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arXiv:1907.04537 · Physical Review A

Heuristic construction of codeword stabilized codes

Alex Rigby, Jan Corne Olivier, Peter Jarvis

Published Dec 2, 2019• 4 citations
arXivHTMLPDF

Abstract

The family of codeword stabilized codes encompasses the stabilizer codes as well as many of the best known nonadditive codes. However, constructing optimal $n$-qubit codeword stabilized codes is made difficult by two main factors. The first of these is the exponential growth with $n$ of the number of graphs on which a code can be based. The second is the NP-hardness of the maximum clique search required to construct a code from a given graph. We address the second of these issues through the use of a heuristic clique finding algorithm. This approach has allowed us to find $((9,97\ensuremath{\le}K\ensuremath{\le}100,2))$ and $((11,387\ensuremath{\le}K\ensuremath{\le}416,2))$ codes, which are larger than any previously known codes. To address the exponential growth of the search space, we demonstrate that graphs that give large codes typically yield clique graphs with a large number of nodes. The number of such nodes can be determined relatively efficiently, and we demonstrate that $n$-node graphs yielding large clique graphs can be found using a genetic algorithm. This algorithm uses a crossover operation based on spectral bisection that we demonstrate to be superior to more standard crossover operations. Using this genetic algorithm approach, we have found ((13, 18, 4)) and ((13, 20, 4)) codes that are larger than any previously known code. We also consider codes for the amplitude damping channel. We demonstrate that for $n\ensuremath{\le}9$, optimal codeword stabilized codes correcting a single amplitude damping error can be found by considering standard form codes that detect one of only three of the ${3}^{n}$ possible equivalent error sets. By combining this error-set selection with the genetic algorithm approach, we have found ((11, 68)) and ((11, 80)) codes capable of correcting a single amplitude damping error and ((11, 4)), ((12, 4)), ((13, 8)), and ((14, 16)) codes capable of correcting two amplitude damping errors.

