[Paper Review] Randomized benchmarking with random quantum circuits

Markus R. Heinrich, Martin Kliesch+1 more|arXiv (Cornell University)|Dec 12, 2022

Quantum Computing Algorithms and Architecture8 citations

TL;DR

This paper establishes general theoretical guarantees for non-uniform randomized benchmarking (RB) protocols using random quantum circuits, showing that filtered RB with random circuits in linear depth can reliably estimate gate fidelity with sample efficiency. It proves that subdominant decays from non-uniformity are negligible under fast-mixing conditions, enabling robust, SPAM-robust fidelity estimation for NISQ devices with minimal assumptions on noise and gate sets.

ABSTRACT

In its many variants, randomized benchmarking (RB) is a broadly used technique for assessing the quality of gate implementations on quantum computers. A detailed theoretical understanding and general guarantees exist for the functioning and interpretation of RB protocols if the gates under scrutiny are drawn uniformly at random from a compact group. In contrast, many practically attractive and scalable RB protocols implement random quantum circuits with local gates randomly drawn from some gate-set. Despite their abundance in practice, for those non-uniform RB protocols, general guarantees for gates from arbitrary compact groups under experimentally plausible assumptions are missing. In this work, we derive such guarantees for a large class of RB protocols for random circuits that we refer to as filtered RB. Prominent examples include linear cross-entropy benchmarking, character benchmarking, Pauli-noise tomography and variants of simultaneous RB. Building upon recent results for random circuits, we show that many relevant filtered RB schemes can be realized with random quantum circuits in linear depth, and we provide explicit small constants for common instances. We further derive general sample complexity bounds for filtered RB. We show filtered RB to be sample-efficient for several relevant groups, including protocols addressing higher-order cross-talk. Our theory for non-uniform filtered RB is, in principle, flexible enough to design new protocols for non-universal and analog quantum simulators.

Motivation & Objective

• To close the theoretical gap in non-uniform randomized benchmarking (RB) protocols that use random quantum circuits instead of uniformly sampled group elements.
• To provide general, experimentally plausible guarantees for filtered RB schemes—such as linear XEB and character benchmarking—when implemented with random circuits.
• To show that random circuits of linear depth can realize filtered RB with negligible subdominant decays due to non-uniformity.
• To derive explicit sample complexity bounds for filtered RB under minimal assumptions, demonstrating sample efficiency for key groups including the Pauli group and local unitary 3-designs.
• To enable the design of new RB protocols for non-universal and analog quantum simulators through a flexible theoretical framework.

Proposed method

• Introduces the concept of 'filtered RB' as a generalization of standard RB, where a filter function selects specific matrix elements of the channel to analyze.
• Uses spectral gap analysis of random circuit generators to bound the convergence rate to a 2-design, ensuring subdominant decays are negligible.
• Applies recent results on random circuits to show that linear-depth local random circuits mix sufficiently fast to suppress non-uniformity effects.
• Derives sample complexity bounds for filtered RB by relating them to the spectral properties of the channel and the filter function.
• Computes explicit constants for sufficient sequence lengths in common cases (e.g., Pauli group, local 3-designs) to guide experimental implementation.
• Demonstrates that sample complexity is preserved when replacing uniform unitaries with random circuits, ensuring efficiency is maintained.

Experimental results

Research questions

• RQ1 Can filtered RB protocols using random quantum circuits provide reliable, interpretable estimates of gate fidelity under minimal assumptions on noise and gate sets?
• RQ2 What circuit depth is sufficient to ensure that non-uniformity-induced subdominant decays are negligible in filtered RB signals?
• RQ3 How does the sample complexity of filtered RB compare between uniformly distributed unitaries and random circuits?
• RQ4 Can the framework be extended to design new RB protocols for non-universal or analog quantum simulators?
• RQ5 What filter functions can further reduce required circuit depth while maintaining theoretical guarantees?

Key findings

• Filtered RB with random quantum circuits in linear depth can reliably estimate gate fidelity, as the subdominant decay from non-uniformity is negligible when the circuit mixes sufficiently fast.
• For random circuits that form approximate 2-designs, a circuit depth linear in the number of qudits is sufficient to suppress non-uniformity effects and extract the relevant decay parameter.
• The sample complexity of filtered RB is preserved when using random circuits instead of uniformly drawn unitaries, ensuring sample efficiency for groups including the Pauli group and local unitary 3-designs.
• Explicit, small constants are derived for sufficient sequence lengths in common cases, enabling direct experimental guidance.
• The framework is general enough to support new protocols for non-universal and analog quantum simulators, with potential applications in XEB and related benchmarking tasks.
• The theory is robust under minimal assumptions and can, in principle, be extended to non-Markovian noise, though such extensions are left for future work.