Decoding small surface codes with feedforward neural networks

Abstract

Surface codes reach high error thresholds when decoded with known algorithms, but the decoding time will likely exceed the available time budget, especially for near-term implementations. To decrease the decoding time, we reduce the decoding problem to a classification problem that a feedforward neural network can solve. We investigate quantum error correction and fault tolerance at small code distances using neural network-based decoders, demonstrating that the neural network can generalize to inputs that were not provided during training and that they can reach similar or better decoding performance compared to previous algorithms. We conclude by discussing the time required by a feedforward neural network decoder in hardware. I. INTRODUCTIONQuantum computing has emerged as a solution to accelerate various calculations using systems governed by quantum mechanics. Such calculations are believed to take exponential time to perform using classical computers. Initial applications where quantum computing will be useful are simulation of quantum physics [1], cryptanalysis [2, 3] and unstructured search [4], and there is a growing set of other quantum algorithms [5].Simple quantum algorithms have been shown to scale better than classical algorithms [6-8] for small test cases, though larger computers are required to solve real-world problems. The main obstacle to scalability is that the required quantum operations (state preparations, singleand two-qubit unitary gates, and measurements) are subject to external noise, therefore quantum algorithms cannot run with perfect fidelity. This requires quantum computers to use active error correction [9] to achieve scalability, which in turn requires a classical co-processor to infer which corrections to make, given a stream of measurement results as input. If this co-processor is slow, performance of the quantum computer may be degraded (though recent results [10] suggest that this may be mitigated).The remainder of this paper is organized as follows. In Section II, we outline the relevant aspects of quantum error correction and fault tolerance. We discuss the need for a fast classical coprocessor in Section III. In Section IV, we give a brief summary of existing techniques to perform decoding quickly, and follow this in Section V with the introduction of a new technique based on feedforward neural networks. We examine the accuracy of the proposed decoder in Section VI, and conclude by discussing its speed in Section VII. II. QUANTUM ERROR CORRECTIONWhile it is often possible to decrease the amount of noise affecting a quantum operation using advanced control techniques [11,12], their analog nature suggests that some imperfection will always remain. This has driven the development of algorithmic techniques to protect quantum states and computations from noise, which are called quantum error correction and fault tolerance, respectively.

Neural network decoder for topological color codes with circuit level noise

“…Generally, a decoder executes a classical algorithm that determines the operator P(t)äΠ n (the so-called Pauli frame) which transforms y ñ | ( ) t L back into the logical qubit space   =  L 0 . Equivalently (with minimal overhead), a decoder may keep track of logical parity bits  p that determine whether the Pauli frame of a 'simple decoder' [38] commutes with a set of chosen logical operators for each logical qubit.…”

“…No a priori knowledge of the error model is required. Machine learning approaches have been previously shown to be successful for the families of surface and toric codes [37][38][39][40][41], and applications to color codes are now being investigated [42][43][44]. We adapt the recurrent neural network of [39] to decode color codes with distances up to 7, fully incorporating the information from flag qubits.…”

Neural Decoder for Topological Codes using Pseudo-Inverse of Parity Check Matrix

“…Researchers have previously studied the perspective of quantum error correction as a classification problem using neural networks [7,10,15]. As mentioned before, we model our decoder as a two-step process.…”

“…Our final error correction is L E. Refer Eqs. (5), (7), and (8). Note that the H-inverse decoder in step-one need not always give us pure error.…”

“…Torlai and Melko were the first to propose a decoder for surface codes based on neural networks [6]. Since then, many other researchers have applied neural networks to study a variety of problems in the context of decoding [6][7][8][9][10][11][12][13][14][15].…”