Random Circuit Sampling: Fourier Expansion and Statistics

TL;DR

This work advances the statistical analysis of noisy random circuit sampling by applying Fourier--Walsh expansion to quantify how readout and gate errors attenuate different Fourier degrees, refining the ${\cal F}_{XEB}$ fidelity estimator through degree-specific measures $\Lambda_k$. By developing fast Fourier-based estimators for degree-$k$ contributions and linking them to fidelity statistics ($U$, $V$, ${\rm MLE}$, $\phi_{ro}$), the authors analyze both Google's 2019 quantum supremacy data and a range of simulations (Weber QVM, IBM Fake Guadalupe, and neutral-atom experiments). A two-parameter noise model $s T_{(1-2q)}({\cal P}_C(x))+(1-s)/M$ captures readout and gate-noise effects, revealing that readout noise dominantly suppresses high-degree Fourier coefficients, with gate noise amplifying this decay in some simulators. The framework proves robust across diverse datasets and noise models, offering a scalable approach to characterizing NISQ devices and informing interpretations of quantum-supremacy demonstrations and fidelity assessments. Overall, Fourier analysis provides a principled, quantitative lens to dissect noise in RCS and to compare experimental data with noisy simulations, with potential applicability to a wide spectrum of NISQ experiments and future fault-tolerant benchmarks.

Abstract

Considerable effort in experimental quantum computing is devoted to noisy intermediate scale quantum computers (NISQ computers). Understanding the effect of noise is important for various aspects of this endeavor including notable claims for achieving quantum supremacy and attempts to demonstrate quantum error correcting codes. In this paper we use Fourier methods combined with statistical analysis to study the effect of noise. In particular, we use Fourier analysis to refine the linear cross-entropy fidelity estimator. We use both analytical methods and simulations to study the effect of readout and gate errors, and we use our analysis to study the samples of Google's 2019 quantum supremacy experiment.

Figures (12)

• Figure 1: A histogram of the Fourier--Walsh coefficients $\widehat{P}(S)$ for different Google files.
• Figure 2: Simulation of the effect of various noise models on the degree $k$ Fourier-Walsh contributions $\Lambda_k$ as defined in equation \ref{['e:Lambda']}. The noiseless model is the Google noise model, with $\phi=1$. For all the other models we took $\phi=0.3862$ when $n=12$, and $\phi=0.332$ when $n=14$. The Google model is given in \ref{['e:gnm']}. The symmetric readout model is given in \ref{['e:secondary']} with $q=0.038$. The asymmetric readout model is detailed in RSK22, and in this simulation we took the probability $q_1$ that 1 is read as 0 is 0.055 and the probability $q_2$ that 0 is read as 1 is 0.023. The simulation is based on sampling $N=500,000$ samples from the corresponding model, where the base probabilities are the first Google file, with $n=12$ or $n=14$. The solid black line is given by Equation \ref{['eq:reference_readout']}.
• Figure 3: The decay of degree $k$ Fourier-Walsh contribution coefficients averaged over the ten experimental (full) circuits for the Google experimental data for $n=12,16,18,22,24,26$. The degree-$k$ Fourier contributions $\Lambda_k$ are defined in equation \ref{['e:Lambda']}. The solid black line is based on the symmetric readout noise model given by Equation \ref{['eq:reference_readout']}, with $\phi=0.3862, 0.2828,0.2207, 0.1554,0.1256,0.1024$ for $n=12,16,18,22,24,26.$ Extreme values of $k$ are unstable.
• Figure 4: The decay of degree $k$ Fourier coefficients for simulation with Google’s full noise model of the Google circuits for 12 and 14 qubits. The main decrease in the Fourier contribution is based on the readout errors and it appears that the gate errors contribute an additional effect of a similar nature. The solid black line describes the theoretical effect of readout errors \ref{['eq:reference_readout']} under the assumption that every gate error leads to a uniform random bitstring. (It is based on $q=0.038$, $\phi_g=0.516$ for $n=12$ and $\phi_g= 0.462$ for $n=14$. The values were derived from Table \ref{['t:QVM']} using Equation eq:phi_ro). The solid red line is based on the best fit: For $n=12$, $q=0.053$, and $s=0.627$, for $n=14$, $q=0.047$ and $s=0.532$ and. Extreme values of $k$, are unstable and are omitted. See Section \ref{['s:stab']} for the full data and discussion.
• Figure 5: The effect of gate errors on the degree $k$ Fourier-Walsh contribution $\Lambda_k$. This computation is based on simulations with Cirq of the Google files with 12 qubits with 1-gate depolarization errors on all 1-gates (including those for the calibration). The error rate is adjusted so that the overall fidelity is similar to the Google experimental fidelity. For these simulations the best fit for the parameters $s$ and $q$ in the model given by Equation \ref{['e:noise-model-rho']} was (averaged on 10 simulation files) $s=0.4207$ and $q=0.013$, for $n=14$ we have $s=0.3673$, $q=0.014$. (Extreme values of $k$, are unstable and are omitted.)

Theorems & Definitions (1)

• Proposition 2.1