.GITHUB_TOKEN }} # For authentication with GitHub Actions token - run: julia --project=docs/ docs/make.jl + run: julia --project=docs docs/make.jl From 8081763678526ef45aad3bc600776298fc306a2a Mon Sep 17 00:00:00 2001 From: Pedro Maciel Xavier Date: Mon, 17 Apr 2023 16:19:43 -0300 Subject: [PATCH 30/38] Enhance text [incomplete] --- docs/src/booklet/2-qubo.md | 34 ++++++++++++++++++++++------------ 1 file changed, 22 insertions(+), 12 deletions(-) diff --git a/docs/src/booklet/2-qubo.md b/docs/src/booklet/2-qubo.md index 196a855f..2f84fcb2 100644 --- a/docs/src/booklet/2-qubo.md +++ b/docs/src/booklet/2-qubo.md @@ -1,31 +1,41 @@ # QUBO ## Definition -QUBO stands for _**Q**uadratic **U**nconstrained **B**inary **O**ptimization_. - -So every QUBO model has the following characteristics: -- Its objective function is at most quadratic -- There are no constraints -- All variables are binary (0 or 1) - -We can represent a QUBO as follows. +**Q**uadratic **U**nconstrained **B**inary **O**ptimization, as its name suggests, refers to the global minimization or maximization of a given quadratic polynomial over binary variables whose domain is not constrained in any way. +A common mathematical presentation is given by their quadratic matrix form, i.e. ```math \begin{array}{rl} - \min & \mathbf{x}' Q\,\mathbf{x} \\ - \mathrm{s.t.} & \mathbf{x} \in \mathbb{B}^{n} + \min_{\mathbf{x}} & \mathbf{x}' Q\,\mathbf{x} \\ + \textrm{s.t.} & \mathbf{x} \in \mathbb{B}^{n} \end{array} ``` +where ``Q \in \mathbb{R}^{n \times n}`` is symmetric and ``\mathbb{B} = \lbrace{0, 1}\rbrace``. + ## OK, but why QUBO? +Mathematically speaking, there is a notorious equivalence between QUBO and [Max-Cut](https://en.wikipedia.org/wiki/Maximum_cut) problems, e.g. for every QUBO instance there is an information preserving Max-Cut reformulation and vice versa. +This statement has two immediate implications: +1. In the general case, solving QUBO globally is NP-Hard. +2. It is a simple yet expressive mathematical programming framework. + +Implication 1. tells us that such problems are computationally intractable and that heuristics and metaheuristics are to be employed instead of exact methods. +No 2. relates to the fact that we are able to represent many other optimization models by means of the QUBO formalism. + +Moreover, [Ising Model](https://en.wikipedia.org/wiki/Ising_model). +```math +\begin{array}{rl} + \min_{\mathbf{s}} & \mathbf{h}'\mathbf{s} + \mathbf{s}' J\,\mathbf{s} \\ + \textrm{s.t.} & \mathbf{s} \in \lbrace{-1,+1}\rbrace^{n} +\end{array} +``` -QUBOs are suited for representing non-convex global optimization problems. With that said, the significant advances in computing systems and algorithms specialized for sampling QUBOs have contributed to their popularity. Some of the paradigms that stand out for running QUBOs are quantum gate-based optimization algorithms (QAOA and VQE), quantum annealers and hardware-accelerated platforms (Coherent Ising Machines and Simulated Bifurcation Machines). -![D-Wave Washington 1000Q](../assets/figures/quantum-chip.png) +[![D-Wave Washington 1000Q](../assets/figures/quantum-chip.png)](https://commons.wikimedia.org/wiki/File:D-Wave-Washington-1000Q.jpg) ```@docs ToQUBO.isqubo From b27628fea56e3c914cb948e76a79220d49b1674d Mon Sep 17 00:00:00 2001 From: Pedro Maciel Xavier Date: Tue, 18 Apr 2023 01:17:14 -0300 Subject: [PATCH 31/38] Rewrite `QUBO` chapter --- docs/src/booklet/2-qubo.md | 19 ++++++++----------- 1 file changed, 8 insertions(+), 11 deletions(-) diff --git a/docs/src/booklet/2-qubo.md b/docs/src/booklet/2-qubo.md index 2f84fcb2..fe29d3e7 100644 --- a/docs/src/booklet/2-qubo.md +++ b/docs/src/booklet/2-qubo.md @@ -1,7 +1,7 @@ # QUBO ## Definition -**Q**uadratic **U**nconstrained **B**inary **O**ptimization, as its name suggests, refers to the global minimization or maximization of a given quadratic polynomial over binary variables whose domain is not constrained in any way. +**Q**uadratic **U**nconstrained **B**inary **O**ptimization, as its name suggests, refers to the global minimization or maximization of a given quadratic polynomial over binary variables whose domain is that often reliesany way. on the laws of physics to compute solution candidates. A common mathematical presentation is given by their quadratic matrix form, i.e. ```math @@ -15,14 +15,15 @@ where ``Q \in \mathbb{R}^{n \times n}`` is symmetric and ``\mathbb{B} = \lbrace{ ## OK, but why QUBO? Mathematically speaking, there is a notorious equivalence between QUBO and [Max-Cut](https://en.wikipedia.org/wiki/Maximum_cut) problems, e.g. for every QUBO instance there is an information preserving