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arXiv:1703.01517 · Physical Review Research

Linear-time maximum likelihood decoding of surface codes over the quantum erasure channel

Nicolas Delfosse, Gilles Zémor

Published Jul 9, 2020• 89 citations
arXivHTMLPDF

Abstract

This paper presents a decoding algorithm for the correction of erasure or qubit loss using surface codes. The peeling decoder identifies an optimal correction and it can be implemented in linear time. It corrects up to 50% of qubit loss.
DOI
10.1103/physrevresearch.2.033042

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We propose a decoding algorithm for color codes over the quantum erasure channel, which is linear-time maximum likelihood (ML) when the set of erased qubits satisfies a certain condition called trimmability. Two methods are proposed for general erasure sets, either by extending the erasure set to make it trimmable, or by inactivating some vertices. The former is linear time but not ML, while the latter is ML but not linear time. Numerical results are provided to assess the error correction performance and the complexity of both methods.
The quantum erasure channel models phenomena such as loss or leakage of qubits. Using quantum codes, we can recover from such errors. In this paper, we are interested in studying the performance of two-dimensional color codes over the quantum erasure channel. Our approach makes use of the local equivalence between color codes and surface codes. We propose a variety of decoding algorithms for color codes over the erasure channel. First, we propose algorithms that decode by projecting the erasures on the color to surface codes. Then, instead of directly decoding on the color code or on the equivalent copies of surface codes, we decode jointly on the color code and the equivalent surface codes. We observe a threshold of 44.3% for the color code on the square octagonal lattice.
The problem of recovering from qubit erasures has recently gained attention as erasures occur in many physical systems such as photonic systems, trapped ions, superconducting qubits and circuit quantum electrodynamics. While several linear-time decoders for error correction are known, their errorcorrecting capability is limited to half the minimum distance of the code, whereas erasure correction allows one to go beyond this limit. As in the classical case, stopping sets pose a major challenge in designing efficient erasure decoders for quantum LDPC codes. In this paper, we show through simulation, that an attractive alternative here, is the use of quantum expander codes in conjunction with the peeling decoder that has linear complexity. We also discuss additional techniques including small-set-flip decoding, that can be applied following the peeling operation, to improve decoding performance and their associated complexity.
We describe two implementations of the optimal error correction algorithm known as the maximum likelihood decoder (MLD) for the 2D surface code with a noiseless syndrome extraction. First, we show how to implement MLD exactly in time O(n2)O(n^2)O(n2), where nnn is the number of code qubits. Our implementation uses a reduction from MLD to simulation of matchgate quantum circuits. This reduction however requires a special noise model with independent bit-flip and phase-flip errors. Secondly, we show how to implement MLD approximately for more general noise models using matrix product states (MPS). Our implementation has running time O(nχ3)O(nχ^3)O(nχ3) where χχχ is a parameter that controls the approximation precision. The key step of our algorithm, borrowed from the DMRG method, is a subroutine for contracting a tensor network on the two-dimensional grid. The subroutine uses MPS with a bond dimension χχχ to approximate the sequence of tensors arising in the course of contraction. We benchmark the MPS-based decoder against the standard minimum weight matching decoder observing a significant reduction of the logical error probability for χ≥4χ\ge 4χ≥4.
In this work, we develop a reduced complexity maximum likelihood (ML) decoder for quantum low-density parity-check (QLDPC) codes over erasures. Our decoder combines classical inactivation decoding, which integrates peeling with symbolic guessing, with a new dual peeling procedure. In the dual peeling stage, we perform row operations on the stabilizer matrix to efficiently reveal stabilizer generators and their linear combinations whose support lies entirely on the erased set. Each such stabilizer identified allows us to freely fix a bit in its support without affecting the logical state of the decoded result. This removes one degree of freedom that would otherwise require a symbolic guess, reducing the number of inactivated variables and decreasing the size of the final linear system that must be solved. We further show that dual peeling combined with standard peeling alone, without inactivation, is sufficient to achieve ML for erasure decoding of surface codes. Simulations across several QLDPC code families confirm that our decoder matches ML logical failure performance while significantly reducing the complexity of inactivation decoding, including more than a 20% reduction in symbolic guesses for the B1 lifted product code at high erasure rates.
Quantum networks encounter unavoidable channel noises and erasure errors, presenting a huge obstacle in designing protocols that attain both high reliability and efficiency. Typically, quantum networks fall into two categories: those utilize quantum entanglements for quantum teleportation, and those directly transfer the actual quantum messages. In this paper, we present SurfNet, a quantum network that inherits the main advantages from both categories. It employs surface codes as logical qubits for encoding messages, and utilizes two parallel communication channels to fault-tolerantly transfer each surface code in a modular manner. Our approach of using surface codes can timely correct both operational and photon loss errors within the network, and the integration of the two channels within the network can greatly improve network throughput. For the implementation of SurfNet, we propose a novel network architecture, designed to better integrate surface codes into quantum networks. We also propose a novel error correction decoder, designed to fully utilize the modular characteristic of surface codes within our network. Simulation results demonstrate that SurfNet with its decoder significantly enhances the communication fidelity within quantum networks.