Bayesian inference of general noise-model parameters from the syndrome statistics of surface codes

TL;DR

The paper tackles online estimation of general noise models for the surface code from syndrome statistics. It couples a tensor-network surface-code simulator with Bayesian inference, employing Markov chain Monte Carlo (MCMC) for stationary noise and sequential Monte Carlo (SMC) for time-varying noise to estimate noise parameters ${\\boldsymbol{\\alpha}}$ from observed syndrome data $\\mathbf{m}$, with $p(\\boldsymbol{\\alpha}|\\mathbf{m}) \\\propto p(\\mathbf{m}|\\boldsymbol{\\alpha})p(\\boldsymbol{\\alpha})$. It demonstrates parameter recovery across several models (one- and two-parameter, uniform, nonuniform) and tracks time-varying amplitude-damping noise, showing improved decoder performance when the TN-based noise estimates are used, though some parameters (e.g., in generalized amplitude damping) remain difficult to estimate. The work highlights the potential of online, general-noise estimation to enhance decoder accuracy without extra quantum overhead and lays groundwork for extending to phenomenological, circuit-level, and non-Markovian noise in fault-tolerant quantum computing.

Abstract

The performance of error correction in the surface code can be enhanced by leveraging the knowledge of the noise model for physical qubits. To provide accurate noise information to the decoder in parallel with quantum computation, an adaptive estimation of the noise model based on syndrome measurement statistics is an effective approach. While noise model estimation based on syndrome measurement statistics is well-established for Pauli noise, it remains unexplored for more complex and realistic scenarios such as amplitude damping which cannot be represented as a Pauli channel. In this paper, we propose Bayesian inference methods for general noise models, integrating a tensor network simulator of surface code, which can efficiently simulate various noise models, with Monte Carlo sampling techniques. For stationary noise, we propose a method based on the Markov chain Monte Carlo. For time-varying noise, which is a more realistic scenario, we introduce another method based on the sequential Monte Carlo. We present numerical results of applying our proposed methods to various noise models, such as static, time-varying, and nonuniform cases, and evaluate their performance in detail.

Figures (15)

• Figure 1: The conceptual figure of key ideas in our work. We propose a novel noise model estimation method based on syndrome measurement results using the TN and Monte Carlo methods. The proposed method can be applied to a broader range of noise models than just the Pauli noise models without any additional quantum overhead.
• Figure 2: Rotated surface code with $d=5$. The number of physical qubits and stabilizer generators is 25 and 24, respectively. This figure only shows the physical qubits but not the qubits for syndrome measurements.
• Figure 3: (a) TN representation of projectors $\Pi_{\pm g}=(I\pm g)/2$. (b) TN representation of $d=5$ initial code state $\ket{0}_L$. It can be made from $\ket{0}^{\otimes 25}$ by operating the $X$ stabilizer projectors. It is because $\ket{0}^{\otimes 25}$ is already stabilized by the products of $Z$. (c) A conceptual picture of the TN diagram of the likelihood $p(\bm{m}|\bm{\alpha})$ calculation of $d=3$ surface code.
• Figure 4: Conceptual diagram of updates of weighted particles for one cycle results in SMC.
• Figure 5: Results of estimation for the one-parameter uniform noise models: (a) amplitude damping, (b) phase damping, and (c) systematic rotation.