Max-Cut reformulation and vice versa. -This statement has two immediate implications: +This statement if followed by two immediate implications: 1. In the general case, solving QUBO globally is NP-Hard. 2. It is a simple yet expressive mathematical programming framework. Implication 1. tells us that such problems are computationally intractable and that heuristics and metaheuristics are to be employed instead of exact methods. No 2. relates to the fact that we are able to represent many other optimization models by means of the QUBO formalism. -Moreover, [Ising Model](https://en.wikipedia.org/wiki/Ising_model). +The [Ising Model](https://en.wikipedia.org/wiki/Ising_model), on the other hand, is a mathematical abstraction to describe statistical interactions within mechanical systems. +Its _Hamiltonian energy function_ leads to an optimization formulation in terms of the ``\pm 1`` _spin_ values of their states, written as ```math \begin{array}{rl} @@ -31,14 +32,10 @@ Moreover, [Ising Model](https://en.wikipedia.org/wiki/Ising_model). \end{array} ``` -With that said, the significant advances in computing systems and algorithms specialized for sampling QUBOs have contributed to their popularity. - -Some of the paradigms that stand out for running QUBOs are quantum gate-based optimization algorithms (QAOA and VQE), quantum annealers and hardware-accelerated platforms (Coherent Ising Machines and Simulated Bifurcation Machines). +where ``J \in \mathbb{R}^{n \times n}`` is strictly upper triangular and ``\mathbf{h} \in \mathbb{R}``. [![D-Wave Washington 1000Q](../assets/figures/quantum-chip.png)](https://commons.wikimedia.org/wiki/File:D-Wave-Washington-1000Q.jpg) -```@docs -ToQUBO.isqubo -ToQUBO.toqubo -ToQUBO.toqubo! -``` +The Ising reformulation alternative draws the bridge between QUBO problems and specialized devices that rely on the laws of physics to compute solution candidates. +Some of the paradigms that stand out in this context are quantum gate-based optimization algorithms (QAOA and VQE), quantum annealers and hardware-accelerated platforms (Coherent Ising Machines and Simulated Bifurcation Machines). +The significant advances in these computing systems contributed to the growing popularity of the model across the literature. From bfa79e1b4b1300634eaa2681a78ff5b7681a0b56 Mon Sep 17 00:00:00 2001 From: Pedro Maciel Xavier Date: Tue, 18 Apr 2023 02:25:50 -0300 Subject: [PATCH 32/38] Fix text --- docs/src/booklet/2-qubo.md | 21 +++++++++++++-------- 1 file changed, 13 insertions(+), 8 deletions(-) diff --git a/docs/src/booklet/2-qubo.md b/docs/src/booklet/2-qubo.md index fe29d3e7..2f75ef79 100644 --- a/docs/src/booklet/2-qubo.md +++ b/docs/src/booklet/2-qubo.md @@ -1,8 +1,8 @@ # QUBO ## Definition -**Q**uadratic **U**nconstrained **B**inary **O**ptimization, as its name suggests, refers to the global minimization or maximization of a given quadratic polynomial over binary variables whose domain is that often reliesany way. on the laws of physics to compute solution candidates. -A common mathematical presentation is given by their quadratic matrix form, i.e. +**Q**uadratic **U**nconstrained **B**inary **O**ptimization, as the name suggests, refers to the global minimization or maximization of a quadratic polynomial on binary variables. +A common presentation, the quadratic matrix form, is written as ```math \begin{array}{rl} @@ -12,18 +12,20 @@ A common mathematical presentation is given by their quadratic matrix form, i.e. ``` where ``Q \in \mathbb{R}^{n \times n}`` is symmetric and ``\mathbb{B} = \lbrace{0, 1}\rbrace``. +Note that, since ``x^{2} = x`` holds for ``x \in \mathbb{B}``, the linear terms of the objective function are stored in the main diagonal of ``Q``. ## OK, but why QUBO? Mathematically speaking, there is a notorious equivalence between QUBO and [Max-Cut](https://en.wikipedia.org/wiki/Maximum_cut) problems, e.g. for every QUBO instance there is an information preserving Max-Cut reformulation and vice versa. -This statement if followed by two immediate implications: +This statement is followed by two immediate implications: + 1. In the general case, solving QUBO globally is NP-Hard. 2. It is a simple yet expressive mathematical programming framework. Implication 1. tells us that such problems are computationally intractable and that heuristics and metaheuristics are to be employed instead of exact methods. No 2. relates to the fact that we are able to represent many other optimization models by means of the QUBO formalism. -The [Ising Model](https://en.wikipedia.org/wiki/Ising_model), on the other hand, is a mathematical abstraction to describe