Adding layerwise folding as tutorial to mitiq. (#1894)

* Adding layerwise folding as tutorial to mitiq.

* fixing typing issue for python3.10.

* Update docs/source/examples/layerwise-folding.md

Co-authored-by: Nathan Shammah <[email protected]>

* Fixing doc warnings. Adding thumbnail.

* Update docs/source/examples/layerwise-folding.md

Co-authored-by: Andrea Mari <[email protected]>

* Responding to diff comments.

* Update docs/source/examples/layerwise-folding.md

Co-authored-by: nate stemen <[email protected]>

* Update docs/source/examples/layerwise-folding.md

Co-authored-by: nate stemen <[email protected]>

* Moving motivational material to top.

* Removing duplicated content from tutorial.

---------

Co-authored-by: vrusso <[email protected]>
Co-authored-by: Nathan Shammah <[email protected]>
Co-authored-by: Andrea Mari <[email protected]>
Co-authored-by: nate stemen <[email protected]>

docs/source/conf.py

@@ -334,6 +334,7 @@ def get_incollection_template(self, e):
  "examples/hamiltonians": "_static/vqe-cirq-pauli-sum-mitigation-plot.png",
  "examples/braket_mirror_circuit": "_static/mirror-circuits.png",
  "examples/maxcut-demo": "_static/max-cut.png",
+ "examples/layerwise-folding": "_static/layerwise.png",
  "examples/cirq-ibmq-backends": "_static/cirq-mitiq-ibmq.png",
  "examples/pennylane-ibmq-backends": "_static/zne-pennylane.png",
  "examples/ibmq-backends": "_static/ibmq-gate-map.png",

docs/source/examples/examples.md

@@ -14,6 +14,7 @@ ZNE with PyQuil: Improving VQE <vqe-pyquil-demo.ipynb>
 ZNE with Cirq: Energy landscape of a variational circuit <simple-landscape-cirq.md>
 ZNE with Braket: Energy landscape of a variational circuit <simple-landscape-braket.md>
 ZNE with Qiskit: Energy landscape of a variational circuit <simple-landscape-qiskit.md>
+ZNE with Qiskit: Layerwise folding <layerwise-folding.md>
 ZNE with Qiskit: Quantum simulation of quantum many body scars <quantum_simulation_scars_ibmq.md>
 ZNE with Cirq: Solving MaxCut with QAOA <maxcut-demo.md>
 ZNE with Cirq: Hamiltonian simulation with Pauli gates<hamiltonians.md>

