Noise-mitigated randomized measurements and self-calibrating shadow estimation

TL;DR

This work introduces an error-mitigated method of randomized measurements, giving rise to a robust shadow estimation procedure, and demonstrates that, if properly used, easily accessible data from randomized benchmarking schemes already provide such valuable diagnostic information to inform about the noise dynamics and to assist in quantum learning procedures.

Abstract

Randomized measurements are increasingly appreciated as powerful tools to estimate properties of quantum systems, e.g., in the characterization of hybrid classical-quantum computation. On many platforms they constitute natively accessible measurements, serving as the building block of prominent schemes like shadow estimation. In the real world, however, the implementation of the random gates at the core of these schemes is susceptible to various sources of noise and imperfections, strongly limiting the applicability of protocols. To attenuate the impact of this shortcoming, in this work we introduce an error-mitigated method of randomized measurements, giving rise to a robust shadow estimation procedure. On the practical side, we show that error mitigation and shadow estimation can be carried out using the same session of quantum experiments, hence ensuring that we can address and mitigate the noise affecting the randomization measurements. Mathematically, we develop a picture derived from Fourier-transforms to connect randomized benchmarking and shadow estimation. We prove rigorous performance guarantees and show the functioning using comprehensive numerics. More conceptually, we demonstrate that, if properly used, easily accessible data from randomized benchmarking schemes already provide such valuable diagnostic information to inform about the noise dynamics and to assist in quantum learning procedures.

Figures (6)

• Figure 1: An unknown state is subjected to random Clifford sequences, followed by measurements in the computational basis. Using randomized benchmarking, decay parameters are being estimated to establish the calibrated frame operator $\widehat{\mathcal{M}}$. The calibration procedure can be part of the estimation or run separately, in four variants.
• Figure 2: (a) Application of the RB-calibrated shadow estimation scheme for gate-independent noise applied to 8 qubits, for amplitude damping and dephasing, each with parameter $p\in [0,1]$ shown on the horizontal axis. (b) The performance of the scheme for gate-dependent noise applied on 5 qubits. We consider a simple gate-dependent noise model in which CNOT gates are followed by two-qubit depolarizing noise with parameter $p$ shown on the horizontal axis. In both cases, the task is to estimate the state fidelity with a GHZ state. The noise-free target value is shown by the black dashed line. Error bars show one bootstrap standard deviation of the median-of-means estimator. See \ref{['sec:numerics_details']} for details on the numerical simulation.
• Figure 3: Simulation results for 5 qubits and local depolarizing noise. In line with the results of \ref{['eq:local_depol_values']}, the calibration using the Clifford RB parameter overcompensates the effect of the noise, since $\lambda_\mathrm{adj}(\Lambda_\mathrm{dep,loc}) < \lambda_Z(\Lambda_\mathrm{dep,loc})$. See \ref{['sec:numerics_details']} for details on the numerical simulation.
• Figure 4: Simulation results for 5 qubits and local bit flip noise. See \ref{['sec:numerics_details']} for details on the numerical simulation.
• Figure 5: The performance for a 5 qubit system and amplitude damping noise. See \ref{['sec:numerics_details']} for details on the numerical simulation.

Theorems & Definitions (22)

• Lemma 1: Error bias for the uncalibrated frame operator RobustShadows
• Lemma 2: Fitting model for CNOT-dihedral RB
• proof
• Lemma 3: Fitting model for shadow CNOT-dihedral scheme
• Lemma 4: Estimated frame operator for circuit shadows
• proof : Proof of \ref{['shadow_dihedral_fitting_model']}
• proof : Proof of \ref{['lemma:circuit_M_estimator']}
• Lemma 5: Bias for the calibrated frame operator
• Lemma 6: Calibration for global depolarizing channel, cnf. koh2022ClassicalShadowsNoise
• proof