statistical interactions within mechanical systems. -Its _Hamiltonian energy function_ leads to an optimization formulation in terms of the ``\pm 1`` _spin_ values of their states, written as +The [Ising Model](https://en.wikipedia.org/wiki/Ising_model), on the other hand, is a mathematical abstraction to describe statistical interactions within mechanical systems with interesting properties for encoding combinatorial problems. +Its _Hamiltonian_ leads to an optimization formulation in terms of the _spin_ values of their states, given by ```math \begin{array}{rl} @@ -32,10 +34,13 @@ Its _Hamiltonian energy function_ leads to an optimization formulation in terms \end{array} ``` -where ``J \in \mathbb{R}^{n \times n}`` is strictly upper triangular and ``\mathbf{h} \in \mathbb{R}``. +with strictly upper triangular ``J \in \mathbb{R}^{n \times n}`` and ``\mathbf{h} \in \mathbb{R}``. [![D-Wave Washington 1000Q](../assets/figures/quantum-chip.png)](https://commons.wikimedia.org/wiki/File:D-Wave-Washington-1000Q.jpg) -The Ising reformulation alternative draws the bridge between QUBO problems and specialized devices that rely on the laws of physics to compute solution candidates. -Some of the paradigms that stand out in this context are quantum gate-based optimization algorithms (QAOA and VQE), quantum annealers and hardware-accelerated platforms (Coherent Ising Machines and Simulated Bifurcation Machines). +The Ising reformulation alternative draws the bridge between QUBO problems and hardware solvers which aim to find the absolute or approximate ground states of the Ising model[^Mohseni2022]. +Some of the paradigms that stand out in this context are quantum gate-based optimization algorithms (QAOA and VQE), quantum annealers, hardware-accelerated platforms (Coherent Ising Machines and Simulated Bifurcation Machines) and physics-inspired methods (Simulated Annealing, Parallel Tempering). The significant advances in these computing systems contributed to the growing popularity of the model across the literature. + +[^Mohseni2022]: Mohseni, N., McMahon, P. L. & Byrnes, T. **Ising machines as hardware solvers of combinatorial optimization problems**. _Nat Rev Phys 4_, 363–379 (2022). [{arXiv}](https://arxiv.org/pdf/2204.00276.pdf) + \ No newline at end of file From bc0ac56653ca46a0c64843effa305c62948327a6 Mon Sep 17 00:00:00 2001 From: Pedro Maciel Xavier Date: Tue, 18 Apr 2023 10:26:27 -0300 Subject: [PATCH 33/38] Rephrase --- docs/src/booklet/2-qubo.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/docs/src/booklet/2-qubo.md b/docs/src/booklet/2-qubo.md index 2f75ef79..6bec1f28 100644 --- a/docs/src/booklet/2-qubo.md +++ b/docs/src/booklet/2-qubo.md @@ -38,7 +38,7 @@ with strictly upper triangular ``J \in \mathbb{R}^{n \times n}`` and ``\mathbf{h [![D-Wave Washington 1000Q](../assets/figures/quantum-chip.png)](https://commons.wikimedia.org/wiki/File:D-Wave-Washington-1000Q.jpg) -The Ising reformulation alternative draws the bridge between QUBO problems and hardware solvers which aim to find the absolute or approximate ground states of the Ising model[^Mohseni2022]. +The Ising reformulation alternative draws the bridge between QUBO problems and devices designed to sample global or approximate ground states of the Ising Hamiltonian with high probability[^Mohseni2022]. Some of the paradigms that stand out in this context are quantum gate-based optimization algorithms (QAOA and VQE), quantum annealers, hardware-accelerated platforms (Coherent Ising Machines and Simulated Bifurcation Machines) and physics-inspired methods (Simulated Annealing, Parallel Tempering). The significant advances in these computing systems contributed to the growing popularity of the model across the literature. From 1123961eadbe36dd09520ed1c780c4035536f7fd Mon Sep 17 00:00:00 2001 From: pedroripper Date: Wed, 19 Apr 2023 13:09:14 -0300 Subject: [PATCH 34/38] Update docs --- docs/src/booklet/3-pbo.md | 6 ++++++ src/model/virtual.jl | 6 ++++++ 2 files changed, 12 insertions(+) diff --git a/docs/src/booklet/3-pbo.md b/docs/src/booklet/3-pbo.md index 77617401..aade0726 100644 --- a/docs/src/booklet/3-pbo.md +++ b/docs/src/booklet/3-pbo.md @@ -30,6 +30,12 @@ ToQUBO.PBO.quadratize! Currently, `ToQUBO` has two reduction algorithms, one for negative and another for positive terms. +For these quadratization techniques, we have introduced a flag `StableQuadratization()`. + +```@docs +ToQUBO.Attributes.StableQuadratization +``` + ```@docs ToQUBO.PBO.NTR_KZFD ToQUBO.PBO.PTR_BG diff --git a/src/model/virtual.jl b/src/model/virtual.jl index f0c384d1..1cd8fc26 100644 --- a/src/model/virtual.jl +++ b/src/model/virtual.jl @@ -340,7 +340,13 @@ end @doc raw""" Arithmetic() +Let ``x \in [a, b] \subset \mathbb{Z}, n = b - a, \mathbf{y} \in \mathbb{B}^{n}``. + +```math +x = \xi(\mathbf{y}) = a + \sum_{i = 1}^{N - 1} i\,y_{i} + \left({ n - \frac{N\,(N - 1)}{2}}\right)\,y_{N} +``` +Where ``N = \left\lceil{ \frac{1}{2} {\sqrt{1 + 8 n}} - \frac{1}{2} }\right\rceil`` is the number of bits. """ struct Arithmetic <: LinearEncoding end function encode!