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arXiv:2510.13026

Simulation-Free Fidelity Estimation via Quantum Output Order Statistics

Tobias Micklitz

Published Oct 14, 2025• 0 citations
arXivHTMLPDF

Abstract

We introduce a simulation-free method to estimate the fidelity of large quantum circuits based on the order statistics of measured output probabilities from highly entangled, chaotic states. The approach requires only the highest-probability output bitstrings -- the most frequently observed measurement outcomes -- and builds on exact analytical results for the order statistics of Haar-random quantum states derived here. Analyzing their modification under depolarizing noise, we propose a scalable fidelity estimator, validated on Google's 12-qubit Sycamore experiment and further supported by numerical simulations. We demonstrate its practicality for intermediate-scale quantum circuits, where cross-entropy benchmarking is costly and direct fidelity estimation is difficult.

DOI
10.48550/arxiv.2510.13026

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We introduce a simulation-free method to estimate the fidelity of large quantum circuits based on the order statistics of measured output probabilities from highly entangled, chaotic states. The approach requires only the highest-probability output bitstrings -- the most frequently observed measurement outcomes -- and builds on exact analytical results for the order statistics of Haar-random quantum states derived here. Analyzing their modification under depolarizing noise, we propose a scalable fidelity estimator, validated on Google's 12-qubit Sycamore experiment and further supported by numerical simulations. We demonstrate its practicality for intermediate-scale quantum circuits, where cross-entropy benchmarking is costly and direct fidelity estimation is difficult.
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We study the problem of generating independent samples from the output distribution of Google's Sycamore quantum circuits with a target fidelity, which is believed to be beyond the reach of classical supercomputers and has been used to demonstrate quantum supremacy. We propose a method to classically solve this problem by contracting the corresponding tensor network just once, and is massively more efficient than existing methods in generating a large number of uncorrelated samples with a target fidelity. For the Sycamore quantum supremacy circuit with 53 qubits and 20 cycles, we have generated 1×10^{6} uncorrelated bitstrings s which are sampled from a distribution P[over ^](s)=|ψ[over ^](s)|^{2}, where the approximate state ψ[over ^] has fidelity F≈0.0037. The whole computation has cost about 15 h on a computational cluster with 512 GPUs. The obtained 1×10^{6} samples, the contraction code and contraction order are made public. If our algorithm could be implemented with high efficiency on a modern supercomputer with ExaFLOPS performance, we estimate that ideally, the simulation would cost a few dozens of seconds, which is faster than Google's quantum hardware.
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Efficient fidelity estimation of multiqubit quantum states is crucial to many applications in quantum information processing. However, to estimate the infidelity εεε with multiplicative precision, conventional estimation protocols require (order) 1/ε21/ε^21/ε2 different circuits in addition to 1/ε21/ε^21/ε2 samples, which is quite resource-intensive for high-precision fidelity estimation. Here we introduce an efficient estimation protocol by virtue of common randomized measurements (CRM) integrated with shadow estimation based on the Clifford group, which only requires 1/ε1/ε1/ε circuits. Moreover, in many scenarios of practical interest, in the presence of depolarizing or Pauli noise for example, our protocol only requires a constant number of circuits, irrespective of the infidelity εεε and the qubit number. For large and intermediate quantum systems, quite often one circuit is already sufficient. In the course of study, we clarify the performance of CRM shadow estimation based on the Clifford group and 4-designs and highlight its advantages over standard and thrifty shadow estimation.
We propose a general tensor network method for simulating quantum circuits. The method is massively more efficient in computing a large number of correlated bitstring amplitudes and probabilities than existing methods. As an application, we study the sampling problem of Google's Sycamore circuits, which are believed to be beyond the reach of classical supercomputers and have been used to demonstrate quantum supremacy. Using our method, employing a small computational cluster containing 60 graphical processing units (GPUs), we have generated one million correlated bitstrings with some entries fixed, from the Sycamore circuit with 53 qubits and 20 cycles, with linear cross-entropy benchmark (XEB) fidelity equals 0.739, which is much higher than those in Google's quantum supremacy experiments.