Fast Decoding Algorithm for Subspace and Subsystem Color Codes and Local Equivalence of Topological Phases

Topological codes introduced by Kitaev [9] are among the most promising family of quantum error correcting codes that could lead to realistic quantum computer architectures. Many features of these codes explain why they can operate reliably in the presence of a considerable amount of noise, with a threshold of nearly 1% [10]. First, it is possible to implement many gates on the qubits encoded in these codes in a purely topological or transversal fashion. Both of these methods are important because they directly result in fault-tolerant gates. Second, the syndrome extraction involves measurements of only four neighboring qubits on a two-dimensional lattice. This is to be contrasted with, e.g., concatenated codes where the number of qubits involved in each syndrome measurement grows linearly with the size of the code, thus increasing the complexity of the error correction procedure and lowering the threshold. The most general method to protect quantum information is not of a subspace code, but a subsystem code. Recently one of us introduced the family of topological subsystem color codes (TSCC) [2], which pushes the features of topological codes to their extreme. Indeed, the syndrome extraction for these codes requires only twoqubit measurements, as simple as it can possibly get. Additionally, the entire Clifford group can be performed in a topological fashion on the qubits encoded in these codes. For these reasons, TSCC may well be the simplest and most efficient means of achieving fault tolerance. To be of any practical use, any error-correcting code must have an efficient decoder—an algorithm that finds the most likely recovery given the measured error syndrome. Recently, two of us have conceived a decoding algorithm for Kitaev’s toric code (KTC) that is exponentially faster than previously known decoding algorithms (run time log ` rather than ` where ` is the linear size of the torus) [7]. Note that the decoding runtime is a crucial factor for fault tolerance; proofs of the threshold theorem usually assume instantaneous classical side-computation to assist the error-correction procedure. Our algorithm is also very flexible, it enables various tradeoffs between complexity and error-correction performances. In particular, we were able to demonstrate that it can achieve a higher error-correction error threshold than what was achievable by previously known decoding algorithms [6, 8]. Here, we demonstrate how this fast decoding algorithm, as well as other decoding algorithms, can be applied to a wider class of topological error correcting codes. We do this by demonstrating a local equivalence between various codes. More precisely, we consider the family of topological color codes (TCC) [4] and their subsystem cousins TSCC [2, 3] and show that they can be locally mapped to a number of copies of KTC. That is, local in the sense that an operator with support on qubits contained within a region of finite radius r is mapped to an operator with support within a region of radius c + r, where c is some constant. As a consequence of this local equivalence, we can, for the purpose of decoding, treat a topological code as a certain number of copies of KTC, and use any decoding algorithm suitable for KTC to complete the decoding on each of these copies. Crucially, the noise model induced on KTC remains essentially local: an error model that is independent on each qubit will be mapped to an error model with some short range correlations on a length scale c, but no long range correlation. Because the existence of an error threshold is essentially a large scale property of a system, this decoding strategy should also produce a finite error threshold, albeit with a different critical error probability that depends on the value of c and other microscopic details of the mapping. The importance of this decoding strategy for the TSCC can be stressed by the fact that, to our knowledge, it provides the first example of a family of subsystem codes with an efficient decoding algorithm. The general construction of Bacon and Casaccino [1] for instance can produce subsystem codes with sparse gauge operators, a close analogue of classical LDPC codes, but unfortunately have no known decoding algorithm. Another motivation comes from Bravyi’s recent proof that geometrically local subsystem codes are much more powerful than geometrically local subspaces codes in that they can achieve better tradeoffs between minimal distances and encoding rates [5]. Our fast decoding algorithm makes use of methods of statistical physics and classical coding theory, namely renormalization group methods and belief propagation [11]. Details of this decoding algorithm can be found in [7]. Here, we will focus on the local mapping between topological codes. To understand the intuition behind these mapping, it is useful to think of a subspace error correcting code as a local Hamiltonian