Easier randomizing gates provide more accurate fidelity estimation

TL;DR

This paper tackles the problem that interleaved benchmarking using multi-qubit Clifford randomization can yield highly biased or unphysical estimates of an interleaved gate’s fidelity in the presence of coherent errors. It introduces a theoretical framework to quantify systematic errors, uses unitarity-based XRB bounds to tighten these estimates, and demonstrates that cycle benchmarking with Pauli (or Local Clifford) randomization dramatically improves accuracy and data efficiency. The authors provide numerical simulations and experimental results across three platforms, showing substantial reductions in bias and enabling data reuse through CER-compatible analyses. The work has practical impact by guiding the choice of randomization groups for fast, reliable gate calibration and benchmarking in heterogeneous quantum hardware, with implications for scalable quantum computing calibration pipelines.

Abstract

Accurate benchmarking of quantum gates is crucial for understanding and enhancing the performance of quantum hardware. A standard method for this is interleaved benchmarking, a technique which estimates the error on an interleaved target gate by comparing cumulative error rates of randomized sequences implemented with the interleaved gate and without it. In this work, we show both numerically and experimentally that the standard approach of interleaved randomized benchmarking (IRB), which uses the multi-qubit Clifford group for randomization, can produce highly inaccurate and even physically impossible estimates for the error on the interleaved gate in the presence of coherent errors. Fortunately we also show that interleaved benchmarking performed with cycle benchmarking, which randomizes with single qubit Pauli gates, provides dramatically reduced systematic uncertainty relative to standard IRB, and further provides as host of additional benefits including data reusability. We support our conclusions with a theoretical framework for bounding systematic errors, extensive numerical results comparing a range of interleaved protocols under fixed resource costs, and experimental demonstrations on three quantum computing platforms.

Figures (7)

• Figure 1: Circuits for interleaved protocol pairs for each of four different protocols with distinct randomizing/twirling groups which we label: Haar, Clifford, Local Clifford and Pauli; see also their descriptions in Table \ref{['tab:TwirlingGroups']}. In each pair, the interleaved and reference protocols are denoted by the twirling group name followed by ($\mathcal{G}$) and by ($\mathcal{I}$), respectively, which specifies where the gate $\mathcal{G}$ or no gate (viz $\mathcal{I}$) is interleaved in between the twirling gates. Please note that the notation $(\mathcal{I})$ does not mean that there is an identity gate or delay inserted in lieu of the interleaved gate, but denotes that the reference protocol is identical to the interleaved protocol except that it has no interleaved gate. For Haar, the twirling gates $\mathcal{U}_i$ are drawn uniformly at random from $\mathbb{SU}(4)$ according to the Haar measure and $\mathcal{U}_{m+1}$ denotes the correction gate $\mathbb{SU}(4)$ required in motion reversal circuits per the original RB protocol Emerson_2005. For Clifford, the twirling gates $\mathcal{C}_i$ are drawn at random from $\mathbb{C}_2$ where $\mathbb{C}_2$ is the 2-qubit Clifford group and $\mathcal{C}_{m+1}$ is the correction gate PhysRevLett.106.180504. For Local Clifford, twirling gates are drawn uniformly at random from the tensor product of 1-qubit Clifford groups $\mathbb{C}_1^{\otimes2}$ with $\mathcal{C}_{i,j}\in \mathbb{C}_1$ at time step $i$ on qubit $j$, $\mathcal{C}_{m+1,j}$ are the single-qubit correction gates and $\mathcal{C}_{m+1}$ is the 2-qubit correction gate required for motion reversal of the interleaved sequence PhysRevLett.123.030503Polloreno_2025mckay2311benchmarking. For Pauli, twirling gates are drawn uniformly at random from the tensor product of 1-qubit Pauli group $\mathbb{P}_1^{\otimes2}$ with $\mathcal{P}_{i,j}\in \mathbb{P}_1$ at time step $i$ on qubit $j$, where in these circuits $\mathcal{C}_{P,j}$ are local Clifford operators are used only for generating the state preparation and measurement in the +1 eigen state of a random Pauli $P$, which is the cycle benchmarking scheme proposed in Erhard_2019.
• Figure 2: Comparison of systematic uncertainty as a function of the different twirling groups defined in Table \ref{['tab:TwirlingGroups']}, showing that Pauli and Local Clifford twirling groups have the smallest systematic uncertainties, while keeping the number of shots fixed. The error model comprises coherent error on the two-qubit gates of the form $e^{i\theta_2 ZZ}$ with $\theta_2=10^{\circ}$ and coherent error on the single-qubit gates of the form $e^{i\theta_1 Z}$ with $\theta_1=1^{\circ}$. The theoretical infidelity is the obtained using Eqn. \ref{['eq:abstract infidelity']} under a naive (identity) choice of gauge. Even though Local Clifford group is larger than Pauli group, due to our chosen decomposition and error model on single-qubit gates as detailed in \ref{['subsection:numerical results']}, both have almost same error magnitude and provides almost the same accuracy in the estimate.
• Figure 3: Numerical plot showing the size of the systematic uncertainty for varying average single-qubit gate error rate. The error model is the same as the one used in Fig. \ref{['fig:sys uncertainty as function of twirling group']} with $\theta_2$ fixed at $10^{\circ}$ and varying $\theta_1$ so as to vary the average single-qubit gate error rate along the x-axis. The other simulation parameters are the same as Fig. \ref{['fig:sys uncertainty as function of twirling group']} which is a typical instance of this plot.
• Figure 4: Numerical plot highlighting the importance of systematic uncertainties for a natural (overrotation) coherent error model, where we see that the theoretical infidelity value is well outside the statistical bar of the estimates of the protocols such as Haar and Clifford, but is within the systematic bound. This figures also highlights how interference of errors can lead to negative (unphysical) estimate of process infidelity, as is observed for the Haar protocol, an effect we also see experimentally as described in the next section. The overrotation error model is of the form $U^{1+\delta}$ where $U$ is the target unitary, with $\delta=0.05$ for two-qubit gates and $\delta=0.01$ for single-qubit gates. Other simulation parameters are the same as in Fig. \ref{['fig:sys uncertainty as function of twirling group']}. The triangles beside each estimate are the infidelities computed using the self-consistent gauge (SCG) for each twirling group; we see that the SCG yields an almost negligible difference to the naive "error on the gate" method for computing the theoretical infidelity. See Appendix \ref{['Optimal gauge']} for more details on how to construct the self-consistent gauge.
• Figure 5: Numerical plot showing how coherent errors between random twirling gates and fixed hard gate can constructively and destructively interfere to give entirely different infidelity estimates using Clifford protocols. For the destructive interference shown by yellow open circle we use the coherent error model of the form $e^{i\theta YY}$ and for constructive interference shown by yellow closed circle we use $e^{-i\theta YY}$ on the random clifford gates and $e^{i \theta XZ }$ on the interleaved CNOT gates with $\theta=10\degree$. The process infidelity of the reference experiments corresponding to both error models of the form $e^{i\theta YY}$ and $e^{-i\theta YY}$ are same as it only depends on the magnitude of $\theta$. Notice that in the destructive interference case the inferred interleaved CNOT is negative (unphysical) with the lower and upper systematic bound collapsing to a single value using Eqn. \ref{['eq:systematic-bound']}. We also observe similar unphysical estimate due to interference of coherent errors in experimental data, as shown in Fig. \ref{['fig:expeiments']}.