Decay Rates in Interleaved Benchmarking with Single-Qubit References

Abstract

Cross-entropy benchmarking (XEB) with single-qubit reference sequences is widely used to characterize multi-qubit gates in large-scale quantum processors, despite the lack of a rigorous theoretical justification. Here we show that the commonly employed additive single-qubit errors approximation underlying this approach breaks down and leads to a systematic overestimation of gate fidelities. We derive an analytical expression for the joint decay of simultaneous single-qubit reference sequences and introduce a refined expression for the interleaved gate fidelity estimation. Experiments on a superconducting quantum processor validate the theory and demonstrate that fidelities obtained using XEB with single-qubit references agree with those extracted from standard interleaved randomized benchmarking (IRB), while achieving higher precision due to reduced reference-sequence errors. Our results establish theoretical foundation for the single-qubit-based XEB and show that, with appropriate post-processing, it enables a reliable and robust approach for entangling gates benchmarking without the need for multi-qubit Clifford reference sequences.

Figures (4)

• Figure 1: Simulation of the XEB experiment. Teal dots show the data obtained from sequences composed of the two-qubit Clifford group, while red triangles correspond to simultaneous single-qubit Clifford gates, both under the same local noise model. The simulated points closely follow the theoretical predictions for $F_\text{multi}$ [Eq. (\ref{['eq:error_clif']})] and $F_\text{single}$ [Eq. (\ref{['eq:error_sq']})], respectively, demonstrating that the choice of the averaging gate set affects the decay even for identical noise.
• Figure 2: Simultaneous single-qubit cross-entropy benchmarking for (a) two and (b) three qubits. Teal points show the experimentally measured fidelities, error bars are obtained using a bootstrap technique. The teal solid curve represents the theoretical dependence calculated via Eq. (\ref{['eq:error_sq']}) using depolarizing parameters $p_i$ extracted from individual single-qubit benchmarking. The dashed red curve corresponds to the additive model $F=(1-\sum_i e_i)^m$, illustrating the substantial deviation from the experimentally observed decay.
• Figure 3: Cumulative distributions of output bitstring probabilities generated by randomly sampled simultaneous single-qubit and two-qubit arbitrary (a) unitary and (b) Clifford operations. The dashed curves in both plots correspond to depth-4 single-qubit reference sequences with an interleaved CZ gate.
• Figure 4: Comparison of Clifford-based standard interleaved randomized benchmarking (IRB) with a multi-qubit reference sequence and cross-entropy benchmarking (XEB) with a single-qubit reference. The interleaved CZ gate fidelity calculated using the refined formula (\ref{['eq:int_1Q_sim']}) is consistent with that obtained from the IRB experiment, while achieving higher accuracy due to reduced errors in the reference sequence. Uncertainties are estimated using bootstrap resampling.