Interleaved Randomized Benchmarking (IRB)

The interleaved randomized benchmarking routine allows us to estimate the gate fidelity of single qubit Clifford gates. To demonstrate this routine, consider device noise modelled by an amplitude damping channel with decay probability \(\gamma=0.01\)

import cirq
import numpy as np
import supermarq

decay_prob = 0.025
noise = cirq.AmplitudeDampingChannel(gamma=decay_prob)
simulator = cirq.DensityMatrixSimulator(noise=noise)

We can calculate the average fidelity of this channel as:

\[\begin{split}\begin{align} \overline{F} &= \int\langle\psi|\mathcal{N_\gamma}(|\psi\rangle\langle\psi|)|\psi\rangle d\psi\\ &=\int\langle\psi|K_0|\psi\rangle\langle\psi|K_0^\dagger|\psi\rangle + \langle\psi|K_1|\psi\rangle\langle\psi|K_1^\dagger|\psi\rangle d\psi\\ & =\frac{1}{4\pi}\int_0^\pi\int_0^{2\pi}\left|\begin{pmatrix}\cos\frac{\theta}{2}&e^{-i\phi}\sin\frac{\theta}{2}\end{pmatrix}\begin{pmatrix}1 & 0 \\0 & \sqrt{1 - \gamma}\end{pmatrix}\begin{pmatrix}\cos\frac{\theta}{2}\\e^{i\phi}\sin\frac{\theta}{2}\end{pmatrix}\right|^2\sin\theta \\ & + \left|\begin{pmatrix}\cos\frac{\theta}{2}&e^{-i\phi}\sin\frac{\theta}{2}\end{pmatrix}\begin{pmatrix}0 & \sqrt{\gamma} \\0 & 0\end{pmatrix}\begin{pmatrix}\cos\frac{\theta}{2}\\e^{i\phi}\sin\frac{\theta}{2}\end{pmatrix}\right|^2\sin\theta d\phi d\theta \\ &=\frac{1}{4\pi}\int_0^\pi\int_0^{2\pi}\left|\cos^2\frac{\theta}{2}+\sqrt{1-\gamma}\sin^2\frac{\theta}{2}\right|^2\sin\theta + \left|\sqrt{\gamma}e^{i\phi}\sin\frac{\theta}{2}\cos\frac{\theta}{2}\right|^2\sin\theta d\phi d\theta \\ &=\frac{1}{2}\int_0^\pi\left(\cos^4\frac{\theta}{2}+(1-\gamma)\sin^4\frac{\theta}{2}+\frac{\sqrt{1-\gamma}}{2}\sin^2\theta + \frac{\gamma}{4}\sin^2\theta\right)\sin\theta d\theta \\ &=\frac{1}{2}\int_0^\pi\sin\theta\cos^4\frac{\theta}{2}+(1-\gamma)\sin\theta\sin^4\frac{\theta}{2}+\frac{\gamma+2\sqrt{1-\gamma}}{4}\sin^3\theta d\theta \\ &=\frac{1}{2}\left(\frac{2}{3} + (1-\gamma)\frac{2}{3} + \frac{\gamma+2\sqrt{1-\gamma}}{4}\frac{4}{3}\right) \\ &=\frac{1}{2}\left(\frac{4}{3} - \frac{\gamma}{3} + \frac{2\sqrt{1-\gamma}}{3}\right) \\ &=\frac{2}{3}-\frac{\gamma}{6} + \frac{\sqrt{1-\gamma}}{3}. \end{align}\end{split}\]

Thus we have a gate error

\[\frac{1}{3}+\frac{\gamma}{6} - \frac{\sqrt{1-\gamma}}{3}\]

expected_gate_error = 1 / 3 + decay_prob / 6 - np.sqrt(1 - decay_prob) / 3

experiment = supermarq.qcvv.IRB(interleaved_gate=cirq.ops.SingleQubitCliffordGate.Y)
experiment.prepare_experiment(100, [1, 5, 10, 15])
experiment.run_with_simulator(simulator=simulator)

if experiment.collect_data():
    results = experiment.analyze_results(plot_results=True)
    print(results)

IRBResults(target='Local simulator', total_circuits=800, experiment_name='IRB', rb_layer_fidelity=0.9835345476930657, rb_layer_fidelity_std=0.0004865849120610916, irb_layer_fidelity=0.9672464523458026, irb_layer_fidelity_std=0.000336442324359281, average_interleaved_gate_error=0.008280388007451123, average_interleaved_gate_error_std=0.0002973777530802783)

print(f"Expected gate error: {expected_gate_error:.6f}")
print(
    f"Measured gate error: {results.average_interleaved_gate_error:.6f} +/- {results.average_interleaved_gate_error_std:.6f}"
)

Expected gate error: 0.008360
Measured gate error: 0.008280 +/- 0.000297