( From f3eb784c04897249d7c175219f1b36eb0557e19b Mon Sep 17 00:00:00 2001 From: pedroripper Date: Wed, 19 Apr 2023 15:54:38 -0300 Subject: [PATCH 35/38] Update docs --- docs/src/booklet/3-pbo.md | 15 +++++++++------ src/lib/pbo/quadratization.jl | 13 +++++++++---- src/model/virtual.jl | 4 ++-- 3 files changed, 20 insertions(+), 12 deletions(-) diff --git a/docs/src/booklet/3-pbo.md b/docs/src/booklet/3-pbo.md index aade0726..d7376962 100644 --- a/docs/src/booklet/3-pbo.md +++ b/docs/src/booklet/3-pbo.md @@ -30,17 +30,20 @@ ToQUBO.PBO.quadratize! Currently, `ToQUBO` has two reduction algorithms, one for negative and another for positive terms. -For these quadratization techniques, we have introduced a flag `StableQuadratization()`. - -```@docs -ToQUBO.Attributes.StableQuadratization -``` - ```@docs ToQUBO.PBO.NTR_KZFD ToQUBO.PBO.PTR_BG ``` +### Stable Quadratization + +The quadratization of a PBF does not guarantee that the resulting function will always be the same, as the order of terms can be different each time. This can be an issue in some situations where a deterministic output is required. + +With said that, we have introduced the concept of Stable Quadratization, where the terms of the PBF are sorted, guaranteeing that the resulting PBF will be the same every time. +We have defined it as an attribute of the compiler, with the [`ToQUBO.Attributes.StableQuadratization`](@ref) flag. + + + ### A Primer on Submodularity A set function ``f : 2^{S} \to \mathbb{R}`` is said to be submodular if diff --git a/src/lib/pbo/quadratization.jl b/src/lib/pbo/quadratization.jl index f5d99bdb..ceae52bd 100644 --- a/src/lib/pbo/quadratization.jl +++ b/src/lib/pbo/quadratization.jl @@ -25,11 +25,14 @@ aux(::Integer)::Vector{S} Negative Term Reduction NTR-KZFD (Kolmogorov & Zabih, 2004; Freedman & Drineas, 2005) +Let ``f(\mathbf{x}) = x_{1} x_{2} \dots x_{k}``. + ```math -f(x_{1}, \dots, x_{k}) = - x_{1} \cdots x_{k} \to -\mathcal{Q}\left\lbrace{f}\right\rbrace(x_{1}, \dots, x_{k}; z) = (k - 1) z - \sum_{i = 1}^{k} x_{i} z +\mathcal{Q}\left\lbrace{f}\right\rbrace(\mathbf{x}; z) = (k - 1) z - \sum_{i = 1}^{k} x_{i} z ``` +where ``\mathbf{x} \in \mathbb{B}^k`` + !!! info Introduces a new variable ``z`` and no non-submodular terms. """ struct NTR_KZFD <: QuadratizationMethod end @@ -70,12 +73,14 @@ end Positive Term Reduction PTR-BG (Boros & Gruber, 2014) +Let ``f(\mathbf{x}) = x_{1} x_{2} \dots x_{k}``. + ```math -f(x_{1}, \dots, x_{k}) = x_{1} x_{2} \dots x_{k} \to -\mathcal{Q}\left\lbrace{f}\right\rbrace(x_{1}, \dots, x_{k}; z_{1}, \dots, z_{k-2} ) = \left[{ +\mathcal{Q}\left\lbrace{f}\right\rbrace(\mathbf{x}; \mathbf{z}) = \left[{ \sum_{i = 1}^{k-2} z_{i} \left({ k - i - 1 + x_{i} + \sum_{j = i+1}^{k} x_{j} }\right) }\right] + x_{k-1} x_{k} ``` +where ``\mathbf{x} \in \mathbb{B}^k`` and ``\mathbf{z} \in \mathbb{B}^{k-2}`` !!! info Introduces ``k - 2`` new variables ``z_{1}, \dots, z_{k-2}`` and ``k - 1`` non-submodular terms. diff --git a/src/model/virtual.jl b/src/model/virtual.jl index 1cd8fc26..c6aa1541 100644 --- a/src/model/virtual.jl +++ b/src/model/virtual.jl @@ -484,10 +484,10 @@ end A *sequential encoding* is one of the form ```math -\xi(\mathbf{y}) = \sum_{i = 1}^{n} \gamma_{i} \left({y_{i + 1} \circast y_{i}}\right) +\xi(\mathbf{y}) = \sum_{i = 1}^{n} \gamma_{i} \left({y_{i + 1} \ast y_{i}}\right) ``` -where ``\mathbf{y} \in \mathbb{B}^{n + 1}`` and ``\circast`` is a binary operator. +where ``\mathbf{y} \in \mathbb{B}^{n + 1}`` and ``\ast`` is a binary operator. """ abstract type SequentialEncoding <: Encoding end function encode!