DOI
10.1103/physreva.100.062303

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•Jan 1, 2021•0 citations
The family of codeword stabilized codes encompasses the stabilizer codes as well as many of the best known nonadditive codes. However, constructing optimal nnn-qubit codeword stabilized codes is made difficult by two main factors. The first of these is the exponential growth with nnn of the number of graphs on which a code can be based. The second is the NP-hardness of the maximum clique search required to construct a code from a given graph. We address the second of these issues through the use of a heuristic clique finding algorithm. This approach has allowed us to find ((9,97\ensuremath≤K\ensuremath≤100,2))((9,97\ensuremath{\le}K\ensuremath{\le}100,2))((9,97\ensuremath≤K\ensuremath≤100,2)) and ((11,387\ensuremath≤K\ensuremath≤416,2))((11,387\ensuremath{\le}K\ensuremath{\le}416,2))((11,387\ensuremath≤K\ensuremath≤416,2)) codes, which are larger than any previously known codes. To address the exponential growth of the search space, we demonstrate that graphs that give large codes typically yield clique graphs with a large number of nodes. The number of such nodes can be determined relatively efficiently, and we demonstrate that nnn-node graphs yielding large clique graphs can be found using a genetic algorithm. This algorithm uses a crossover operation based on spectral bisection that we demonstrate to be superior to more standard crossover operations. Using this genetic algorithm approach, we have found ((13, 18, 4)) and ((13, 20, 4)) codes that are larger than any previously known code. We also consider codes for the amplitude damping channel. We demonstrate that for n\ensuremath≤9n\ensuremath{\le}9n\ensuremath≤9, optimal codeword stabilized codes correcting a single amplitude damping error can be found by considering standard form codes that detect one of only three of the 3n{3}^{n}3n possible equivalent error sets. By combining this error-set selection with the genetic algorithm approach, we have found ((11, 68)) and ((11, 80)) codes capable of correcting a single amplitude damping error and ((11, 4)), ((12, 4)), ((13, 8)), and ((14, 16)) codes capable of correcting two amplitude damping errors.
The codeword stabilized (CWS) quantum code formalism presents a unifying approach to both additive and nonadditive quantum error-correcting codes [IEEE Trans. Inf. Theory 55, 433 (2009)]. This formalism reduces the problem of constructing such quantum codes to finding a binary classical code correcting an error pattern induced by a graph state. Finding such a classical code can be very difficult. Here, we consider an algorithm which maps the search for CWS codes to a problem of identifying maximum cliques in a graph. While solving this problem is in general very hard, we provide three structure theorems which reduce the search space, specifying certain admissible and optimal ((n,K,d)) additive codes. In particular, we find that the re does not exist any ((7,3,3)) CWS code though the linear programming bound does not rule it out. The complexity of the CWS-search algorithm is compared with the contrasting method introduced by Aggarwal and Calderbank [IEEE Trans. Inf. Theory 54, 1700 (2008)].
We discuss a method to adapt the codeword stabilized (CWS) quantum code framework to the problem of finding asymmetric quantum codes. We focus on the corresponding Pauli error models for amplitude damping noise and phase damping noise. In particular, we look at codes for Pauli error models that correct one or two amplitude damping errors. Applying local Clifford operations on graph states, we are able to exhaustively search for all possible codes up to length 999. With a similar method, we also look at codes for the Pauli error model that detect a single amplitude error and detect multiple phase damping errors. Many new codes with good parameters are found, including nonadditive codes and degenerate codes.
In this study we present a method that adapts the codeword stabilized (CWS) quantum code framework to the problem of finding asymmetric quantum codes. Making use of the the corresponding Pauli error models for amplitude and phase-damping models, we focus on codes that correct one or two amplitude-damping errors. As a result, we are able to exhaustively search for all possible codes up to length 9 by applying local Clifford operations on graph states. With a similar method, we also look at codes for the Pauli error models that detect a single amplitude error and detect multiple phase damping errors. Many new codes with good parameters are found, including non-additive codes and degenerate codes.
We construct families of high performance quantum amplitude damping codes. All of our codes are nonadditive and most modestly outperform the best possible additive codes in terms of encoded dimension. One family is built from nonlinear error-correcting codes for classical asymmetric channels, with which we systematically construct quantum amplitude damping codes with parameters better than any prior construction known for any block lengthn≥ 8 exceptn=2r-1. We generalize this construction to employ classical codes overGF(3) with which we numerically obtain better performing codes up to length 14. Because the resulting codes are of the codeword stabilized (CWS) type, conceptually simple (though potentially computationally expensive) encoding and decoding circuits are available.
Noise is a major obstacle in the development of practical schemes for quantum computation and communication. Similar to the case of classical communication, this noise can be protected against by employing a code, which provides a means for encoding quantum states prior to transmission and allows for errors to be inferred, and hopefully corrected, by a decoder at the receiver. Unfortunately, designing good codes and decoders is typically a difficult problem. This thesis focuses on developing low-complexity heuristic approaches to three such problems: the design of modified belief propagation decoders for quantum low-density parity-check codes, the design of stabilizer codes for asymmetric channels, and the design of codeword stabilized codes. Quantum low-density parity-check codes are stabilizer codes with low-weight generators. Such codes permit low-complexity decoding via the use of belief propagation, which is an iterative message passing algorithm that takes place on a factor graph defined by the code. However, the performance of such a decoder is limited both by code structure and the degenerate nature of quantum errors. To overcome these limitations, at least in part, a number of modifications to belief propagation are developed. Central among these is the augmented decoder, which in the case of a decoding error, iteratively reattempts decoding using modified factor graph. This heuristic modification simply involves the duplication of a randomly selected subset of the graph’s check nodes, which are in one-to-one correspondence with the code’s stabilizer generators. Across a range of codes, it is shown that the decoders developed perform as well as or better than other modified decoders presented in literature. For a number of channels of physical interest, phase-flip errors occur far more frequently than bit-flip errors. When transmitting across these so-called asymmetric channels, the decoding error rate can be minimized by tailoring the code used to the channel. However, assessing the performance of codes on a given channel is made difficult by the #P-completeness of optimal decoding. To address this complexity, it is shown that the decoding error rate can be accurately approximated using only a small fraction of the possible errors caused by the channel. This approximation is then used to identify a number of cyclic stabilizer codes that perform well on two different asymmetric channels. To further build on this, a heuristic is demonstrated for assessing code performance based on the decoding error rate of an associated classical code. The complexity of calculating this classical error rate is relatively low, and it is shown that it can be used as the basis for a hill-climbing search algorithm. Such searches have yielded a large number of highly performant codes satisfying various structure constraints. The family of codeword stabilized codes encompasses both the stabilizer codes as well as many of the best known nonadditive codes. Constructing a standard form codeword stabilized code is a matter of selecting a simple undirected graph and a binary classical code. This makes designing optimal codes difficult as the number of possible graphs grows exponentially with code length, and the clique search required to construct the classical code is NP-hard. To address the exponential growth of the search space, a heuristic is developed for assessing graphs. This heuristic is then employed by a genetic algorithm that also makes use of a novel crossover operation based on spectral bisection, which is show to be superior to more standard crossover operations. With a graph selected, it is demonstrated that the complexity of the clique search required to construct the associated classical code can be mitigated through the use of a heuristic clique finding algorithm. A number of best known codes are presented that have been found using this approach.