docs/source/examples/layerwise-folding.md

@@ -0,0 +1,291 @@
+---
+jupytext:
+ text_representation:
+ extension: .md
+ format_name: myst
+ format_version: 0.13
+ jupytext_version: 1.11.1
+kernelspec:
+ display_name: Python 3
+ language: python
+ name: python3
+---
+
+# ZNE with Qiskit: Layerwise folding
+
+
+This tutorial shows an example of how to mitigate noise on IBMQ backends using
+layerwise folding in contrast with global folding.
+
+One may ask why folding by layer is potentially beneficial to consider. One
+reason is that applying global folding will increase the length of the entire
+circuit while layerwise folding on a subset of only the noisiest layers will
+increase the circuit by a smaller factor. 
+
+If running a circuit on hardware is bottle-necked by the cost of running a long
+circuit, this technique could potentially be used to arrive at a better result
+(although not as good as global folding) but with less monetary cost.
+
+More information on the layerwise folding technique can be found in 
+*Calderon et al. Quantum (2023)* {cite}`Calderon_2023_Quantum`.
+
+
+- [ZNE with Qiskit: Layerwise folding](#zne-with-qiskit-layerwise-folding)
+ - [Setup](#setup)
+ - [Helper functions](#helper-functions)
+ - [Define circuit to analyze](#define-circuit-to-analyze)
+ - [Total variational distance metric](#total-variational-distance-metric)
+ - [Impact of single vs. multiple folding](#impact-of-single-vs-multiple-folding)
+ - [Executor](#executor)
+ - [Global folding with linear extrapolation](#global-folding-with-linear-extrapolation)
+ - [Layerwise folding with linear extrapolation](#layerwise-folding-with-linear-extrapolation)
+
++++
+
+## Setup
+
+```{code-cell} ipython3
+from typing import Dict, List, Optional
+import numpy as np
+import os
+import cirq
+import qiskit
+import matplotlib.pyplot as plt
+
+from mitiq import zne
+from mitiq.zne.scaling.layer_scaling import layer_folding, get_layer_folding
+from mitiq.interface.mitiq_qiskit.qiskit_utils import initialized_depolarizing_noise
+from mitiq.interface.mitiq_qiskit.conversions import to_qiskit
+from mitiq.interface.mitiq_cirq.cirq_utils import sample_bitstrings
+
+from cirq.contrib.svg import SVGCircuit
+from qiskit_ibm_provider import IBMProvider
+
+# Default to a simulator.
+backend = qiskit.Aer.get_backend("qasm_simulator")
+noise_model = initialized_depolarizing_noise(noise_level=0.02)
+
+shots = 10_000
+```
+
+## Helper functions
+
+The following function will return a list of circuits where the ith element in
+the list is a circuit with layer "i" folded `num_folds` number of times. This
+will be useful when analyzing how much folding increases the noise on a given
+layer.
+
+```{code-cell} ipython3
+def apply_num_folds_to_all_layers(circuit: cirq.Circuit, num_folds: int = 1) -> List[cirq.Circuit]:
+ """List of circuits where ith circuit is folded `num_folds` times."""
+ return [
+ layer_folding(circuit, [0] * i + [num_folds] + [0] * (len(circuit) - i))
+ for i in range(len(circuit))
+ ]
+```
+
+For instance, consider the following circuit.
+
+```{code-cell} ipython3
+# Define a basic circuit for
+q0, q1 = cirq.LineQubit.range(2)
+circuit = cirq.Circuit(
+ [cirq.ops.H(q0)],
+ [cirq.ops.CNOT(q0, q1)],
+ [cirq.measure(cirq.LineQubit(0))],
+)
+print(circuit)
+```
+
+Let us invoke the `apply_num_folds_to_all_layers` function as follows.
+
+```{code-cell} ipython3
+folded_circuits = apply_num_folds_to_all_layers(circuit, num_folds=2)
+```
+
+Note that the first element of the list is the circuit with the first layer of
+the circuit folded twice.
+
+```{code-cell} ipython3
+print(folded_circuits[0])
+```
+
+Similarly, the second element of the list is the circuit with the second layer
+folded.
+
+```{code-cell} ipython3
+print(folded_circuits[1])
+```
+
+
+## Define circuit to analyze
+
+We will use the following circuit to analyze, but of course, you could use
+other circuits here as well.
+
+```{code-cell} ipython3
+circuit = cirq.Circuit([cirq.X(cirq.LineQubit(0))] * 10, cirq.measure(cirq.LineQubit(0)))
+print(circuit)
+```
+
+## Total variational distance metric
+
+An $i$-inversion can be viewed as a local perturbation of the circuit. We want
+to define some measure by which we can determine how much such a perturbation
+affects the outcome.
+
+Define the quantity:
+
+$$
+p(k|C) = \langle \langle k | C | \rho_0 \rangle \rangle
+$$
+
+as the probability distribution over measurement outcomes at the output of a
+circuit $C$ where $k \in B^n$ with $B^n$ being the set of all $n$-length bit
+strings where $\langle \langle k |$ is the vectorized POVM element that
+corresponds to measuring bit string $k$. 
+