( From 29408966edf2309882be720d5673c19cb4bc220c Mon Sep 17 00:00:00 2001 From: Pedro Maciel Xavier Date: Wed, 19 Apr 2023 20:41:21 -0300 Subject: [PATCH 36/38] Add encoding expansions --- docs/src/booklet/4-encoding.md | 24 ++++++++++-------- src/model/virtual.jl | 46 ++++++++++++++++++++++++++++------ 2 files changed, 52 insertions(+), 18 deletions(-) diff --git a/docs/src/booklet/4-encoding.md b/docs/src/booklet/4-encoding.md index 62accdf4..7b08fc7c 100644 --- a/docs/src/booklet/4-encoding.md +++ b/docs/src/booklet/4-encoding.md @@ -2,18 +2,21 @@ ## Variable Encoding -As you should already know, QUBO models are comprised only of binary variables. So when we are reformulating general optimization problems, one important step is to encode variables into binary ones. +As you should already know, QUBO models are comprised only of binary variables. +So when we are reformulating general optimization problems, one important step is to encode variables into binary ones. -`ToQUBO` currently implements 6 encoding techniques. Each method introduces a different number of variables, quadratic terms and linear terms. Also, they differ in the range of their coefficients($\Delta$). +`ToQUBO` currently implements 6 encoding techniques. +Each method introduces a different number of variables, quadratic terms and linear terms. +Also, they differ in the magnitude of their coefficients ``\Delta``. -| Encoding | Binary Variables | # Linear terms | # Quadratic terms | $\Delta$ | -|:---------------------:|:------------------:|:----------------:|:-------------------:|:----------:| -| Binary | $O(log \ n)$ | $O(log \ n)$ | - | $O(n)$ | -| Unary | $O(n)$ | $O(n)$ | - | $O(1)$ | -| One-Hot | $O(n)$ | $O(n)$ | $O(n^2)$ | $O(n)$ | -| Domain-Wall | $O(n)$ | $O(n)$ | $O(n)$ | $O(n)$ | -| Bounded-Coefficient | $O(n)$ | $O(n)$ | - | $O(1)$ | -| Arithmetic Prog | $O(\sqrt{n})$ | $O(\sqrt{n})$ | - | $O(\sqrt{n})$ | +| Encoding | Binary Variables | # Linear terms | # Quadratic terms | ``\Delta`` | +|:---------------------:|:------------------:|:----------------:|:-------------------:|:----------------:| +| Binary | ``O(log \ n)`` | ``O(log \ n)`` | - | ``O(n)`` | +| Unary | ``O(n)`` | ``O(n)`` | - | ``O(1)`` | +| One-Hot | ``O(n)`` | ``O(n)`` | ``O(n^2)`` | ``O(n)`` | +| Domain-Wall | ``O(n)`` | ``O(n)`` | ``O(n)`` | ``O(n)`` | +| Bounded-Coefficient | ``O(n)`` | ``O(n)`` | - | ``O(1)`` | +| Arithmetic Prog | ``O(\sqrt{n})`` | ``O(\sqrt{n})`` | - | ``O(\sqrt{n})`` | ### Linear Encoding Methods @@ -23,7 +26,6 @@ ToQUBO.Mirror ```@docs ToQUBO.LinearEncoding -ToQUBO.Linear ToQUBO.Binary ToQUBO.Unary ToQUBO.Arithmetic diff --git a/src/model/virtual.jl b/src/model/virtual.jl index c6aa1541..d9726205 100644 --- a/src/model/virtual.jl +++ b/src/model/virtual.jl @@ -216,12 +216,31 @@ end @doc raw""" Unary() +## Integer Let ``x \in [a, b] \subset \mathbb{Z}, n = b - a, \mathbf{y} \in \mathbb{B}^{n}``. ```math -x = \xi(\mathbf{y}) = a + \sum_{j = 1}^{b - a} y_{j} +\xi{[a, b]}(\mathbf{y}) = a + \sum_{j = 1}^{b - a} y_{j} ``` -""" struct Unary <: LinearEncoding end + +## Real +Given ``n \in \mathbb{N}`` for ``x \in [a, b] \subset \mathbb{R}``, + +```math +\xi{[a, b]}(\mathbf{y}) = a + \frac{b - a}{n} \sum_{j = 1}^{n} y_{j} +``` + +### Encoding error +Given ``\tau > 0``, for the expected encoding error to be less than or equal to ``\tau``, at least + +```math +n \ge 1 + \frac{b - a}{4 \tau} +``` + +binary variables become necessary. + +""" +struct Unary <: LinearEncoding end function encode!( model::VirtualModel{T}, @@ -273,15 +292,28 @@ end @doc raw""" Binary() -Binary Expansion within the closed interval ``[\alpha, \beta]``. +## Integer +Let ``x \in [a, b] \subset \mathbb{Z}``, ``n = \left\lceil \log_{2}(b - a) + 1 \right\rceil``, ``\mathbf{y} \in \mathbb{B}^{n}``. -For a given variable ``x \in [\alpha, \beta]`` we approximate it by +```math +\xi{[a, b]}(\mathbf{y}) = a + \left(b - a - 2^{n - 1} + 1\right) y_{n} + \sum_{j = 1}^{n - 1} 2^{j - 1} y_{j} +``` -```math -x \approx \alpha + \frac{(\beta - \alpha)}{2^{n} - 1} \sum_{i=0}^{n-1} {2^{i}\, y_i} +## Real +Given ``n \in \mathbb{N}`` for ``x \in [a, b] \subset \mathbb{R}``, + +```math +\xi{[a, b]}(\mathbf{y}) = a + \frac{b - a}{2^{n} - 1} \sum_{j = 1}^{n} 2^{j - 1} y_{j} +``` + +### Encoding error +Given ``\tau > 0``, for the expected encoding error to be less than or equal to ``\tau``, at least + +```math +n \ge \log_{2} \left[1 + \frac{b - a}{4 \tau}\right] ``` -where ``n`` is the number of bits and ``y_i \in \mathbb{B}``. +binary variables become necessary. """ struct Binary <: LinearEncoding end function encode!