Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A blockBP decoder for the surface code

Aviad Kaufmann, Itai Arad

TL;DR

This work introduces blockBP, a belief-propagation–based contraction method for tensor-network representations of degenerate quantum maximal likelihood decoding (DQMLD) in the surface code. By partitioning the 2D tensor network into blocks and using boundary MPS messages with bond dimension χ, blockBP delivers parallelizable, near real-time capable DQMLD decoding. Numerical results in the code-capacity model show substantial improvements over the MWPM decoder for various code distances, with larger block sizes extending the regime of improved performance, and near-term potential for real-time hardware integration. The approach also lays groundwork for extensions to circuit-level noise and other 2D codes, and invites further enhancements via scheduling strategies or machine-learning–informed BP refinements.

Abstract

We present a new decoder for the surface code, which combines the accuracy of the tensor-network decoders with the efficiency and parallelism of the belief-propagation algorithm. Our main idea is to replace the expensive tensor-network contraction step in the tensor-network decoders with the blockBP algorithm - a recent approximate contraction algorithm, based on belief propagation. Our decoder is therefore a belief-propagation decoder that works in the degenerate maximal likelihood decoding framework. Unlike conventional tensor-network decoders, our algorithm can run efficiently in parallel, and may therefore be suitable for real-time decoding. We numerically test our decoder and show that for a large range of lattice sizes and noise levels it delivers a logical error probability that outperforms the Minimal-Weight-Perfect-Matching (MWPM) decoder, sometimes by more than an order of magnitude.

A blockBP decoder for the surface code

TL;DR

This work introduces blockBP, a belief-propagation–based contraction method for tensor-network representations of degenerate quantum maximal likelihood decoding (DQMLD) in the surface code. By partitioning the 2D tensor network into blocks and using boundary MPS messages with bond dimension χ, blockBP delivers parallelizable, near real-time capable DQMLD decoding. Numerical results in the code-capacity model show substantial improvements over the MWPM decoder for various code distances, with larger block sizes extending the regime of improved performance, and near-term potential for real-time hardware integration. The approach also lays groundwork for extensions to circuit-level noise and other 2D codes, and invites further enhancements via scheduling strategies or machine-learning–informed BP refinements.

Abstract

We present a new decoder for the surface code, which combines the accuracy of the tensor-network decoders with the efficiency and parallelism of the belief-propagation algorithm. Our main idea is to replace the expensive tensor-network contraction step in the tensor-network decoders with the blockBP algorithm - a recent approximate contraction algorithm, based on belief propagation. Our decoder is therefore a belief-propagation decoder that works in the degenerate maximal likelihood decoding framework. Unlike conventional tensor-network decoders, our algorithm can run efficiently in parallel, and may therefore be suitable for real-time decoding. We numerically test our decoder and show that for a large range of lattice sizes and noise levels it delivers a logical error probability that outperforms the Minimal-Weight-Perfect-Matching (MWPM) decoder, sometimes by more than an order of magnitude.
Paper Structure (15 sections, 1 theorem, 21 equations, 7 figures, 1 algorithm)

This paper contains 15 sections, 1 theorem, 21 equations, 7 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Theoretical Background
  3. Maximal likelihood and degenerate maximal likelihood decoding in stabilizer codes
  4. The surface code
  5. The blockBP decoder for the surface code
  6. The Tensor Network based DQMLD decoder
  7. BP decoders for quantum codes
  8. Using BP for tensor network contraction
  9. The blockBP algorithm
  10. Description of the blockBP decoder
  11. Numerical simulations
  12. Fast convergence to the right coset.
  13. Faster convergence for $\epsilon\to 0$.
  14. Summary and conclusions
  15. Proof of Lemma \ref{['lem:Bethe']}

Key Result

Lemma 3.1

Let $\mathcal{T}$ be a TN defined on a graph $G=(V,E)$, and let $m_{u\to v}(x_e)$ be the set of converged BP messages. Let $\mathcal{F}_{Bethe}(\mathcal{T})$ the Bethe free-energy of the BP fixed point. Then where $\{\hat{m}_{u\to v}\}$ are a rescaling of the fixed-point messages, normalized such that

Figures (7)

  • Figure 1: (a) The lattice structure of a $d=3$ surface code. The qubits sit on the edges of the square lattice with $d^2$ horizontal edges and $(d-1)^2$ vertical edges. Given an Pauli operator $f_{\underline{s}}$ that corresponds to the syndrome ${\underline{s}}$ and a logical operator $\bar{L}\in \{\bar{I}, \bar{X}, \bar{Y}, \bar{Z}\}$, the lattice is mapped to a $(2d-1)\times(2d-1)$ square tensor network $T_{\underline{s}}(\bar{L})$, whose contraction value gives the coset probability $\pi(f_{\underline{s}} \bar{L}\mathcal{G})$. See Sec. \ref{['sec:surface-to-TN']}. (b) The check operators (stabilizers) of the surface code: Plaquette operators $B_p$ are the product of Pauli $X$ operators sitting on the edges surrounding a plaquette $p$. Site operators $A_u$ are the product of Pauli $Z$ operators on the qubits that sit on the edges adjacent to a site $u$.
  • Figure 2: Sketch of the BP algorithm for tensor networks. (a) a tensor network with 5 tensors. Each neighboring tensors exchange messages between them. (b) The outgoing message from $T_1$ to $T_3$ is obtained by contracting $T_1$ with all the incoming messages to $T_1$ from the previous round except for the incoming message from $T_3$.
  • Figure 3: Sketch of the blockBP algorithm for 2D tensor networks. (a) The 2D TN is partitioned into disjoint blocks of tensors. (b) The update rule for an outgoing message is the same as in regular BP, only that now the incoming messages are MPSs, and the contraction is done approximately using boundary MPS method, which results in an outgoing MPS message.
  • Figure 4: The dependence of the logical probability error $P_L$ on the depolarization noise strength for $d=5,9, \ldots, 25$, as given by the blockBP, bMPS and MWPM decoders.
  • Figure 5: The logical error probability $P_L$ as a function of the code distance $d$ for a physical depolarization rates of $\epsilon=0.08, 0.09, 0.10, 0.11$ and for the different decoders.
  • ...and 2 more figures

Theorems & Definitions (1)

  • Lemma 3.1