Multi-path Summation for Decoding 2D Topological Codes

Abstract

Fault tolerance is a prerequisite for scalable quantum computing. Architectures based on 2D topological codes are effective for near-term implementations of fault tolerance. To obtain high performance with these architectures, we require a decoder which can adapt to the wide variety of error models present in experiments. The typical approach to the problem of decoding the surface code is to reduce it to minimum-weight perfect matching in a way that provides a suboptimal threshold error rate, and is specialized to correct a specific error model. Recently, optimal threshold error rates for a variety of error models have been obtained by methods which do not use minimum-weight perfect matching, showing that such thresholds can be achieved in polynomial time.It is an open question whether these results can also be achieved by minimum-weight perfect matching. In this work, we use belief propagation and a novel algorithm for producing edge weights to increase the utility of minimum-weight perfect matching for decoding surface codes. This allows us to correct depolarizing errors using the rotated surface code, obtaining a threshold of 17.76 ± 0.02%. This is larger than the threshold achieved by previous matching-based decoders (14.88±0.02%), though still below the known upper bound of ∼ 18.9%.A surface code [9-11] is supported on a square array of qubits, with stabilisers defined on individual square tiles, see Figure 1. Specifically, we focus on the rotated surface code, with the boundary conditions from Figure 1. Earlier constructions involve tiling a surface with nontrivial topology, such as a torus [8], or using an alternate open boundary condition with smooth and rough boundaries [10]. We choose to study the rotated code, since it requires the fewest physical qubits per logical qubit.

Parallelized quantum error correction with fracton topological codes

“…The algorithm takes a graph with weighted edges and returns a subset of the edges of the original graph such that each vertex has exactly one incident edge and the sum of the weights of the edges that are returned is minimal. We use this algorithm by assigning each defect a vertex of the graph, and adding edges to the graph that are weighted to approximate the logarithm of the likelihood that a Pauli-error caused the pair of defects that are connected by the edge [14,[45][46][47][48][49]. Most interestingly, all the decoders we present use matching subroutines on the defects lying on L×L planes of the system rather than its full volume.…”

Quantum message-passing algorithm for optimal and efficient decoding

“…The notion of belief propagation has been applied in the quantum setting in many different previous works; here we comment on their relation to the present work. Most closely related is the use of belief propagation for decoding quantum information subject to Pauli errors, as studied by Poulin in [13,14], as well as many others since [15][16][17][18][19][20][21][22]. Here, however, classical BP is sufficient: The task is to infer which error occured from the classical syndrome information, which only involves the classical conditional probability of the former given the latter.…”

Machine-learning-assisted correction of correlated qubit errors in a topological code

“…Both the neural network decoder and the improved blossom decoder perform below the optimal maximum-likelihood decoder. Several approaches exist to reach the optimal limit, we mention the incorporation of X-Z correlations via a belief propagation algorithm [37], and approaches based on renormalization group methods or Monte Carlo methods [28,38,39].…”

“…The existence of algorithms [28,33,[36][37][38][39] that improve on the blossom decoder does not diminish the appeal of machine learning decoders, since these offer a flexibility to different types of topological codes that a dedicated decoder lacks.…”