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src/content/blog/2024_lre.md

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+---
+title: "Adding Layerwise Richardson Extrapolation in Mitiq"
+author: 
+day: 28
+month: 11
+year: 2024
+tags:
+    - Mitiq
+    - 
+---
+
+Layerwise Richardson Extrapolation (LRE) [^1] is a multivariate extension of Zero Noise Extrapolation (ZNE)
+where the noise of each layer is treated as an independent variable.
+
+![](/images/lre_workflow_steps.png)
+
+The general workflow of LRE comprises of creating multiple layerwise noise scaled circuits that are executed on a backend and then
+extrapolated independently. ZNE allows a choice of which layer to scale based on the chosen scale factor while the
+noise scaled circuits in LRE are created in a specific pattern dependent on factors like the number of layers in the input circuit,
+degree of the multivariate extrapolating polynomial and the fold multiplier as the scale factor to control how many layers are inserted
+via unitary folding.
+
+The mitiq module `mitiq.lre` can be used in two ways.
+
+- [`execute_with_lre`](https://mitiq.readthedocs.io/en/stable/apidoc.html#mitiq.lre.lre.execute_with_lre) can be used directly to get the error-mitigated expectation value.
+
+```py
+
+# Get the error mitigated expectation value directly
+mitigated = execute_with_lre(
+    circuit,
+    execute,
+    degree=degree,
+    fold_multiplier=fold_multiplier,
+)
+
+```
+
+- To allow more control over the process, use [`multivariate_layer_scaling`](https://mitiq.readthedocs.io/en/stable/apidoc.html#mitiq.lre.multivariate_scaling.layerwise_folding.multivariate_layer_scaling) to create the intermediary noise scaled circuits and [`multivariate_richardson_coefficients`](https://mitiq.readthedocs.io/en/stable/apidoc.html#mitiq.lre.inference.multivariate_richardson.multivariate_richardson_coefficients) to get the coefficients of linear combination used by multivariate extrapolation. The latter are combined with the noise scaled expectation values to estimate the error-mitigated value through LRE.
+
+```py
+# Step by step application
+# noise scaled circuits
+noise_scaled_circuits = multivariate_layer_scaling(circuit, degree, fold_multiplier)
+
+# noise scaled expectation values
+noise_scaled_exp_values = [execute(circuit) for circuit in noise_scaled_circuits]
+
+# combine the results
+mitigated = sum(
+    exp_val * coeff
+    for exp_val, coeff in zip(noise_scaled_exp_values, coefficients)
+)
+
+```
+
+Additional information is available in the [LRE User Guide](https://mitiq.readthedocs.io/en/stable/guide/lre.html) in Mitiq's documentation.
+
+------------------------------------------------------
+[^1]: Russo, Vincent and Mari, Andrea, _Quantum error mitigation by layerwise Richardson extrapolation_ (2024) [arXiv:2402.04000](https://arxiv.org/abs/2402.04000).