( From cad8e12736b55f203b54d8e1e8e534d8db76945b Mon Sep 17 00:00:00 2001 From: Pedro Maciel Xavier Date: Wed, 19 Apr 2023 21:09:41 -0300 Subject: [PATCH 37/38] Update text --- docs/src/booklet/4-encoding.md | 29 ++++++++++++++--------------- 1 file changed, 14 insertions(+), 15 deletions(-) diff --git a/docs/src/booklet/4-encoding.md b/docs/src/booklet/4-encoding.md index 7b08fc7c..0bdf08c3 100644 --- a/docs/src/booklet/4-encoding.md +++ b/docs/src/booklet/4-encoding.md @@ -1,8 +1,8 @@ # Encoding Methods -## Variable Encoding +## Variables -As you should already know, QUBO models are comprised only of binary variables. +As you may already know, QUBO models are comprised only of binary variables. So when we are reformulating general optimization problems, one important step is to encode variables into binary ones. `ToQUBO` currently implements 6 encoding techniques. @@ -11,7 +11,7 @@ Also, they differ in the magnitude of their coefficients ``\Delta``. | Encoding | Binary Variables | # Linear terms | # Quadratic terms | ``\Delta`` | |:---------------------:|:------------------:|:----------------:|:-------------------:|:----------------:| -| Binary | ``O(log \ n)`` | ``O(log \ n)`` | - | ``O(n)`` | +| Binary | ``O(\log n)`` | ``O(\log n)`` | - | ``O(n)`` | | Unary | ``O(n)`` | ``O(n)`` | - | ``O(1)`` | | One-Hot | ``O(n)`` | ``O(n)`` | ``O(n^2)`` | ``O(n)`` | | Domain-Wall | ``O(n)`` | ``O(n)`` | ``O(n)`` | ``O(n)`` | @@ -19,22 +19,20 @@ Also, they differ in the magnitude of their coefficients ``\Delta``. | Arithmetic Prog | ``O(\sqrt{n})`` | ``O(\sqrt{n})`` | - | ``O(\sqrt{n})`` | -### Linear Encoding Methods +### Linear Encoding ```@docs -ToQUBO.Mirror -``` - -```@docs -ToQUBO.LinearEncoding ToQUBO.Binary ToQUBO.Unary ToQUBO.Arithmetic ToQUBO.OneHot ``` -### Sequential Encoding Methods ```@docs -ToQUBO.SequentialEncoding +ToQUBO.Mirror +``` + +### Sequential Encoding +```@docs ToQUBO.DomainWall ``` @@ -43,13 +41,14 @@ ToQUBO.DomainWall ToQUBO.Bounded ``` -## Constraint Encoding +## Constraints -A QUBO model is unconstrained. So when `ToQUBO` is reformulating a problem, it needs to encode all constraints into the objective function loosing as little information as possible. +A QUBO model is unconstrained. So when `ToQUBO` is reformulating a problem, it needs to encode all constraints into the objective function losing as little information as possible. -As constraints are introduced into the objective function, we need to make sure that they won't be violated. In order to do that, `ToQUBO` multiplies the encoded constraint by a large penalty $\rho$, so that any violation would result in an infeasible solution to the problem. +As constraints are introduced into the objective function, we need to make sure that they won't be violated. +In order to do that, `ToQUBO` multiplies the encoded constraint by a large penalty ``\rho``, so that any violation would result in a sub-optimal solution to the problem. -Sometimes, moving a constraint to the objective fuction might introduce higher-order terms (degree > 2). If that is the case, `ToQUBO` needs to reduce it back to a quadratic function. +Sometimes, the encoding process might introduce higher-order terms, demanding `ToQUBO` to reduce the offending polynomials back to a quadratic form. As of today, `ToQUBO` provides encoding for the following constraints: From 65b044130f15982971e614bcead79f820e839198 Mon Sep 17 00:00:00 2001 From: Pedro Maciel Xavier Date: Thu, 20 Apr 2023 14:04:23 -0300 Subject: [PATCH 38/38] Add explanation on encoding error --- docs/make.jl | 10 ++- docs/src/booklet/4-encoding.md | 29 ++++++++- src/compiler/constraints.jl | 11 ---- src/model/virtual.jl | 113 +++++++++++++++++++++------------ 4 files changed, 108 insertions(+), 55 deletions(-) diff --git a/docs/make.jl b/docs/make.jl index 4a289630..101429f1 100644 --- a/docs/make.jl +++ b/docs/make.jl @@ -11,14 +11,20 @@ makedocs(; doctest = true, clean = true, format = Documenter.HTML( + sidebar_sitename = false, + mathengine = Documenter.KaTeX( + Dict( + :macros => Dict( + raw"\set" => raw"\left\lbrace{#1}\right\rbrace" + ) + ) + ), assets = [ "assets/extra_styles.css", "assets/favicon.ico", asset("https://tikzjax.com/v1/fonts.css"; class = :css), asset("https://tikzjax.com/v1/tikzjax.js"; class = :js), ], - mathengine = Documenter.KaTeX(), - sidebar_sitename = false, ), sitename = "ToQUBO.jl", authors = "Pedro Maciel Xavier and Pedro Ripper and Tiago Andrade and Joaquim Dias Garcia and David E. Bernal Neira", diff --git a/docs/src/booklet/4-encoding.md b/docs/src/booklet/4-encoding.md index 0bdf08c3..b46d64a8 100644 --- a/docs/src/booklet/4-encoding.md +++ b/docs/src/booklet/4-encoding.md @@ -18,11 +18,10 @@ Also, they differ in the magnitude of their coefficients ``\Delta``. | Bounded-Coefficient | ``O(n)`` | ``O(n)`` | - | ``O(1)`` | | Arithmetic Prog | ``O(\sqrt{n})`` | ``O(\sqrt{n})`` | - | ``O(\sqrt{n})`` | - ### Linear Encoding ```@docs -ToQUBO.Binary ToQUBO.Unary +ToQUBO.Binary ToQUBO.Arithmetic ToQUBO.OneHot ``` @@ -41,6 +40,32 @@ ToQUBO.DomainWall ToQUBO.Bounded ``` +### Encoding Error +Let ``\set{x_{i}}_{i \in [k]}`` be the collection of ``k`` evenly spaced samples from the discretization of an interval ``[a, b] \subseteq \mathbb{R}``. + +The representation error for a given point ``x`` with respect to ``\set{x_{i}}_{i \in [k]}`` is + +```math +e_{k}(x) = \min_{i \in [k]} \left|x - x_{i}\right| +``` + +Assuming that ``x`` behaves as a uniformly distributed random variable, the expected absolute encoding error is + +```math +\begin{align*} +\mathbb{E}\left[{e_{k}(x)}\right] &= \frac{1}{b - a} \int_{a}^{b} e_{k}(x) ~\mathrm{d}x \\ + &= \frac{1}{4} \frac{b - a}{k - 1} +\end{align*} +``` + +Thus, for encoding methods that rely on the regular division of an interval, it is possible to define the number of samples ``k`` necessary to limit the expected error according to an upper bound ``\tau``, that is, + +```math +\mathbb{E}\left[{e_{k}(x)}\right] \le \tau \implies k \ge 1 + \frac{b - a}{4 \tau} +``` + +This allows the compiler to automatically infer the number of bits to allocate for an encoded variable given the tolerance factor. + ## Constraints A QUBO model is unconstrained. So when `ToQUBO` is reformulating a problem, it needs to encode all constraints into the objective function losing as little information as possible. diff --git a/src/compiler/constraints.jl b/src/compiler/constraints.jl index 16cc4ae5..92bf3540 100644 --- a/src/compiler/constraints.jl +++ b/src/compiler/constraints.jl @@ -265,17 +265,6 @@ end ::MOI.SOS1{T}, ::AbstractArchitecture, ) where {T} - -Turns constraints of the form - -```math -\begin{array}{rl} -\text{s.t} & \sum{\mathbf{x}} \leq \min \mathbf{x} -\end{array} -``` - -into - """ function toqubo_constraint( model::VirtualModel{T}, diff --git a/src/model/virtual.jl b/src/model/virtual.jl index d9726205..beaf999e 100644 --- a/src/model/virtual.jl +++ b/src/model/virtual.jl @@ -9,7 +9,8 @@ Maps newly created virtual variable `v` within the virtual model structure. It f 1. Maps `v`'s source to it in the model's `source` mapping. 2. For every one of `v`'s targets, maps it to itself and adds a binary constraint to it. 2. Adds `v` to the end of the model's `varvec`. -""" function encode! end +""" +function encode! end @doc raw""" # Variable Expansion methods: @@ -157,7 +158,8 @@ Every linear encoding ``\xi`` is of the form \xi(\mathbf{y}) = \alpha + \sum_{i = 1}^{n} \gamma_{i} y_{i} ``` -""" abstract type LinearEncoding <: Encoding end +""" +abstract type LinearEncoding <: Encoding end function VirtualVariable{T}( e::LinearEncoding, @@ -191,7 +193,8 @@ end Mirror() Mirrors binary variable ``x \in \mathbb{B}`` with a twin variable ``y \in \mathbb{B}``. -""" struct Mirror <: LinearEncoding end +""" +struct Mirror <: LinearEncoding end function encode!(model::VirtualModel{T}, e::Mirror, x::Union{VI,Nothing}) where {T} return encode!(model, e, x, ones(T, 1)) @@ -199,7 +202,8 @@ end @doc raw""" Linear() -""" struct Linear <: LinearEncoding end +""" +struct Linear <: LinearEncoding end function encode!( model::VirtualModel{T}, @@ -217,7 +221,7 @@ end Unary() ## Integer -Let ``x \in [a, b] \subset \mathbb{Z}, n = b - a, \mathbf{y} \in \mathbb{B}^{n}``. +Let ``x \in [a, b] \subset \mathbb{Z}``, ``n = b - a`` and ``\mathbf{y} \in \mathbb{B}^{n}``. ```math \xi{[a, b]}(\mathbf{y}) = a + \sum_{j = 1}^{b - a} y_{j} @@ -293,7 +297,7 @@ end Binary() ## Integer -Let ``x \in [a, b] \subset \mathbb{Z}``, ``n = \left\lceil \log_{2}(b - a) + 1 \right\rceil``, ``\mathbf{y} \in \mathbb{B}^{n}``. +Let ``x \in [a, b] \subset \mathbb{Z}``, ``n = \left\lceil \log_{2}(b - a) + 1 \right\rceil`` and ``\mathbf{y} \in \mathbb{B}^{n}``. ```math \xi{[a, b]}(\mathbf{y}) = a + \left(b - a - 2^{n - 1} + 1\right) y_{n} + \sum_{j = 1}^{n - 1} 2^{j - 1} y_{j} @@ -314,7 +318,8 @@ n \ge \log_{2} \left[1 + \frac{b - a}{4 \tau}\right] ``` binary variables become necessary. -""" struct Binary <: LinearEncoding end +""" +struct Binary <: LinearEncoding end function encode!( model::VirtualModel{T}, @@ -372,14 +377,29 @@ end @doc raw""" Arithmetic() -Let ``x \in [a, b] \subset \mathbb{Z}, n = b - a, \mathbf{y} \in \mathbb{B}^{n}``. +## Integer +Let ``x \in [a, b] \subset \mathbb{Z}``, ``n = \left\lceil{ \frac{1}{2} {\sqrt{1 + 8 (b - a)}} - \frac{1}{2} }\right\rceil`` and ``\mathbf{y} \in \mathbb{B}^{n}``. + +```math +\xi{[a, b]}(\mathbf{y}) = a + \left( {b - a - \frac{n (n - 1)}{2}} \right) y_{n} + \sum_{j = 1}^{n - 1} j y_{j} +``` + +## Real +Given ``n \in \mathbb{N}`` for ``x \in [a, b] \subset \mathbb{R}``, + +```math +\xi{[a, b]}(\mathbf{y}) = a + \frac{b - a}{n (n + 1)} \sum_{j = 1}^{n} j y_{j} +``` + +### Encoding error +Given ``\tau > 0``, for the expected encoding error to be less than or equal to ``\tau``, at least ```math -x = \xi(\mathbf{y}) = a + \sum_{i = 1}^{N - 1} i\,y_{i} + \left({ n - \frac{N\,(N - 1)}{2}}\right)\,y_{N} +n \ge \frac{1}{2} \left[ 1 + \sqrt{3 + \frac{(b - a)}{2 \tau})} \right] ``` -Where ``N = \left\lceil{ \frac{1}{2} {\sqrt{1 + 8 n}} - \frac{1}{2} }\right\rceil`` is the number of bits. -""" struct Arithmetic <: LinearEncoding end +""" +struct Arithmetic <: LinearEncoding end function encode!