Simulating general noise nearly as cheaply as Pauli noise

TL;DR

The work presents stratified importance sampling to extend stabilizer (Clifford) simulations to general noise beyond Pauli, including coherent errors. By decomposing noise into stabilizer channels and stratifying configurations by the number of faults, the method achieves substantial variance reduction and enables efficient estimation of circuit performance metrics $F$ under weak, real-device noise. Demonstrations on Steane code, rotated surface codes up to distance $d=15$, and a Clifford-noise-reduction protocol show accurate results with practical runtimes, highlighting that nonunitary noise costs are close to Pauli-noise costs while coherent noise remains more demanding but still feasible. This provides a practical tool for beyond-Pauli error analysis of quantum error-correcting codes and Clifford-based protocols, with direct implications for evaluating real-device performance and fault-tolerance thresholds.

Abstract

Stabilizer simulation of Clifford quantum circuits - error-correction circuits, Clifford subroutines, etc. - on classical computers has played a central role in our understanding of circuit performance. The stabilizer description, however, restricts the accessible noise one can incorporate into the simulation to Pauli-type noise. More general noise, including coherent errors, may have more severe impact on circuit performance than Pauli noise; yet, such general noise have been difficult to access, much less investigate fully, in numerical simulations. Here, through the use of stratified importance sampling, we show how general noise can be simulated within the stabilizer formalism in reasonable time, with non-unitary noise being nearly as cheap as Pauli noise. Unitary (or coherent) noise can require an order of magnitude more time for the simulation, but nevertheless completes in very reasonable times, a drastic improvement over past approaches that typically fail to converge altogether. Our work thus enables detailed beyond-Pauli understanding of circuit performance in the presence of real device noise, which is rarely Pauli in nature. Among other examples, we present direct simulation results for the performance of the popular rotated planar surface codes under circuit-level general noise, previously available only in limited situations and/or through mappings to efficiently simulatable physical models.

Figures (7)

• Figure 1: The 7-qubit Steane code under depolarizing, amplitude-damping, and $Z$-rotation coherent [Unitary(Z)] noise. Leftmost plot: Logical infidelity computed using our stratified sampling approach, together with an exact state-vector simulation for the $Z$-rotation noise. Right histograms and plots: The histograms give $P_\gamma(k)$ for the $k$ values with significant probability of occurrence for $\epsilon_\textrm{ref}$; the lines give the logical fidelity $F_k$ value for given $k$, with the error bars and number of samples (numerical values at the top of each histogram bar) to achieve a 10% confidence interval around the estimated $F$ value with a 99% confidence level.
• Figure 2: Rotated planar surface code. The checkerboard is a surface-code patch for $d=5$, encoding one logical qubit. Red(Blue) squares (or triangles) are for the $X(Z)$ stabilizers. The logical $X(Z)$ operator is the $X(Z)$ operator on every physical qubit along a single column(row), as indicated by the red(blue) vertical(horizontal) line. The circuit shown measures the $XXXX$ stabilizer for detecting $Z$ errors, with the qubit labels 1--4 referring to the four data qubits around the ancillary qubit. Noise $\mathcal{E}$ is inserted at the indicated locations. The corresponding circuit measuring the $ZZZZ$ stabilizer for detecting $X$ errors, done synchronously, is identical, except that the target and control qubits are swapped for CNOTs and the two Hadamard $H$ gates on the ancillary qubit are replaced by identity gates; the qubit labels 1--4 also refer to the data qubits around the $Z$-ancillary qubit in a different order. Data qubits in the bulk participate in CNOT gates in one of these circuits in every CNOT time step; those on the boundaries have additional identity gates---with insertions of $\mathcal{E}$---if they participate in no CNOT gate in that time step.
• Figure 3: Logical infidelity versus physical infidelity for different noise channels, for the rotated surface code of code distances $d=3,5,7,9,11,15$, involving, respectively, $2d^2-1=17,49, 97,161, 241,449$ qubits. The lines with the smallest gradients are for $d=3$; the line gradients increase monotonically with $d$. The different colored groups of lines correspond to different noise channels: depolarizing noise (red), amplitude-damping noise(green), worst/best (i.e., largest/smallest logical infidelity for given physical infidelity) random non-unitary noise (purple/blue), and worst/best random unitary noise (cyan/orange). The error bars, corresponding to 10% confidence interval with a 99% confidence level, are shown only for the smallest and largest $d$ values; intermediate $d$ values have similar error-bar sizes.
• Figure 4: Reference ranges for rejection sampling, for $>95\%$ acceptance probability, across the physical channel infidelity range of interest. Each horizontal bar corresponds to a single noise channel: unitary channels in the top half (with unitary-$Z$, $Y$, and $X$ channels as the bottom three); nonunitary channels in the bottom half (with amplitude-damping and depolarizing channels as the bottom two). Each colored segment uses a single reference distribution for rejection sampling. The channels marked Unitary($\ell$), $\ell=0,1,\ldots 9$ are randomly generated, as are the nonunitary channels marked Nonunitary$(J|\ell)$, with $J$ Kraus operators; see main text for the channel generation procedure.
• Figure 5: The CliNR protocol of Ref. delfosse2025 under depolarizing, amplitude-damping, and unitary-$Z$ rotation noise, simulated using stratified sampling. For each noise, we plot the logical infidelity for the target circuit $\textsc{Cir}$ against the number of qubits $n$ involved in $\textsc{Cir}$, for three scenarios: no fault-detection protocol ('None'), CliNR with $w_G\leq 2$ ['CliNR(2)'], and CliNR with $w_G\leq 4$ ['CliNR(4)']. For all noise channels, the physical channel (worst-case) infidelity is $10^{-4}$.