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arXiv:2309.15651 · Physical Review Research

Group twirling and noise tailoring for multiqubit controlled phase gates

Guoding Liu, Ziyi Xie, Zitai Xu, Xiongfeng Ma

Published Dec 2, 2024• 7 citations
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Abstract

Group twirling is crucial in quantum information processing, particularly in randomized benchmarking and randomized compiling. While protocols based on Pauli twirling have been effectively crafted to transform arbitrary noise channels into Pauli channels for Clifford gates—thereby facilitating efficient benchmarking and mitigating worst-case errors—the lack of practical twirling groups for multiqubit non-Clifford gates remains a challenge. To address this gap, we study the issue of finding twirling groups for generic quantum gates, focusing on a widely used circuit structure in randomized benchmarking or randomized compiling. Specifically, for multiqubit controlled phase gates, which are essential in quantum algorithms and directly implementable in practice, we determine optimal twirling groups within the realm of classically replaceable unitary operations. Contrasting with the local Pauli twirling group for Clifford gates, the optimal groups for such gates contain nonlocal operations, highlighting the overhead of tailoring noise in global non-Clifford contexts. We design new benchmarking procedures for multiqubit controlled phase gates based on the optimal twirling groups. Our simulation results show that our scheme can improve the precision and accuracy of benchmarking in small-scale systems. Published by the American Physical Society 2024

