Improved decoding algorithms for surface codes under independent bit-flip and phase-flip errors

TL;DR

The paper advances exact decoding for toric, planar, and rotated surface codes under independent X/Z noise by introducing geometry-preserving reductions of SMW to minimum-weight perfect matching on decorated graphs and by developing FKT-based SMLC decoders via Pfaffian evaluations. Planar separators enable $O(n^{3/2}\log n)$ time for SMW on toric/planar/rotated codes and place planar/rotated SMLC in $\mathrm{NC}$ with $O(n^{\omega/2})$ arithmetic (bit complexity $\tilde{O}(n^{\omega/2+1})$); toric SMLC remains polynomial with $O(n^{3})$ arithmetic due to genus-1 complications. The framework rests on Fisher gadgets, dual-cycle representations, and MacWilliams duality, connecting decoding to classical combinatorial routines such as MWPM, Pfaffian evaluations, and determinant computations. Extensions to depolarizing noise and open questions about Pfaffian signs and simple $D$-joins highlight future directions toward fully efficient joint decoding under correlated noise. Overall, the work provides a unified, geometry-conscious approach that substantially speeds exact decoding and reveals deep connections between quantum error correction, planar graph theory, and algebraic methods.

Abstract

We study exact decoding for the toric code and for planar and rotated surface codes under the standard independent \(X/Z\) noise model, focusing on Separate Minimum Weight (SMW) decoding and Separate Most Likely Coset (SMLC) decoding. For the SMW decoding problem, we show that an \(O(n^{3/2}\log n)\)-time decoder is achievable for surface and toric codes, improving over the \(O(n^{3}\log n)\) worst-case time of the standard approach based on complete decoding graphs. Our approach is based on a local reduction of SMW decoding to the minimum weight perfect matching problem using Fisher gadgets, which preserves planarity for planar and rotated surface codes and genus~\(1\) for the toric code. This reduction enables the use of Lipton--Tarjan planar separator methods and implies that SMW decoding lies in \(\mathrm{NC}\). For SMLC decoding, we show that the planar surface code admits an exact decoder with \(O(n^{3/2})\) algebraic complexity and that the problem lies in \(\mathrm{NC}\), improving over the \(O(n^{2})\) algebraic complexity of Bravyi \emph{et al.} Our approach proceeds via a dual-cycle formulation of coset probabilities and an explicit reduction to planar Pfaffian evaluation using Fisher--Kasteleyn--Temperley constructions. The same complexity measures apply to SMLC decoding of the rotated surface code. For the toric code, we obtain an exact polynomial-time SMLC decoder with \(O(n^{3})\) algebraic complexity. In addition, while the SMLC formulation is motivated by connections to statistical mechanics, we provide a purely algebraic derivation of the underlying duality based on MacWilliams duality and Fourier analysis. Finally, we discuss extensions of the framework to the depolarizing noise model and identify resulting open problems.

Figures (14)

• Figure 1: Left: The toric code lattice $\mathcal{X}$ for $L=3$. The orange edges on the lower side and their incident vertices are identified with those on the upper side. Likewise, the green edges on the left side are identified with those on the right side. One $X$-type check is shown in blue and one $Z$-type checks in red. Two noncontractible cycles $a$ and $b$ of minimum length that are not homotopic to each other are shown in orange and green. Right: The dual lattice $\mathcal{X}^\ast$ is shown in blue.
• Figure 2: Left: The cell complex $\mathcal{X}$ of the $L\times (L+1)$ rectangular lattice for $L=5$. The edges of its $1$-dimensional subcomplex $\mathcal{A}$ are shown with dashed lines, and the vertices of $\mathcal{A}$ are indicated by boxes. Two $X$-type checks are shown in blue and two $Z$-type checks in red. A minimal-length relative cycle $a$ connecting a vertex on the left side of $\mathcal{A}$ to a vertex on the right side is shown in green. Right: The dual lattice is shown in blue.
• Figure 3: The top row shows a rotated surface code for odd $L$: $L = 5$. The bottom row shows a rotated surface code for even $L$: $L = 6$. The cell complex $\mathcal{X}$ and its subcomplex $\mathcal{A}$ are shown on the left. The edges of $\mathcal{A}$ are drawn with dashed lines. The vertices of $\mathcal{A}$ are the endpoints of these edges. Among them, the vertices indicated by boxes are those incident to edges outside $\mathcal{A}$; these are the ones relevant for decoding. A minimal-length relative cycle $a$ that is not a relative boundary is shown in green. The dual lattice is shown in blue on the right.
• Figure 4: (a) Even degree-2 Fisher gadget. (b) Odd degree-2 Fisher gadget: It has one internal vertex shown in blue. (d) Odd degree-3 Fisher gadget: Any perfect matching requires an odd number of vertices to be matched externally, as shown in (c) where the perfect matching edges are shown in red. (e) Even degree-3 Fisher gadget: Introducing the additional edge flips the parity. The gadget has one internal vertex shown in blue. (f) Odd degree-4 Fisher gadget: It is constructed by gluing together two odd degree-3 gadgets along the internal vertex shown in blue. (g) Even degree-4 Fisher gadget: Obtained by adding an edge that flips the parity. (h) Even nonplanar degree-4 gadget: Although not planar, it behaves like the gadget in (g) and has smaller size. Its complete internal structure allows excluding any even number of vertices from the internal matching, thereby forcing an even number of vertices to be externally matched. (i) Odd degree-$k$ Fisher gadget, for $k \ge 3$: It is obtained by gluing $k-2$ odd degree-3 gadgets. It follows by induction that this gadget has odd parity. The gadget has $k-3$ internal vertices shown in blue and a total of $3(k-2)$ internal edges. (j) Even degree-$k$ Fisher gadget, for $k \ge 3$: It is obtained by adding a parity-flipping edge to the odd degree-$k$ gadget. It has $k-2$ internal vertices and $3k-5$ internal edges. The parity-flipping edge is shown in red. It may be added on the boundary or in the interior of the gadget, as illustrated by two realizations of the gadget.
• Figure 5: Left: A portion of the graph $G$ of the toric code, where the $X$-syndrome defects are shown in filled black circles. The $D$-join edges are shown in red. Right: the decorated graph $G_D$ using one vertex per defect in $D$, and the even degree-4 nonplanar gadget for vertices outside $D$. The perfect matching edges are shown in red.

Theorems & Definitions (24)

• Theorem 3.1: Dissolving degree-$3$ defects in the toric code
• Theorem 3.2: Dissolving degree-$3$ defects in the planar surface code
• Theorem 3.3: Dissolving degree-$3$ defects in the rotated surface code
• Lemma 4.1: Finding an Augmenting Path Gabow75LiptonTarjan80; see also GMG86
• Theorem 4.3
• Theorem 4.4
• Lemma 5.1
• Theorem 5.3
• Lemma 5.6: Dual-code coset probabilities
• Lemma 5.7