( model::VirtualModel{T}, @@ -433,21 +453,24 @@ end @doc raw""" OneHot() -The one-hot encoding is a linear technique used to represent a variable -``x \in \{ \gamma_{j} \}_{j \in [n]}``. +The one-hot encoding is a linear technique used to represent a variable ``x \in \set{\gamma_{j}}_{j \in [n]}``. + +The associated encoding function is combined with a constraint assuring that only one and exactly one of the expansion's variables ``y_{j}`` is activated at a time. + +```math +\xi[\set{\gamma_{j}}_{j \in [n]}](\mathbf{y}) = \sum_{j = 1}^{n} \gamma_{j} y_{j} ~\textrm{s.t.}~ \sum_{j = 1}^{n} y_{j} = 1 +``` -The encoding function is combined with a constraint assuring that only -one and exactly one of the expansion's variables ``y_{j}`` is activated -at a time. +When a variable is encoded following this approach, a penalty term of the form ```math -\begin{array}{rl} -x = \xi(\mathbf{y}) = & \sum_{j = 1}^{n} \gamma_{j} y_{j} \\ - \mathrm{s.t.} & \sum_{j = 1}^{n} y_{j} = 1 -\end{array} +\rho \left[ \sum_{j = 1}^{n} y_{j} - 1 \right]^{2} ``` -""" struct OneHot <: LinearEncoding end +is added to the objective function. + +""" +struct OneHot <: LinearEncoding end function VirtualVariable{T}( e::OneHot, @@ -516,11 +539,12 @@ end A *sequential encoding* is one of the form ```math -\xi(\mathbf{y}) = \sum_{i = 1}^{n} \gamma_{i} \left({y_{i + 1} \ast y_{i}}\right) +\xi[\set{\gamma_{j}}_{j \in [n]}](\mathbf{y}) = \sum_{j = 1}^{n} \gamma_{j} \left({y_{j + 1} \ast y_{j}}\right) ``` where ``\mathbf{y} \in \mathbb{B}^{n + 1}`` and ``\ast`` is a binary operator. -""" abstract type SequentialEncoding <: Encoding end +""" +abstract type SequentialEncoding <: Encoding end function encode!( model::VirtualModel{T}, @@ -539,17 +563,18 @@ end @doc raw""" DomainWall() -The Domain Wall[^Chancellor2019] encoding method is a sequential approach that requires only -``n - 1`` bits to represent ``n`` distinct values. +The Domain Wall[^Chancellor2019] encoding method is a sequential approach that requires ``n - 1`` bits to represent ``n`` distinct values. -!!! table "Encoding Analysis" - | | bits | linear | quadratic | ``\Delta`` | - | :-: | :--: | :----: | :-------: | :--------: | - | Domain Wall | ``n - 1`` | ``n`` | | ``O(n)`` | +```math +\xi{[\set{\gamma_{j}}_{j \in [n]}]}(\mathbf{y}) = \sum_{j = 1}^{n} \gamma_{j} (y_{j} - y_{j + 1}) ~\textrm{s.t.}~ \sum_{j = 1}^{n} y_{j} \oplus y_{j + 1} = 1, y_{1} = 1, y_{n + 1} = 0 +``` + +where ``\mathbf{y} \in \mathbb{B}^{n + 1}``. [^Chancellor2019]: Nicholas Chancellor, **Domain wall encoding of discrete variables for quantum annealing and QAOA**, *Quantum Science Technology 4*, 2019. -""" struct DomainWall <: SequentialEncoding end +""" +struct DomainWall <: SequentialEncoding end function VirtualVariable{T}( e::DomainWall, @@ -603,16 +628,20 @@ end @doc raw""" Bounded{E,T}(μ::T) where {E<:Encoding,T} -The bounded-coefficient encoding method[^Karimi2019] consists in limiting the magnitude of the -coefficients in the encoding expansion to a parameter ``\mu``. +The bounded-coefficient encoding method[^Karimi2019] consists in limiting the magnitude of the coefficients in the encoding expansion to a parameter ``\mu``. [^Karimi2019]: Karimi, S. & Ronagh, P. **Practical integer-to-binary mapping for quantum annealers**. *Quantum Inf Process 18, 94* (2019). [{doi}](https://doi.org/10.1007/s11128-019-2213-x) +This can be applied to the [`Unary`](@ref), [`Binary`](@ref) and [`Arithmetic`](@ref) encoding schemas, as discussed below. + + Bounded{Unary,T}(μ::T) where {T} + +Given ``\mu > 0``, let ``x \in [a, b] \subset \mathbb{Z}`` and ``n = b - a``. + Bounded{Binary,T}(μ::T) where {T} -## Rationale -Let ``x \in [a, b] \subset \mathbb{Z}`` and ``n = b - a``. +Given ``\mu > 0``, let ``x \in [a, b] \subset \mathbb{Z}`` and ``n = b - a``. First, @@ -641,6 +670,14 @@ and \epsilon = n - 2^{k} + 1 - r \times \mu ``` +Therefore, + +```math +\xi_{\mu}{[a, b]}(\mathbf{y}) = \sum_{j = 1} \gamma_{j} y_{j} +``` + +where + ```math \gamma_{j} = \left\lbrace\begin{array}{cl} 2^{j} & \text{if } 1 \le j \le k \\ @@ -649,12 +686,8 @@ and \end{array}\right. ``` - Bounded{Unary,T}(μ::T) where {T} - -Let ``x \in [a, b] \subset \mathbb{Z}`` and ``n = b - a``. - - -""" struct Bounded{E<:LinearEncoding,T} <: LinearEncoding +""" +struct Bounded{E<:LinearEncoding,T} <: LinearEncoding μ::T function Bounded{E,T}(μ::T) where {E,T}

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