[Paper Review] A framework for randomized benchmarking over compact groups

Linghang Kong|arXiv (Cornell University)|Nov 19, 2021

Quantum Computing Algorithms and Architecture17 references4 citations

TL;DR

This paper extends randomized benchmarking (RB) to continuous compact Lie groups, such as the full unitary group, by generalizing matrix perturbation theory to infinite-dimensional Fourier spaces. It proves that under small noise, the output decays as a linear combination of matrix exponentials, enabling fully randomized benchmarking (FRB) to estimate gate fidelities uniformly across all quantum gates, including non-Clifford and continuous gate sets.

ABSTRACT

Characterization of experimental systems is an essential step in developing and improving quantum hardware. A collection of protocols known as Randomized Benchmarking (RB) was developed in the past decade, which provides an efficient way to measure error rates in quantum systems. In a recent paper (arxiv:2010.07974), a general framework for RB was proposed, which encompassed most of the known RB protocols and overcame the limitation on error models in previous works. However, even this general framework has a restriction: it can only be applied to a finite group of gates. This does not meet the need posed by experiments, in particular the demand for benchmarking non-Clifford gates and continuous gate sets on quantum devices. In this work we generalize the RB framework to continuous groups of gates and show that as long as the noise level is reasonably small, the output can be approximated as a linear combination of matrix exponential decays. As an application, we numerically study the fully randomized benchmarking protocol (i.e. RB with the entire unitary group as the gate set) enabled by our proof. This provides a unified way to estimate the gate fidelity for any quantum gate in an experiment.

Motivation & Objective

• Address the limitation of existing randomized benchmarking (RB) frameworks, which are restricted to finite groups and cannot benchmark non-Clifford or continuous gate sets.
• Enable a unified framework for estimating gate fidelity across all quantum gates, including those in continuous families, by generalizing RB to compact Lie groups.
• Overcome the restriction of previous general RB frameworks that required finite gate sets, particularly for experimental needs involving continuous gate sets like XY- and fSim-gates.
• Establish theoretical foundations for fully randomized benchmarking (FRB) on the full unitary group, allowing interleaved FRB to benchmark arbitrary gates.
• Provide a rigorous mathematical extension of matrix perturbation theory to infinite-dimensional Hilbert spaces to support the generalization of RB to continuous groups.

Proposed method

• Generalize a matrix perturbation theorem (from [1, Thm. 6]) to infinite-dimensional Hilbert spaces, enabling analysis of spectral separation in the Fourier space of compact groups.
• Use the representation theory of compact Lie groups to decompose the noise channel into irreducible representations, treating the problem in the Fourier domain.
• Prove that under small noise, the survival probability decays as a linear combination of matrix exponentials, with decay rates determined by the eigenvalues of the noise channel in each irreducible representation.
• Apply the generalized perturbation theory to show that the output of randomized benchmarking sequences on compact groups remains approximately exponentially decaying, even with gate-dependent noise.
• Construct the fully randomized benchmarking (FRB) protocol by sampling gates uniformly from the full unitary group, ensuring robustness to SPAM errors and enabling universal gate fidelity estimation.
• Use numerical simulations to validate the theoretical results, comparing FRB stability with Clifford RB and testing interleaved FRB on continuous gate families.

Experimental results

Research questions

• RQ1 Can randomized benchmarking be generalized from finite groups to continuous compact Lie groups, such as the unitary group, while preserving the exponential decay behavior of the survival probability?
• RQ2 Under what conditions does the output of randomized benchmarking on a compact group still exhibit matrix exponential decay when noise is gate-dependent and small?
• RQ3 Can fully randomized benchmarking (FRB) on the full unitary group provide a stable and accurate estimate of gate fidelity for any quantum gate, including non-Clifford and continuous gate sets?
• RQ4 How does the performance of FRB compare to standard Clifford RB in terms of stability and accuracy for benchmarking tasks common to both?
• RQ5 Can interleaved FRB be used to estimate the fidelity of gates in a continuous family, such as XY- or fSim-gates, and how well do numerical simulations match theoretical predictions?

Key findings

• The paper proves that for compact Lie groups, the survival probability in randomized benchmarking decays as a linear combination of matrix exponentials under small noise, generalizing the finite-group result from [1, Thm. 8].
• The framework enables fully randomized benchmarking (FRB) on the full unitary group, allowing universal estimation of gate fidelity for any quantum gate, including non-Clifford and continuous gate sets.
• Numerical simulations show that FRB exhibits comparable stability to Clifford RB when benchmarking gates common to both protocols, validating its robustness.
• Interleaved FRB successfully estimates the fidelity of gates in continuous families, with simulation results matching theoretical predictions with high accuracy.
• The generalized matrix perturbation theorem in infinite-dimensional Hilbert spaces ensures spectral separation and convergence, enabling the decay approximation even with infinitely many irreducible representations.
• The theoretical framework supports the use of RB with continuous gate sets in near-term quantum computing, such as variational quantum algorithms (VQE, QAOA) and devices using native continuous gates like fSim.