A matching decoder for bivariate bicycle codes

TL;DR

This work adopts the minimum-weight perfect matching algorithm, a subroutine invaluable to decoding topological codes, to decode bivariate bicycle codes, and proposes a method to rapidly find a correction using matching on code symmetries.

Abstract

The discovery of new quantum error-correcting codes that encode several logical qubits into relatively few physical qubits motivates the development of efficient and accurate methods of decoding these systems. Here, we adopt the minimum-weight perfect matching algorithm, a subroutine invaluable to decoding topological codes, to decode bivariate bicycle codes. Using the equivalence of bivariate bicycle codes to copies of the toric code, we propose a method we call the 'cylinder trick' to rapidly find a correction using matching on code symmetries. We benchmark our decoder on the gross code family, cyclic hypergraph-product codes, generalized toric codes, and recently proposed directional codes, demonstrating the general applicability of our protocol. For a subset of these codes, we find that our decoder can be significantly improved by augmenting matching with strategies including belief propagation and 'over-matching', thus achieving performance competitive with state-of-the-art approaches.

Figures (11)

• Figure 1: Evaluating decoder performance. The panels show logical error rates versus physical error rates for the bivariate bicycle codes of \ref{['table:codes']} under code-capacity bit-flip noise for different decoders. We develop matching-based 'symatch' and 'simplex-symatch' decoders (blue and red). For codes on the left, our base strategies in blue tend to perform well. For codes on the right, bare symatch and simplex-symatch underperform; we find that augmentation by belief-propagation (BP) significantly improves logical error rates, as shown by the red curves. For all codes, our best decoders are competitive with state-of-the-art methods (green and yellow). All BP subroutines use the min-sum method for a maximum of 1000 iterations.
• Figure 2: A square lattice with periodic boundary conditions, $(M,N, \alpha) = (12,6,0)$. (Top left) Each site $s$ includes a face $f$, a vertex $v$, and two qubits $L$ and $R$. For every site, qubit $L$ ($R$) lies on the horizontal (vertical) edge above (to the right of) the face of the respective site. (Center and right) A Pauli-$Z$ (Pauli-$X$) stabilizer for the gross code is shown in blue (red), supported on a vertex $v$ (face $f$).
• Figure 3: Single Pauli-$X$ and Pauli-$Z$ errors on horizontal $L$ qubits and vertical $R$ qubits, together with their respective syndromes for the gross code. The syndromes of errors appear in a triangular configuration.
• Figure 4: The vertex stabilizers of a BB144 symmetry are marked by red, green and blue dots. All errors violate an even parity of symmetry stabilizers, as shown by $L$ and $R$ errors. The symmetry can be decomposed into right subsymmetries, which are given different colors. The right error violates two vertices of the blue subsymmetry. A left subsymmetry is also shown by dots enclosed by circles.
• Figure 5: The unfrustrated $R_x \times R_y$ unit cell (green) superimposed onto the original lattice (dotted) with $T_x$ and $T_y$ translation generators. A ribbon operator (blue) creates a pair of syndromes (red) that are related by an unfrustrated translation $T_x^{2 R_x}T_y^{R_y}$.