Generalized matching decoders for 2D topological translationally-invariant codes

TL;DR

This work develops a graph-matching approach to decoding general TTI codes and indicates that graph-matching decoders are a viable approach to decoding BB codes and other TTI codes.

Abstract

Two-dimensional topological translationally-invariant (TTI) quantum codes, such as the toric code (TC) and bivariate bicycle (BB) codes, are promising candidates for fault-tolerant quantum computation. For such codes to be practically relevant, their decoders must successfully correct the most likely errors while remaining computationally efficient. For the TC, graph-matching decoders satisfy both requirements and, additionally, admit provable performance guarantees. Given the equivalence between TTI codes and (multiple copies of) the TC, one may then ask whether TTI codes also admit analogous graph-matching decoders. In this work, we develop a graph-matching approach to decoding general TTI codes. Intuitively, our approach coarse-grains the TTI code to obtain an effective description of the syndrome in terms of TC excitations, which can then be removed using graph-matching techniques. We prove that our decoders correct errors of weight up to a constant fraction of the code distance and achieve non-zero code-capacity thresholds. We further numerically study a variant optimized for practically relevant BB codes and observe performance comparable to that of the belief propagation with ordered statistics decoder. Our results indicate that graph-matching decoders are a viable approach to decoding BB codes and other TTI codes.

Figures (13)

• Figure 1: Decoding the 144-qubit BB code from bravyi2024high under i.i.d bit-flip noise using BP, BP-OSD, and the small-code-size adaptation of the cell-matching decoder. Logical error rates are determined through Monte Carlo sampling of error configurations at different physical error rates followed by decoding. For a data point with logical error rate of order $10^{-k}$, $10^{k+2}$ trials were run. Statistical uncertainties are contained within marker size. Vertical lines indicate psuedothresholds for each decoding method, and shaded regions indicate psuedothreshold uncertainty estimated from logical error rate uncertainty.
• Figure 2: Left: Two $Z$-error chains (red) on a $5{\times}5$ TC patch, each producing exactly two excitations at the string endpoints regardless of chain length. Excitations always come in pairs, reducing decoding to a graph matching problem. MWPM (blue arcs) finds the minimum-weight pairing. Right: In a general 2D TTI code, one check (green) can act on more than the four nearest-neighbor qubits and one qubit can participate in three or more $X$ checks, so a single-qubit error can create three or more excitations simultaneously which is equivalent to a hyperedge. Minimum-weight hypergraph matching is NP-hard in general, breaking the naive MWPM approach.
• Figure 3: Schematic illustration of the decoupling theorem, shown as a stack of 2D planes in 3D perspective. Bottom plane: A patch of a general 2D TTI code. Overlapping colored regions represent the entangled TC copies in the 2D TTI code. Middle plane: A constant-depth CNOT circuit that decouples the 2D TTI code into independent TC copies. Top planes: The code decouples into independent TC copies.
• Figure 4: Pipeline of the layer-decoupling decoder. Project: An algebraic change of basis maps the measured syndrome $\sigma$ of the 2D TTI code into $r$ sector syndromes $D_1,\ldots,D_r$, each living on a TC-like coarse lattice whose check violations come in pairs. Decode: Each sector syndrome $D_i$ is decoded independently by matching. Lift: The sector-correction chains are mapped back to physical qubits via the inverse of the decoupling map, yielding a Pauli correction $P$ that clears the original syndrome.
• Figure 5: Pipeline of the cell-matching decoder. (a) The 2D lattice with $3\times 3$ unit cells. The qubits are edges, the $X$ checks are vertices, and the $Z$ checks are plaquettes. Gold boundaries are drawn around sets of 9 $Z$ checks that belong in the same unit cell. A Pauli $X$ error configuration has resulted in some violated $Z$ checks shown in red. (b) After local flushing: local Pauli $X$ operators (orange arrows) move each violated check into the $2\times 2$ basis subcell (purple) within the unit cell that the violated check is in. If the violated check is already in the basis subcell, no flushing is done for the check. (c) Global MWPM on the coarse lattice: there are four possible check-violation types in the $2 \times 2$ basis subcells and three of the four types (top-left, top-right, and bottom-left) are currently present and require matching on independent toric graphs. Each check-violation type is paired by matching edges (green dashed) that corresponds to short Pauli strings that generate pairs of excitations in the coarse lattice.

Theorems & Definitions (56)

• theorem 2.1: Performance guarantees for the layer-decoupling and cell-matching decoders; see Theorems \ref{['thm:decoupling-decoder-finite-size']}, \ref{['thm:time-complexity-decoupling-decoder-finite-size']}, \ref{['thm:cellular-decoder-finite-size']}, and \ref{['thm:time-complexity-cellular-decoder-finite-size']}
• definition 3.1: Quantum CSS Codes
• definition 3.2: Chain Complexes
• definition 3.3
• definition 4.1: Noise model mapping
• definition 4.2: Syndrome mapping
• lemma 4.3: Correctness of lifting
• proof
• theorem 4.4: Layer-decoupling decoder for 2D TTI codes
• proof : Proof of Theorem \ref{['thm:decoupling-decoder-finite-size']}