Benchmarking non-Clifford gates using only Pauli twirling group

TL;DR

This work tackles SPAM-induced distortions in benchmarking quantum gates, with a focus on non-Clifford operations. It introduces Pauli Transfer Character Benchmarking (PTCB), a protocol that uses only Pauli twirling and Pauli-basis SPAM to access products of symmetric Pauli-transfer matrix elements for a quantum channel, aided by a virtual Clifford pair to keep operations local. Building on PTCB, the authors propose a fidelity benchmarking method for gates U with U² = I, and demonstrate how to extract relevant PTM products via exponential fits, yielding a SPAM-robust estimate hatF(Λ) when possible or a direct path to F(Λ) under favorable noise assumptions. Numerical simulations on Toffoli gates validate the approach, showing controllable estimation bias and robustness to SPAM, and illustrating practical pathways toward benchmarking essential non-Clifford components of universal quantum computation.

Abstract

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 $U$ satisfying $U^2=I$. We validate the feasibility of our protocol through numerical simulations applied to Toffoli gates as a concrete example.

Figures (7)

• Figure 1: The schematic diagram of the main contributions of this work. The quantum circuit effectively illustrates the core concept of the PTCB protocol. While the introduction of the Clifford gate $C$ enables access to arbitrary PTM elements of $\tilde{U}$, it simultaneously introduces uncontrolled additional noise. To mitigate this challenge, we incorporate the conjugate gate $C^\dagger$ and combine $C$, $C^\dagger$, and $P_2 P_1^\dagger$ into a single Pauli gate, thereby circumventing the physical implementation of Clifford gates altogether.
• Figure 2: Circuit diagrams for Algorithm \ref{['alg:ptcb']} (3-qubit example). The three circuits are equivalent to each other. (a) The expected effect of the circuit. (b) Group twirling with random Pauli gates. $\Pi_Q$ is also implemented by Eq. \ref{['eq:coefficient']}. (c) Rearrange the gates such that the operation between the two $\tilde{U}$'s remains a Pauli gate. Adjacent Pauli gates (e.g., $P_1 P_0$ and $P_3 P_2^\dagger$) should be compiled into a single Pauli operation.
• Figure 3: Discrepancy between the actual infidelity $F(\Lambda)$ and the estimated infidelity $\hat{F}(\Lambda)$. The right figure is an enlarged version for the region between 0.0246 and 0.0250 within the left figure. For all simulated noise channels, the discrepancy remains below $1 \times 10^{-4}$.
• Figure 4: Discrepancy between the actual infidelity and the infidelity estimated via Eq. \ref{['eq:outer']} for different values of $M$. Results are grouped by actual infidelity to demonstrate protocol performance across different noise levels. As $M$ increases, the estimated infidelity converges toward the actual value. Larger actual infidelity is accompanied by increased estimation variance.
• Figure 5: Discrepancy between $\sqrt{\tilde{\mathcal{U}}_{PQ}\tilde{\mathcal{U}}_{QP}}$ and the square root of the value estimated by PTCB protocol. (a) Impact of repetition count at a fixed SPAM error rate. The label "accurate” denotes direct acquisition of the survival probability without circuit repetition. As $M'$ increases, the estimate converges to the actual value, while the repetition count has negligible impact. (b) Impact of SPAM error rate at a fixed repetition count. As $M'$ increases, the estimate converges to the actual value, while the SPAM error rate has little impact, demonstrating the SPAM-error-free nature of the protocol.