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IEEE International Symposium on Information Theory (ISIT)

Stabilizer codes from modified symplectic forms

Tejas Gandhi, Piyush P. Kurur, Rajat Mittal

Published Jul 1, 2019• 5 citations
Publisher

Abstract

In this paper, we construct explicit families of quantum error correcting codes based on cyclic codes. But for a crucial theoretical idea that we introduce here, namely working with a modified symplectic form as the basis of Weyl commutation, these codes would not have been possible - under the standard symplectic form, there are Galois theoretic no-go theorems on cyclic codes [7], [Corollary IV.5, page 651], [6], [Corollary 5.16, page 51] of certain lengths. More importantly, we circumvent the above no-go theorems without disturbing the one crucial property that cyclic codes have - efficient decoding algorithms. Recall that for general codes, efficient decoding algorithms do not exist if some widely believed complexity theoretic assumptions are true hence this is a crucial property to have. Cyclicity is a basis dependent property and for codes that we construct, is only evident in the modified symplectic setting. If we were to recast this code using the standard symplectic form, there would be a basis change on the underlying vector space that disturbs cyclicity. Hence, this change of perspective that we have here is crucial not only in the construction, where we had to circumvent the no-go theorems that we mentioned above, but also in the design of efficient decoding algorithm. Theoretically our result is also tight: we have a complete characterization (Theorem 6) of the codes that we construct here.

DOI
10.1109/isit.2019.8849639

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In this paper, we construct explicit families of quantum error correcting codes based on cyclic codes. But for a crucial theoretical idea that we introduce here, namely working with a modified symplectic form as the basis of Weyl commutation, these codes would not have been possible - under the standard symplectic form, there are Galois theoretic no-go theorems on cyclic codes [7], [Corollary IV.5, page 651], [6], [Corollary 5.16, page 51] of certain lengths. More importantly, we circumvent the above no-go theorems without disturbing the one crucial property that cyclic codes have - efficient decoding algorithms. Recall that for general codes, efficient decoding algorithms do not exist if some widely believed complexity theoretic assumptions are true hence this is a crucial property to have. Cyclicity is a basis dependent property and for codes that we construct, is only evident in the modified symplectic setting. If we were to recast this code using the standard symplectic form, there would be a basis change on the underlying vector space that disturbs cyclicity. Hence, this change of perspective that we have here is crucial not only in the construction, where we had to circumvent the no-go theorems that we mentioned above, but also in the design of efficient decoding algorithm. Theoretically our result is also tight: we have a complete characterization (Theorem 6) of the codes that we construct here.
Stabilizer codes form an important class of quantum error correcting codes which have an elegant theory, efficient error detection, and many known examples. Constructing stabilizer codes of length nnn is equivalent to constructing subspaces of Fpn×Fpn\mathbb{F}_p^n \times \mathbb{F}_p^nFpn​×Fpn​ which are "isotropic" under the symplectic bilinear form defined by ⟨(a,b),(c,d)⟩=aTd−bTc\left\langle (\mathbf{a},\mathbf{b}),(\mathbf{c},\mathbf{d}) \right\rangle = \mathbf{a}^{\mathrm{T}} \mathbf{d} - \mathbf{b}^{\mathrm{T}} \mathbf{c}⟨(a,b),(c,d)⟩=aTd−bTc. As a result, many, but not all, ideas from the theory of classical error correction can be translated to quantum error correction. One of the main theoretical contribution of this article is to study stabilizer codes starting with a different symplectic form. In this paper, we concentrate on cyclic codes. Modifying the symplectic form allows us to generalize the previous known construction for linear cyclic stabilizer codes, and in the process, circumvent some of the Galois theoretic no-go results proved there. More importantly, this tweak in the symplectic form allows us to make use of well known error correcting algorithms for cyclic codes to give efficient quantum error correcting algorithms. Cyclicity of error correcting codes is a "basis dependent" property. Our codes are no more "cyclic" when they are derived using the standard symplectic forms (if we ignore the error correcting properties like distance, all such symplectic forms can be converted to each other via a basis transformation). Hence this change of perspective is crucial from the point of view of designing efficient decoding algorithm for these family of codes. In this context, recall that for general codes, efficient decoding algorithms do not exist if some widely believed complexity theoretic assumptions are true.
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.
We prove that certain classical cyclic redundancy check codes can be used for classical error correction and not just classical error detection. We extend the idea of classical cyclic redundancy check codes to quantum cyclic redundancy check codes. This allows us to construct quantum stabilizer codes which can correct burst errors where the burst length attains the quantum Reiger bound. We then consider a certain family of quantum cyclic redundancy check codes for which we present a fast linear time decoding algorithm.
The previous two chapters laid out the theoretical framework for quantum error correcting codes, and especially for quantum stabilizer codes. Following Cleve and Gottesman [1], the present chapter will explain how to construct quantum circuits that efficiently encode and decode a quantum stabilizer code. Before entering into that discussion, it proves advantageous to introduce a standard form for quantum stabilizer codes. This is done in Section 4.1. Assuming that a quantum stabilizer code in standard form has been given, Sections 4.2 and 4.3 explain how to, respectively, encode and decode its code­ words. A brief summary of quantum circuits is given in Appendix C.
Orthogonal geometric constructions are the basis of many many quantum error-correcting codes (QEC), but strict orthogonality constraints limit design flexibility and resource efficiency. We introduce a quasi-orthogonal geometric framework for stabilizer codes that relaxes these constraints while preserving the symplectic commutation structure on the binary symplectic space F22\mathbb{F}_{2}^{2}F22​. The approach permits controlled overlap between X- and Z-check supports, leading to quasi-orthogonal Pauli operators and a generalized notion of effective distance defined via induced anti-commutation with logical operators. This relaxation expands the stabilizer design space, enabling codes that approach the Gilbert-Varshamov regime with improved logical rates at moderate distances. Finite-length constructions, including quasi-orthogonal variants of the [[8,3,≈3]][[8,3,\approx 3]][[8,3,≈3]], [[10,4,≈3]][[10,4,\approx 3]][[10,4,≈3]], [[13,1,5]][[13,1,5]][[13,1,5]], and [[29,1,11]][[29,1,11]][[29,1,11]] codes, demonstrate consistent improvements over strictly orthogonal counterparts. Under depolarizing noise with error rates up to p=0.30p=0.30p=0.30, logical error rates, fidelities, and trace distances improve by up to two orders of magnitude. These improvements reflect the increased connectivity of the underlying stabilizer geometry while remaining compatible with standard decoding schemes. The proposed framework offers a principled extension of stabilizer code design through quasi-orthogonal geometric structures.
Existing quantum stabilizer codes constructed from the classic binary codes exclusively belong to the special subclass of Calderbank-Shor-Steane (CSS) codes. This paper fills in the gap by proposing five systematic constructions for non-CSS stabilizer codes, the first four of which are based on classic binary quasi-cyclic low-density parity-check (LDPC) codes and last on classic binary LDPC-convolutional codes. These new constructions exploit structured sparse graphs, make essential use of simple and powerful coding techniques including concatenation, rotation and scrambling, and generate rich classes of codes with a wide range of lengths and rates. We derive the sufficient, and in some cases also the necessary, conditions for each construction to satisfy the general symplectic inner product (SIP) condition, and develop practical decoder algorithms for these codes. The resulting codes are the first classes of non-CSS quantum LDPC codes and non-CSS quantum convolutional codes rooted from classic binary codes (rather than codes in GF(4)), and some of them perform as well as or better than the existing quantum codes.