Simulation-Free Fidelity Estimation via Quantum Output Order Statistics

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.

Figures (6)

• Figure 1: Histogram of bitstring probabilities from a single Haar-random quantum state of $N=6$ qubits ($D=64$), compared to the finite-$D$ Porter-Thomas distribution (dashed line). Fluctuations resemble those of optical speckle, with each realization characterized by a unique fingerprint of ordered bitstring probabilities $p_k$, shown in the bottom panel.
• Figure 2: Upper panel: Decomposition of Porter Thomas distribution for $N=4$ qubits ($D=16$) into its ordered statistical components. Distributions $P_k(x,N)$ transition from Gumbel-like to Gaussian and approximately exponential profiles as the order increases from the largest ($k=1$) to the smallest ($k=16$) probability. Middle panel: Ideal 'fingerprint' of ordered bitstring probabilities $p_k$, constructed from the first moments of the order statistics Eq. \ref{['eq:order_statistics']}, $\langle p_k \rangle = \frac{1}{D} [\psi(D+1) - \psi(k)]$, where $\psi(k)$ is the digamma function. Notice the characteristic spacing between successive values, $\langle p_k\rangle - \langle p_{k-1} \rangle = 1/k$, reflecting the logarithmic structure of the ranked probabilities. Bottom panel: The same fingerprint in the presence of homogeneous depolarizing noise Eq. \ref{['eq:noise_model']} with fidelity $f=0.5$, similar to that reported in the $12$-qubit Sycamore experiment Arute2019. Probabilities larger than the uniform value $1/D$ (red) are compressed by $f<1$, while those smaller (blue) are inflated toward it.
• Figure 3: Illustrative example of the likelihood function $\Lambda(f)$ vs. fidelity $f$ for a single circuit realization ($N=12$) from Google's Sycamore experiment, evaluated at different ranks $k-1,3,5,12,20$. The approximately Gaussian shape near the peak, the narrowing of the distribution with increasing $k$, and the rank-consistent location of the maxima exemplify generic features of the likelihood function across system sizes. These features underpin the rank-averaging strategy used in our estimator and hold more generally beyond the $N=12$ case shown here.
• Figure 4: Number of shots $S$ required for system size $N$ to correctly estimate the fidelity $f = 0.10$ within relative error thresholds $\varepsilon_{\mathrm{rel}}=0.05,0.1,0.2,0.4$. The dashed black line shows the reference scaling $S_N = 2^N/N$. Results are averaged over $M = 1000$ realizations ($M=2000$ for $\varepsilon_{\mathrm{rel}}=0.4$).
• Figure 5: Number of shots $S$ required as a function of system size $N$ to estimate fidelity within a fixed relative error threshold $\varepsilon_{\mathrm{rel}} = 0.10$, for different fidelities $f= 0.1, 0.3, 0.5$. The dashed black line indicates the naive scaling $S_N = 2^N/N$. Results are averaged over $M = 1000$ realizations.