When Clifford benchmarks are sufficient; estimating application performance with scalable proxy circuits

TL;DR

The paper addresses benchmarking large quantum devices by showing that, under the Pauli twirling assumption, Clifford-based proxy circuits can tightly bound the performance of general circuits. It establishes that the diamond distance of any circuit is well approximated by the fidelity of a Cliffordized proxy, and that Clifford fidelities can be estimated robustly against SPAM using two approaches. The authors validate these ideas with simulations and IBM hardware, including a volumetric Cliffordization benchmark up to 65 qubits and comparisons with random circuit sampling. The work provides a practical, verifiable framework for benchmarking near-term quantum processors, linking tractable Clifford simulations to the performance on more complex circuits across scalable regimes.

Abstract

The goal of benchmarking is to determine how far the output of a noisy system is from its ideal behavior; this becomes exceedingly difficult for large quantum systems where classical simulations become intractable. A common approach is to turn to circuits comprised of elements of the Clifford group (e.g., CZ, CNOT, $π$ and $π/2$ gates), which probe quantum behavior but are nevertheless efficient to simulate classically. However, there is some concern that these circuits may overlook error sources that impact the larger Hilbert space. In this manuscript, we show that for a broad class of error models these concerns are unwarranted. In particular, we show that, for error models that admit noise tailoring by Pauli twirling, the diamond norm and fidelity of any generic circuit is well approximated by the fidelities of proxy circuits composed only of Clifford gates. We discuss methods for extracting the fidelities of these Clifford proxy circuits in a manner that is robust to errors in state preparation and measurement and demonstrate these methods in simulation and on IBM Quantum's fleet of deployed heron devices.

Figures (10)

• Figure 1: General layered circuit with prep and measurement. Here each $U$ corresponds to an arbitrary single qubit gate. The ${\mathcal{C}}$ terms denote multiqubit Clifford layers and the ${\mathcal{E}}$ denote error processes. We may marginalize over some subset of the measurement outcomes, and correspondingly may only be sensitive to some subset of the qubit initializations, denoted by the binary vectors $M_j$ and $S_j$ respectively.
• Figure 2: Removing SPAM with a scrambled reference experiment. If we divide the fidelity of the Cliffordized circuit by that of the SPAM reference circuit, we get a SPAM-free estimate of the diamond norm of the target circuit.
• Figure 3: Random brickwork layered circuit. The CZs could easily be substituted with any other 2-qubit entangling Clifford gate.
• Figure 4: Testing the uniformity of Cliffordized circuit infidelities. We selected $K$$n$-qubit test circuits for $n=2$, 3, and 4 (with $K=100$ for $n=2$ and $n=3$, and $K=25$ for $n=4$) and a different Pauli stochastic error model with randomly-selected error rates for each test circuit (details in main text). We then created 500 Cliffordizations of each of those test circuits, and simulated all Cliffordizations under the error model selected for that test circuit. We did this for $K$ test circuits that are disordered (these circuits had independently-sampled random layers) and test circuits that are periodic (these circuits repeated the same pair of randomly-sampled layers many times). We computed the process infidelity $r(\cal C)$ for each Cliffordized circuit, and we computed the mean $\mu_r$ and standard deviation $\sigma_r$ of all 500 Cliffordized circuits corresponding to each test circuit. Here we show a histogram of the coefficients of variation $\sigma_r/\mu_r$ for the Cliffordizations of each test circuit. As predicted by our theory, the variation in the process infidelities $r(\cal C)$ over different Cliffordization of the same test circuit is very small.
• Figure 5: Accuracy of diamond norm predictions from Cliffordization. We tested the accuracy with which the infidelities of Cliffordized circuits can predict the diamond distance error ($d_{\diamond}$) of a target non-Clifford circuit, using the simulated data described in Fig. \ref{['fig:simulations-1']}. We find that the difference between the mean infidelity of the Cliffordized circuits $\mu_r$ and their corresponding target circut's $d_{\diamond}$ is very small, for both classes of target circuit that we simulated: disordered circuits (top) and periodic circuits (bottom). The discrepancy increases as $d_{\diamond}$ increases, as predicted by our theory.