Estimating and decoding coherent errors of QEC experiments with detector error models

TL;DR

This work addresses how to estimate and decode coherent errors in QEC experiments using detector error models (DEMs) learned from syndrome histories. By applying code-capacity, phenomenological, and circuit-level simulations to repetition and surface codes, the authors show that coherent components of noise can be inferred directly from syndrome data and that DEMs capture interference effects and hyperedges not present in Pauli-twirled models. They demonstrate angle estimation from edge statistics (e.g., $p=\sin^2\theta$ and effective doubling to $2\theta$ on boundary edges) and reveal how coherent noise shifts logical-threshold behavior, while decoding with the estimated DEM can reduce logical infidelity without altering the threshold. Overall, the approach eliminates the need for separate device benchmarking for noise characterization and provides a practical pathway to configure decoders that are tuned to actual coherent noise in QEC experiments.

Abstract

Decoders of quantum error correction (QEC) experiments make decisions based on detected errors and the expected rates of error events, which together comprise a detector error model. Here we show that the syndrome history of QEC experiments is sufficient to detect and estimate coherent errors, removing the need for prior device benchmarking experiments. Importantly, our method shows that experimentally determined detector error models work equally well for both stochastic and coherent noise regimes. We model fully-coherent or fully-stochastic noise for repetition and surface codes and for various phenomenological and circuit-level noise scenarios, by employing Majorana and Monte Carlo simulators. We capture the interference of coherent errors, which appears as enhanced or suppressed physical error rates compared to the stochastic case, and also observe hyperedges that do not appear in the corresponding Pauli-twirled models. Finally, we decode the detector error models undergoing coherent noise and find different thresholds compared to detector error models built based on the stochastic noise assumption.

Figures (7)

• Figure 1: Estimating the error angles $\theta$, when data qubits experience a coherent noise channel $e^{-i\theta Z}$. (a) Estimated angles for $d=5$$X$-memory repetition code. The bars within each color correspond to different qubits in the same repetition code. The different colors correspond to a different memory simulation where we set a new value for the rotation angles. (b) Estimated angles for the a distance $d=3$$X$-memory rotated surface code. For boundary edges of the DEM connecting to weight-2 checks, we estimate the cumulative $2\theta$ angle since there is only one detector attached to the two different data qubits, leading to a single boundary edge in the DEM.
• Figure 2: Estimating coherent noise parameters and classical readout errors for a $d=3$$X$-memory rotated surface code and $r=3$ QEC rounds. (a) Relative error in estimating the error rates of time edges, for a uniform readout error rate of $q=0.03$. (b) Estimated angles extracted from the estimated error rates of space-like DEM edges. The bars within each color correspond to a different round index. Each color corresponds to a particular edge in the DEM as color-coded in (c). (c) Coherent noise parameters for the data qubits and DEM structure. To estimate the error rates we used $N=180,000$ shots, and for each shot we corrupted the measurement outcome with probability $q=0.03$ for 100 random realizations.
• Figure 3: Comparing the logical error rate performance under stochastic or coherent noise models for a rotated surface code, when errors are decoded with uniform-weight decoder. (a) Logical error rate as a function of the physical error rate $p=\sin^2\theta$ when data qubits experience $e^{-i\theta Z}$ and measurements experience readout errors with $q=p$. The number of shots used for the decoding is $N=60,000$, and for each one of this shots, we corrupt the measurement outcomes for $100$ different realizations. (b) Logical error rate when data qubits experience stochastic $Z$ errors, and measurements experience readout errors. The number of shots used for the decoding is $N=150,000$.
• Figure 4: Logical error rate as a function of the physical error rate, when data qubits experience $e^{-i\theta Z}$ errors per QEC round, and measurement outcomes are incorrectly recorded to the opposite bit value with probability $q=p=\sin^2\theta$, for an $X$-memory rotated surface code. Our estimated DEM is constructed from $N=200,000$ syndrome data, where for each shot the outcomes are corrupted with probability $q$ for $100$ random realizations. For the decoding, we use a new batch of $N=40,000$ shots, with detection outcomes corrupted again for another $100$ random realizations per shot.
• Figure 5: (a) Distance $d=3$$X$-memory repetition code circuit, where both data and ancilla qubits experience $e^{-i\theta Z}=R_z(2\theta)$ errors at the start of every QEC round. Here we show $r=1$ QEC round. (b) Logical error rate as a function of the physical error rate $p=\sin^2\theta$, for the error model shown in (a) and $r=d$ QEC rounds. The threshold is slightly smaller than $8\%$. The decoding graph is formed based on the estimated error rates. The number of shots used for estimation and decoding is $N=10^6$.