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Physical Review A

Erasure decoding of two-dimensional color codes

Arun B. Aloshious, Pradeep Kiran Sarvepalli

Published Oct 14, 2019• 6 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.100.042312

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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.
Qubit loss is one of the forms of noise encountered in some quantum technologies. Such noise is modeled using the quantum erasure channel. Unlike the depolarizing noise, it is much more tractable, yet the performance of many quantum codes over the erasure channel has not been studied as extensively. In this paper, we study the performance of topological subsystem color codes (TSCCs) over the quantum erasure channel. It is the first such study of TSCCs over the erasure channel. We propose multiple decoding algorithms for TSCC and obtain the highest threshold of about 9.7% for the subsystem color code derived from the square octagon lattice.
In recent years, there have been many studies on local stabilizer codes. Under the assumption of translation and scale invariance Yoshida classified such codes. His result implies that translation invariant 2D color codes are equivalent to copies of toric codes. Independently, Bombin, Duclos-Cianci, and Poulin showed that a local translation invariant 2D topological stabilizer code is locally equivalent to a finite number of copies of Kitaev's toric code. In this paper we focus on 2D topological color codes and relax the assumption of translation invariance. Using a linear algebraic framework we show that any 2D color code (without boundaries) is locally equivalent to two copies of a related surface code. This surface code is induced by color code. We study the application of this equivalence to the decoding of 2D color codes over the bit flip channel as well as the quantum erasure channel. We report the performance of the color code on the square octagonal lattice over the quantum erasure channel. Further, we provide explicit circuits that perform the transformation between 2D color codes and surface codes. Our circuits do not require any additional ancilla qubits.
Fault-tolerant quantum computation relies on scaling up quantum error correcting codes in order to suppress the error rate on the encoded quantum states. Topological codes, such as the surface code or color codes, are leading candidates for practical scalable quantum error correction and require efficient and scalable decoders. In this work, we propose and study the efficiency of a decoder for two-dimensional topological color codes on the 4.8.8 lattice (also known as the square-octagon code), by building on the work of Sarvepalli and Raussendorf [Phys. Rev. A 85, 022317 (2012)], for color codes on hexagonal lattices. The decoder is based on a rescaling approach, in which syndrome information on a part of the qubit lattice is processed locally, and then the lattice is rescaled iteratively to smaller sizes. We find a threshold of 6.0% for code capacity noise.
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.
Topological color codes defined by the 4.8.8 semiregular lattice feature geometrically local check operators and admit transversal implementation of the entire Clifford group, making them promising candidates for fault-tolerant quantum computation. Recently, several efficient algorithms for decoding the syndrome of color codes were proposed. Here, we modify one of these algorithms to account for errors affecting the syndrome, applying it to the family of triangular 4.8.8 color codes encoding one logical qubit. For a three-dimensional bit-flip channel, we report a threshold error rate of 0.0208(1), compared with 0.0305(4) previously reported for an integer-program-based decoding algorithm. When we account for circuit details, this threshold is reduced to 0.00143(1) per gate, compared with 0.00672(1) per gate for the surface code under an identical noise model.
We introduce an efficient decoder of the color code in d ≥ 2 dimensions, the Restriction Decoder, which uses any d -dimensional toric code decoder combined with a local lifting procedure to find a recovery operation. We prove that the Restriction Decoder successfully corrects errors in the color code if and only if the corresponding toric code decoding succeeds. We also numerically estimate the Restriction Decoder threshold for the color code in two and three dimensions against the bit-flip and phase-flip noise with perfect syndrome extraction. We report that the 2D color code threshold p 2D ≈ 10.2 % on the square-octagon lattice is on a par with the toric code threshold on the square lattice.