DOI
10.1103/physrevresearch.6.043221

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Group twirling is crucial in quantum information processing, particularly in randomized benchmarking and randomized compiling. While protocols based on Pauli twirling have been effectively crafted to transform arbitrary noise channels into Pauli channels for Clifford gates—thereby facilitating efficient benchmarking and mitigating worst-case errors—the lack of practical twirling groups for multiqubit non-Clifford gates remains a challenge. To address this gap, we study the issue of finding twirling groups for generic quantum gates, focusing on a widely used circuit structure in randomized benchmarking or randomized compiling. Specifically, for multiqubit controlled phase gates, which are essential in quantum algorithms and directly implementable in practice, we determine optimal twirling groups within the realm of classically replaceable unitary operations. Contrasting with the local Pauli twirling group for Clifford gates, the optimal groups for such gates contain nonlocal operations, highlighting the overhead of tailoring noise in global non-Clifford contexts. We design new benchmarking procedures for multiqubit controlled phase gates based on the optimal twirling groups. Our simulation results show that our scheme can improve the precision and accuracy of benchmarking in small-scale systems. Published by the American Physical Society 2024
Quantum gate benchmarking is unavoidably influenced by state preparation and measurement errors. Randomized benchmarking addresses this challenge by employing group twirling to regularize the noise channel, then provides a characterization of quantum channels that is robust to these errors through exponential fittings. In practice, local twirling gates are preferred due to their high fidelity and experimental feasibility. However, while existing RB methods leveraging local twirling gates are effective for benchmarking Clifford gates, they face fundamental challenges in benchmarking non-Clifford gates. In this work, we solve this problem by introducing the Pauli Transfer Character Benchmarking. This protocol estimates the Pauli transfer matrix elements for a quantum channel using only local Pauli operations. Building on this protocol, we develop a fidelity benchmarking method for non-Clifford gates UUU satisfying U2=IU^2=IU2=I. We validate the feasibility of our protocol through numerical simulations applied to Toffoli gates as a concrete example.
Abstract Quantum noise limits the reliability of noisy intermediate-scale quantum (NISQ) devices. Quantum error mitigation (QEM) provides a near-term path to improved accuracy, with randomized compiling (RC) serving as a key technique for transforming arbitrary gate noise into a stochastic Pauli form. However, its sampling complexity grows exponentially with circuit depth, hindering practical use. We introduce twirling scheduling (TS), which groups gate cycles into configurable units twirled collectively. By incorporating pseudo-twirling for non-Clifford gates, TS reduces the sampling-space dimension, mitigating this exponential scaling. Our theoretical analysis provides rigorous error bounds. Numerical simulations on Quantum Approximate Optimization Algorithm circuits up to 50 layers show that TS reduces estimation error by approximately 62% compared to conventional RC under a very small sampling budget and maintains superior performance when combined with established QEM protocols, including zero-noise extrapolation, Clifford data regression, and virtual distillation. TS thus offers a scalable, practical advance for error-mitigated quantum computation on NISQ hardware.
With the development of controllable quantum systems, fast and practical characterization of multi-qubit gates has become essential for building high-fidelity quantum computing devices. The usual way to fulfill this requirement via randomized benchmarking demands complicated implementation of numerous multi-qubit twirling gates. How to efficiently and reliably estimate the fidelity of a quantum process remains an open problem. This work thus proposes a character-cycle benchmarking protocol and a character-average benchmarking protocol using only local twirling gates to estimate the process fidelity of an individual multi-qubit operation. Our protocols were able to characterize a large class of quantum gates including and beyond the Clifford group via the local gauge transformation, which forms a universal gate set for quantum computing. We demonstrated numerically our protocols for a non-Clifford gate—controlled- ( T X ) and a Clifford gate—five-qubit quantum error-correcting encoding circuit. The numerical results show that our protocols can efficiently and reliably characterize the gate process fidelities. Compared with the cross-entropy benchmarking, the simulation results show that the character-average benchmarking achieves three orders of magnitude improvements in terms of sampling complexity.
Twirling noise affecting quantum gates is essential in understanding and controlling errors, but applicable operations to noise are usually restricted by symmetries inherent in quantum gates. In this work, we propose symmetric Clifford twirling, a Clifford twirling utilizing only symmetric Clifford operators that commute with certain Pauli subgroups. We fully characterize how each Pauli noise is converted through the twirling and show that certain Pauli noise can be scrambled to a noise exponentially close to the global white noise. Moreover, we provide numerical demonstrations for highly structured circuits, such as Trotterized Hamiltonian simulation circuits, that noise effect on typical observables can be described by the global white noise. We further demonstrate that symmetric Clifford twirling and its hardware-efficient variant using only local symmetric Clifford operators can significantly accelerate the scrambling. These findings enable us to mitigate errors in non-Clifford operations with minimal sampling overhead in the early fault-tolerant regime.
The conventional circuit paradigm, utilizing a small set of gates to construct arbitrary quantum circuits, is hindered by significant noise. In the quantum Fourier transform, for instance, the standard gate paradigm employs two CNOT gates for the partial CPhase. In contrast, some quantum computers can directly implement such operations using their native interaction, resulting in less noisy gates. Unfortunately, coherent errors degrade the performance of these gates. In Clifford gates such as the CNOT, these errors can be addressed through randomized compiling (RC). However, RC does not apply to the non-Clifford multi-qubit native implementations described above. The present work introduces and experimentally demonstrates a technique called ‘Pseudo Twirling’ (PST) to address coherent errors. We demonstrate experimentally that integrating PST with the ‘Adaptive KIK’ quantum error mitigation method enables the simultaneous mitigation of noise and coherent errors in multi-qubit non-Clifford gates.
Randomized benchmarking (RB) is the gold standard for experimentally evaluating the quality of quantum operations.The current framework for RB is centered on groups and their representations but this can be problematic.For example, Clifford circuits need up to O(n 2 ) gates and thus Clifford RB cannot scale to larger devices.Attempts to remedy this include new schemes such as linear cross-entropy benchmarking (XEB), cycle benchmarking, and nonuniform RB but they do not fall within the group-based RB framework.In this work, we formulate the universal randomized benchmarking (URB) framework, which does away with the group structure and also replaces the recovery-gate-plus-measurement component with a general "postprocessing" positive operator-valued measurement (POVM).Not only does this framework cover most of the existing benchmarking schemes but it also gives the language for and helps inspire the formulation of new schemes.We specifically consider a class of URB schemes called twirling schemes.For twirling schemes, the postprocessing POVM approximately factorizes into an intermediate channel, inverting maps, and a final measurement.This leads us to study the twirling map corresponding to the gate ensemble specified by the scheme.We prove that if this twirling map is strictly within unit distance of the Haar twirling map in induced diamond norm, the probability of measurement as a function of gate length is a single exponential decay up to small error terms.The core technical tool we use is the matrix perturbation theory of linear operators on quantum channels.