Optimal complexity correction of correlated errors in the surface code

TL;DR

The paper addresses improving error suppression in the surface code under depolarizing noise by exploiting correlations between $X$ and $Z$ errors. It introduces an optimal-complexity decoding approach that remains parallelizable to $O(1)$ per unit area, using correlation-aware edge reweighting in minimum-weight perfect matching and, for full fault tolerance, a pair of correlated 3-D graph decodings enabled by Autotune. The authors demonstrate substantial performance gains over standard MWPM—approximately a factor of $2$ reduction in logical error rates at practical noise levels—and prove computational efficiency, arguing asymptotic optimality for large code distance $d$ at fixed $p \lesssim 2\times 10^{-4}$. The work provides a scalable decoding strategy that integrates with existing topological quantum error correction frameworks and suggests practical routes to hardware implementations and further runtime improvements.

Abstract

The surface code is designed to suppress errors in quantum computing hardware and currently offers the most believable pathway to large-scale quantum computation. The surface code requires a 2-D array of nearest-neighbor coupled qubits that are capable of implementing a universal set of gates with error rates below approximately 1%, requirements compatible with experimental reality. Consequently, a number of authors are attempting to squeeze additional performance out of the surface code. We describe an optimal complexity error suppression algorithm, parallelizable to O(1) given constant computing resources per unit area, and provide evidence that this algorithm exploits correlations in the error models of each gate in an asymptotically optimal manner.

Figures (9)

• Figure 1: Distance 4 surface code. White circles represent data qubits. Each shaded bubble represents an $X$ or $Z$ stabilizer Gott97, and the error-free surface code state can be thought of as the simultaneous +1 eigenstate of all stabilizers. When errors occur, the surface code state becomes the -1 eigenstate of some stabilizers. At least $\lceil d/2 \rceil$ errors must occur for stabilizer measurements, which report whether the underlying state is the +1 or -1 eigenstate of that stabilizer, to be ambiguous and have the potential to lead to a logical error after correction.
• Figure 2: (Color online) a) Distance 4 surface code with two data qubit errors. Stabilizers containing light dots will report -1 eigenstates when measured (indicating a detection event). b) The $X$ stabilizer measurements can, with reasonable effectiveness, be used in isolation to predict the location of $Z$ errors. The minimum weight perfect matching algorithm takes the indicated structure as input, and outputs a pairing of the detection events to one another or a boundary making use of the minimum total number of edges. In this ambiguous case however, the two edges to the left or right of the single detection event will be returned with equal probability, resulting in corresponding $Z$ corrections and success or failure, respectively. c) The $Z$ detection events will be matched with one another, leading to successful correction of the $X$ component of the $Y$ error.
• Figure 3: (Color online) Basic perfect stabilizer measurement correction. Probability of logical $X$ error $p_L$ as a function of the depolarizing error probability $p$ for a range of distances $d=4, \ldots, 100$ when performing basic minimum weight perfect matching only. Referring to the left of the figure, the distance increases top to bottom. Quadratic, cubic, quartic, and quintic lines (dashed) have been drawn through the lowest distance 4, 6, 8, 10 data points obtained, respectively.
• Figure 4: (Color online) Perfect stabilizer measurement correlated error correction. Probability of logical $X$ error $p_L$ as a function of the depolarizing error probability $p$ for a range of distances $d=3, \ldots, 100$ when performing two rounds of minimum weight perfect matching with edge reweighting based on the first round. Referring to the left of the figure, the distance increases top to bottom. Quadratic, cubic, quartic, and quintic lines (dashed) have been drawn through the lowest distance 4, 6, 8, 10 data points obtained, respectively.
• Figure 5: (Color online) a) Distance 4 surface code with two data qubit errors. Stabilizers containing light dots will report -1 eigenstates when measured (indicating a detection event). b) The minimum weight perfect matching algorithm will return the two edges to the left or right of the single detection event with equal probability, resulting in success or failure, respectively. c) The $Z$ detection events can be matched as shown, leading to no helpful reweighting of b).