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arXiv:1708.00617

Stabilizer codes from modified symplectic form

Tejas Gandhi, Piyush P. Kurur, Mittal, Rajat

Published Aug 2, 2017• 1 citation
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Abstract

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 $n$ is equivalent to constructing subspaces of $\mathbb{F}_p^n \times \mathbb{F}_p^n$ which are "isotropic" under the symplectic bilinear form defined by $\left\langle (\mathbf{a},\mathbf{b}),(\mathbf{c},\mathbf{d}) \right\rangle = \mathbf{a}^{\mathrm{T}} \mathbf{d} - \mathbf{b}^{\mathrm{T}} \mathbf{c}$. 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.

DOI
10.48550/arxiv.1708.00617

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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.
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.
Calderbank, Rains, Shor and Sloane (see \cite{Sloane}) showed that error-correction is possible in the context of quantum computations. Quantum stabilizer codes are a class of additive quaternary codes in binary projective spaces, which are self-orthogonal with respect to the symplectic form. A geometric description is given in \cite{Bierbra}, where also the notion of quantum cap is introduced. Quantum caps correspond to the special case of quantum stabilizer codes of distance d=4d=4d=4 when the code is linear over GF(4). In the present paper we review the translation from quantum error-correction to symplectic geometry and study quantum codes in PG(4,4) where we construct complete quantum caps with 20, 29, 30, 32, 33, 34, 36 and 38 points and incomplete quantum caps with 10, 12, 13, 20, 23, 24, 25 and 26 points and we prove the non existence of 11-quantum caps. In particular the quantum caps of sizes 36 and 38 yield positive answers to the existence questions of quantum codes [[36,26,4]][[36,26,4]][[36,26,4]] and [[38,28,4]][[38,28,4]][[38,28,4]] that remained open in the data base \cite{codetable}.
In this paper, we define and study \emph{quantum cyclic codes}, a generalisation of cyclic codes to the quantum setting. Previously studied examples of quantum cyclic codes were all quantum codes obtained from classical cyclic codes via the CSS construction. However, the codes that we study are much more general. In particular, we construct cyclic stabiliser codes with parameters [[5,1,3]][[5,1,3]][[5,1,3]], [[17,1,7]][[17,1,7]][[17,1,7]] and [[17,9,3]][[17,9,3]][[17,9,3]], all of which are \emph{not} CSS. The [[5,1,3]][[5,1,3]][[5,1,3]] code is the well known Laflamme code and to the best of our knowledge the other two are new examples. Our definition of cyclicity applies to non-stabiliser codes as well; in fact we show that the ((5,6,2))((5,6,2))((5,6,2)) nonstabiliser first constructed by Rains\etal~ cite{rains97nonadditive} and latter by Arvind \etal~\cite{arvind:2004:nonstabilizer} is cyclic. We also study stabiliser codes of length 4m+14^m +14m+1 over F2\mathbb{F}_2F2​ for which we define a notation of BCH distance. Much like the Berlekamp decoding algorithm for classical BCH codes, we give efficient quantum algorithms to correct up to $\floor{\frac{d-1}{2}}$ errors when the BCH distance is ddd.
It is well known that n-length stabilizer quantum error correcting codes (QECCs) can be obtained via n-length classical error correction codes (CECCs) over GF(4), that are additive and self-orthogonal with respect to the trace Hermitian inner product. But, most of the CECCs have been studied with respect to the Euclidean inner product. In this paper, it is shown that n-length stabilizer QECCs can be constructed via 3n-length linear CECCs over GF(2) that are self-orthogonal with respect to the Euclidean inner product. This facilitates usage of the widely studied self-orthogonal CECCs to construct stabilizer QECCs. Moreover, classical, binary, self-orthogonal cyclic codes have been used to obtain stabilizer QECCs with guaranteed quantum error correcting capability. This is facilitated by the fact that (i) self-orthogonal, binary cyclic codes are easily identified using transform approach and (ii) for such codes lower bounds on the minimum Hamming distance are known. Several explicit codes are constructed including two pure MDS QECCs.
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.
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.