Fast and Accurate Decoder for the XZZX Code Using Simulated Annealing

TL;DR

This work proposes a simulated annealing (SA) decoder for the XZZX code and suggests that combining SA with the greedy matching initializer is a practical approach toward fault-tolerant quantum computation.

Abstract

The XZZX code is a variant of the surface code tailored to address biased noise in realistic quantum devices. We propose a simulated annealing (SA) decoder for the XZZX code. Our SA decoder is amenable to parallelization because its MCMC updates are simple and local. To initialize SA, we use a recovery configuration produced by our greedy matching decoder. Although $Z$-biased noise is commonly assumed in realistic quantum devices, we instead focus on $Y$-biased noise, where MWPM becomes suboptimal because it neglects correlations induced by $Y$ errors. Our numerical simulations for the code capacity noise model, where only data qubits suffer errors, show that our SA decoder achieves higher accuracy than the MWPM decoder. Furthermore, our SA decoder achieves an accuracy comparable to that of the optimal minimum-energy (MAP-configuration) decoder formulated as an integer programming problem, called the CPLEX decoder. In our greedy matching decoder, we randomize the tie-breaking among equal-weight pairs. This randomness generates a variety of initial configurations for SA, which can lead to faster convergence of our SA decoder. By comparing decoding times of our SA decoder, the CPLEX decoder, and the matrix product state (MPS) decoder, all of which can handle $Y$-biased noise appropriately, we estimate that our SA decoder could be competitive in runtime under an idealized assumption of near-perfect parallel efficiency. These results suggest that combining SA with our greedy matching initializer is a practical approach toward fault-tolerant quantum computation.

Figures (15)

• Figure 1: The XZZX code for the code distance $d=5$: (a) Stabilizer generators $G_f$, showing four-body operators in the bulk and three-body operators at the boundaries. (b) Logical operator $L_Z$. (c) Logical operator $L_X$.
• Figure 2: An example of the error configuration $C$. Circles indicate data qubits with $X$, $Y$, or $Z$ errors; qubits without errors are omitted. The error configuration can be represented as two types of error chains $C_\mathrm{white}$ and $C_\mathrm{gray}$, connecting the white faces and gray faces and drawn with the pink and purple lines on the corresponding decoding graphs $(\mathcal{V}_\mathrm{white}, \mathcal{E}_\mathrm{white})$ and $(\mathcal{V}_\mathrm{gray}, \mathcal{E}_\mathrm{gray})$, respectively. The endpoints of these error chains are captured by the syndrome defects, drawn with the green squares. In both the error chains $C_\mathrm{white}$ and $C_\mathrm{gray}$, the $Z$- and $X$-type Pauli components correspond to operators on data qubits located at the centers of horizontal and vertical edges, respectively. A $Y$ error occurs precisely on a data qubit where both the $Z$- and $X$-type components are present (i.e., where the corresponding horizontal and vertical edges are both occupied).
• Figure 3: Logical error rates under the depolarizing noise for the code distances $d=5,7,9$. Our SA decoder achieves the logical error rates of the CPLEX decoder for all $d$. We employ $N_\mathrm{SA} = 100$, which suffices to attain the logical error rates of the CPLEX decoder. The error bars indicate one standard deviation.
• Figure 4: Logical error rates under the $Y$-biased noise where $p_x : p_y : p_z = 1 : 5 : 1$ for the code distances $d=5,7,9$. Our SA decoder achieves the logical error rates of the CPLEX decoder. We employ $N_\mathrm{SA} = 100$, which suffices to attain the logical error rates of the CPLEX decoder. The error bars indicate one standard deviation.
• Figure 5: Logical error rates under the $Y$-biased noise where $p_x : p_y : p_z = 1 : 10 : 1$ for the code distances $d=5,7,9$. Our SA decoder cannot achieve the logical error rate of the CPLEX decoder for $d=9$, even by $N_\mathrm{SA} = 100$. The error bars indicate one standard deviation.