+The *impact* of applying an inversion is given by
+
+$$
+d \left[p(\cdot|C), p(\cdot|C^{(i)})\right]
+$$
+
+where $d$ is some distance measure. In 
+*Calderon et al. Quantum (2023)* {cite}`Calderon_2023_Quantum` the authors used the total variational distance
+(TVD) measure where
+
+$$
+\eta^{(i)} := \frac{1}{2} \sum_{k} |p(k|C) - p(k|C^{(i)})|.
+$$
+
+```{code-cell} ipython3
+def tvd(circuit: cirq.Circuit, num_folds: int = 1, shots: int = 10_000) -> List[float]:
+ """Compute the total variational distance (TVD) between ideal circuit and folded circuit(s)."""
+ circuit_dist = sample_bitstrings(circuit=circuit, shots=shots).prob_distribution()
+
+ folded_circuits = apply_num_folds_to_all_layers(circuit, num_folds)
+
+ distances: Dict[int, float] = {}
+ for i, folded_circuit in enumerate(folded_circuits):
+ folded_circuit_dist = sample_bitstrings(circuit=folded_circuit, shots=shots).prob_distribution()
+
+ res: float = 0.0
+ for bitstring in circuit_dist.keys():
+ res += np.abs(circuit_dist[bitstring] - folded_circuit_dist[bitstring])
+ distances[i] = res / 2
+
+ return distances
+```
+
+## Impact of single vs. multiple folding 
+
+We can plot the impact of applying layer inversions to the circuit.
+
+```{code-cell} ipython3
+def plot_single_vs_multiple_folding(circuit: cirq.Circuit) -> None:
+ """Plot how single vs. multiple folding impact the error at a given layer."""
+ single_tvd = tvd(circuit, num_folds=1).values()
+ multiple_tvd = tvd(circuit, num_folds=5).values()
+
+ labels = [f"L{i}" for i in range(len(circuit))]
+ x = np.arange(len(labels)) # the label locations
+ width = 0.35 # the width of the bars
+
+ fig, ax = plt.subplots()
+ rects1 = ax.bar(x - width/2, single_tvd, width, label="single")
+ rects2 = ax.bar(x + width/2, multiple_tvd, width, label="multiple")
+
+ # Add some text for labels, title and custom x-axis tick labels, etc.
+ ax.set_xlabel(r"$L_{G_i \theta_i}$")
+ ax.set_ylabel(r"$\eta^{(i)}$")
+ ax.set_title("Single vs. multiple folding")
+ ax.set_xticks(x, labels, rotation=60)
+
+ ax.legend()
+
+ ax.bar_label(rects1, padding=3)
+ ax.bar_label(rects2, padding=3)
+
+ fig.tight_layout()
+
+ plt.show()
+```
+
+```{code-cell} ipython3
+plot_single_vs_multiple_folding(circuit)
+```
+
+As can be seen, the amount of noise on each layer is increased if the number of
+folds on that layer are increased.
+
+## Executor
+
+Next, we define an executor function that will allow us to run our experiment
+
+```{code-cell} ipython3
+def executor(circuit: cirq.Circuit, shots: int = 10_000) -> float:
+ """Returns the expectation value to be mitigated.
+
+ Args:
+ circuit: Circuit to run.
+ shots: Number of times to execute the circuit to compute the expectation value.
+ """
+ qiskit_circuit = to_qiskit(circuit) 
+ job = qiskit.execute(
+ experiments=qiskit_circuit,
+ backend=backend,
+ noise_model=noise_model,
+ basis_gates=noise_model.basis_gates if noise_model is not None else None,
+ optimization_level=0, # Important to preserve folded gates.
+ shots=shots,
+ )
+
+ # Convert from raw measurement counts to the expectation value
+ counts = job.result().get_counts()
+
+ expectation_value = 0.0 if counts.get("0") is None else counts.get("0") / shots
+ return expectation_value
+```
+
+## Global folding with linear extrapolation
+
+First, for comparison, we apply ZNE with global folding on the entire circuit.
+We then compare the mitigated result of applying ZNE with linear extrapolation
+against the unmitigated value.
+
+
+```{code-cell} ipython3
+unmitigated = executor(circuit)
+
+linear_factory = zne.inference.LinearFactory(scale_factors=[1.0, 1.5, 2.0, 2.5, 3.0])
+mitigated = zne.execute_with_zne(circuit, executor, factory=linear_factory)
+
+print(f"Unmitigated result {unmitigated:.3f}")
+print(f"Mitigated result {mitigated:.3f}")
+```
+
+## Layerwise folding with linear extrapolation
+
+For contrast, we apply layerwise folding on only the layer with the most noise
+and use linear extrapolation. As above, we compare the mitigated and
+unmitigated values.
+
+```{code-cell} ipython3
+# Calculate the TVDs of each layer in the circuit (with `num_folds=3`):
+tvds = tvd(circuit, num_folds=3)
+
+# Fold noisiest layer only.
+layer_to_fold = max(tvds, key=tvds.get)
+fold_layer_func = zne.scaling.get_layer_folding(layer_to_fold)
+
+mitigated = zne.execute_with_zne(circuit, executor, scale_noise=fold_layer_func, factory=linear_factory)
+print(f"Mitigated (layerwise folding) result {mitigated:.3f}")
+print(f"Unmitigated result {unmitigated:.3f}")
+```
+
+```{note}
+While doing layerwise folding on the noisiest layer will, on average,
+improve the mitigated value, it still will not eclipse the benefit of doing
+global folding.
+```