Add block encoding with matrix access oracles demo

[soranjh]

Title:
Block encoding with matrix access oracles

Summary:
This demo explains a general block encoding method that can be very efficient for sparse and structured matrices. It complements block encoding with linear combination of unitaries.

Relevant references:

Possible Drawbacks:

Related GitHub Issues:

[github-actions]

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[KetpuntoG]

From what we have discussed in person, the confusion may come since you don't see the clear separation between the two algorithms. It may seem that the spare matrix case is a particular case of Fable but they work differently. For that reason, I would clarify it in the introduction. After showing the structure of this type of encodings, say:

"In this demo we are going to work with two particular algorithms based on oracles: the FABLE method and another method that focuses on sparse matrices."

So here you already start talking about FABLE, you talk about UA with the formulas and say that we will always take the same size |i> and |j>, representing the row and column indexes (I think). Remark that in this algorithm you always take UB as a SWAP gate whose goal is to order the indices in the right way.

Then you get to the sparse matrix block. Here you start by saying that this is a different method that seeks to optimize the algorithm for situations where the matrix is sparse. You also say something like "Unlike the FABLE algorithm, in this case the |j> register will not store the column "j" but will store b(i,j). b(i, j) means [...]. In this way we can greatly improve the efficiency since the number of qubits will not grow with the size of the matrix but with the number of non-empty entries. With this in mind, note that we will find a small difference in the construction of UA, we will simply have to adapt the indexes to the new definition:

[formula]

On the other hand UB is in charge of returning precisely that b(i,j) value we were commenting:

[formula]

[KetpuntoG]

Just some typos :)

solved :)

[KetpuntoG]

Great Job @Jaybsoni ! It was not an easy topic but now it's clear 🚀
You can have the honors of clicking on the merge button 😉