Remaining pages of LRE user guide

docs/source/guide/guide.md

@@ -10,6 +10,7 @@ pec.md
 cdr.md
 shadows.md
 ddd.md
+lre.md
 rem.md
 qse.md
 pt.md

docs/source/guide/lre-1-intro.md

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+---
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+ name: python3
+---
+
+
+# How do I use LRE?
+
+
+LRE works in two steps: generate noise-scaled circuits and apply inference to results from executed circuits.
+
+
+A user has the choice to either use {func}`.execute_with_lre` to combine both steps into one if they are
+only interested in obtaining the mitigated expectation value or splitting the process into two using
+{func}`.multivariate_layer_scaling` and {func}`.multivariate_richardson_coefficients`.
+
+
+```{warning}
+LRE is currently compatible with quantum programs written using `cirq`. Work on making this technique compatible with other frontends is ongoing. 🚧
+```
+
+
+## Problem Setup
+
+
+To use {func}`.execute_with_lre` without any additional options the following are required:
+
+
+- a quantum circuit
+- a method of returning an expectation value from a circuit
+- the degree of the multivariate polynomial extrapolation
+- fold multiplier AKA the scaling gap which is used to generate the scale factor vectors
+
+
+### Define the circuit of interest
+
+
+For simplicity, we define a simple circuit whose ideal execution is identical to the identity operation.
+
+
+```{code-cell} ipython3
+from mitiq import benchmarks
+
+
+circuit = benchmarks.generate_rb_circuits(n_qubits=1, num_cliffords=3)[0]
+
+
+print(circuit)
+```
+
+
+### Define executor for ideal and noisy executions
+
+
+We define an [executor](executors.md) which executes the input circuit subjected to depolarizing noise, and returns the probability of the ground state. By altering the value for `noise_level`, ideal and noisy expectation
+values can be obtained.
+
+
+```{code-cell} ipython3
+import numpy as np
+from cirq import DensityMatrixSimulator, depolarize
+
+
+def execute(circuit, noise_level=0.025):
+   """Default executor for all unit tests."""
+   noisy_circuit = circuit.with_noise(depolarize(p=noise_level))
+   rho = DensityMatrixSimulator().simulate(noisy_circuit).final_density_matrix
+   return rho[0, 0].real
+```
+
+
+Compare the noisy and ideal expectation values:
+
+
+```{code-cell} ipython3
+# Compute the expectation value of the |0><0| observable.
+noisy_value = execute(circuit)
+ideal_value = execute(circuit, noise_level=0.0)
+print(f"Error without mitigation: {abs(ideal_value - noisy_value) :.5f}")
+```
+
+
+## Apply LRE directly
+
+
+With the circuit, and executor defined, we just need to choose the polynomial extrapolation degree as well as the fold multiplier.
+
+
+```{code-cell} ipython3
+from mitiq.lre import execute_with_lre
+
+
+input_degree = 2
+input_fold_multiplier = 3
+
+
+mitigated_result = execute_with_lre(
+    circuit,
+    execute,
+    degree = input_degree,
+    fold_multiplier = input_fold_multiplier,
+)
+
+
+print(f"Error with mitigation (ZNE): {abs(ideal_value - mitigated_result):.{3}}")
+```
+
+
+## Step by step application of LRE
+
+In this section we will walk through what happens in each of the two stages of LRE.
+
+
+### Create noise-scaled circuits
+
+
+We start with creating a number of noise-scaled circuits which we will pass to the executor.
+
+
+```{code-cell} ipython3
+from mitiq.lre import multivariate_layer_scaling
+
+
+noise_scaled_circuits = multivariate_layer_scaling(circuit, input_degree, input_fold_multiplier)
+
+
+print(f"total number of noise-scaled circuits for LRE = {len(noise_scaled_circuits)}")
+```
+An example noise-scaled circuit is shown below:
+
+
+```{code-cell} ipython3
+noise_scaled_circuits[3]
+```
+
+
+### Classical inference
+
+
+Based on the choice of input parameters, a sample matrix created using {func}`.sample_matrix` is used to find the
+coefficients of linear combination required for multivariate Richardson extrapolation (**link theory section here**).
+
+
+```{code-cell} ipython3
+from mitiq.lre import multivariate_richardson_coefficients
+
+coeffs_of_linear_comb = multivariate_richardson_coefficients(
+    circuit,
+    fold_multiplier = input_fold_multiplier,
+    degree = input_degree,
+)
+
+
+print(f"total number of noise-scaled circuits for LRE = {len(noise_scaled_circuits)}")
+print(f"total number of coefficients of linear combination for LRE = {len(coeffs_of_linear_comb)}")
+```
+Each noise scaled circuit has a coefficient of linear combination and a noisy expectation value associated with it.
+
+
+### Combine the results
+
+
+```{code-cell} ipython3
+## execute each noise scaled circuit
+
+noise_scaled_exp_values = []
+
+
+for i in noise_scaled_circuits:
+ noise_scaled_exp_values.append(execute(i))
+
+
+calculated_mitigated_result = np.dot(noise_scaled_exp_values, coeffs_of_linear_comb)
+print(f"Error with mitigation (ZNE): {abs(ideal_value - calculated_mitigated_result):.{3}}")
+```
+The section [](lre-3-options.md) contains more information on other options available in LRE in addition to how to
+control the hyperparameters associated with the LRE options.

docs/source/guide/lre-2-use-case.md

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+---
+
+
+# When should I use LRE?
+
+
+## Advantages
+
+
+Layerwise Richardson Extrapolation is a generalized multivariate extension of the Richardson extrapolation where the univariate
+version is available as an option in [ZNE](zne-3-options.md). Just as in ZNE, LRE can also be applied without a detailed knowledge of the underlying noise model as the effectiveness of the technique depends on the choice of scale factors. Thus, LRE is useful in scenarios where tomography is impractical.
+
+
+The sampling overhead is flexible wherein the cost can be reduced by using larger values for the fold multiplier (used to
+create the noise-scaled circuits) or by chunking a larger circuit into a smaller number of chunks.
+
+
+
+
+## Disadvantages
+
+
+When using a large circuit, the number of noise scaled circuits grows polynomially such that the execution time rises because we require the sample matrix to be a square matrix (**link theory page here**).
+
+
+If one aims to reduce the sampling cost by using a larger fold multiplier, the bias for polynomial extrapolation increases as one moves farther away from the zero-noise limit.
+
+
+Chunking a large circuit with a lower number of chunks to reduce the sampling cost can reduce the performance of LRE. In ZNE parlance, this is equivalent to local folding faring better than global folding in LRE when we use a higher number of chunks in [LRE](lre-3-options.md). 
+
+
+```{note}
+We are currently investigating the issue related to chunking large circuits, as reduced performance has been noticed in our testing.
+```
+

docs/source/guide/lre-3-options.md

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+---
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+  display_name: Python 3
+  language: python
+  name: python3
+---
+
+# What additional options are available when using LRE?
+

docs/source/guide/lre-4-low-level.md

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+---
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+  language: python
+  name: python3
+---
+
+# What happens when I use LRE?

docs/source/guide/lre.md

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+# LRE
+
+
+```{toctree}
+---
+maxdepth: 1
+---
+lre-1-intro.md
+lre-2-use-case.md
+lre-3-options.md
+lre-4-low-level.md
+```