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Allerton Conference on Communication, Control, and Computing

Entanglement-assisted quantum error correcting codes over nice rings

Sang-Jun Lee, Andreas Klappenecker

Published Sep 1, 2014• 2 citations
Publisher

Abstract

Entanglement-assisted quantum error correcting codes have the nice feature that they can be constructed from classical additive codes that do not need to be self-orthogonal - an advantage over stabilizer codes. In the literature, these codes were investigated for finite fields, mostly binary fields. A generalization of the Pauli basis to nice error bases indexed by rings allows one to consider alphabet sizes that are not restricted to powers of a prime. The main goal of this paper is to show how entanglement-assisted quantum error correcting codes over nice rings can be constructed. We develop the rudiments of symplectic geometry over rings and prove that an R-module with antisymmetric bicharacter can be decomposed as an orthogonal direct sum of hyperbolic pairs.

DOI
10.1109/allerton.2014.7028614

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Entanglement-assisted quantum error correcting codes have the nice feature that they can be constructed from classical additive codes that do not need to be self-orthogonal - an advantage over stabilizer codes. In the literature, these codes were investigated for finite fields, mostly binary fields. A generalization of the Pauli basis to nice error bases indexed by rings allows one to consider alphabet sizes that are not restricted to powers of a prime. The main goal of this paper is to show how entanglement-assisted quantum error correcting codes over nice rings can be constructed. We develop the rudiments of symplectic geometry over rings and prove that an R-module with antisymmetric bicharacter can be decomposed as an orthogonal direct sum of hyperbolic pairs.
Quantum error correcting codes play an essential role in protecting quantum information from the noise and the decoherence. Most quantum codes have been constructed based on the Pauli basis indexed by a finite field. With a newly introduced algebraic class called a nice ring, it is possible to construct the quantum codes such that their alphabet sizes are not restricted to powers of a prime.\n\nSubsystem codes are quantum error correcting schemes unifying stabilizer codes, decoherence free subspaces and noiseless subsystems. We show a generalization of subsystem codes over nice rings. Furthermore, we prove that free subsystem codes over a finite chain ring cannot outperform those over a finite field. We also generalize entanglement-assisted quantum error correcting codes to nice rings. With the help of the entanglement, any classical code can be used to derive the corresponding quantum codes, even if such codes are not self-orthogonal. We prove that an R-module with antisymmetric bicharacter can be decomposed as an orthogonal direct sum of hyperbolic pairs using symplectic geometry over rings. So, we can find hyperbolic pairs and commuting generators generating the check matrix of the entanglement-assisted quantum code.\n\nFault-tolerant quantum computation has been also studied over a finite field. Transversal operations are the simplest way to implement fault-tolerant quantum gates. We derive transversal Clifford operations for CSS codes over nice rings, including Fourier transforms, SUM gates, and phase gates. Since transversal operations alone cannot provide a computationally universal set of gates, we add fault-tolerant implementations of doubly-controlled Z gates for triorthogonal stabilizer codes over nice rings.\n\nFinally, we investigate optimal key exchange protocols for unconditionally secure key distribution schemes. We prove how many rounds are needed for the key exchange between any pair of the group on star networks, linear-chain networks, and general networks.
Entanglement-assisted quantum error-correcting codes, which can be seen as a generalization of quantum error-correcting codes, can be constructed from arbitrary classical linear codes by relaxing the self-orthogonality properties and using pre-shared entangled states between the sender and the receiver, and can also improve the performance of quantum error-correcting codes. In this paper, we construct some families of entanglement-assisted quantum maximum-distance-separable codes with parameters [[q2−1a,q2−1a−2d+2+c,d;c]]q[[\frac{{{q^2} - 1}}{a},\frac{{{q^2} - 1}}{a} - 2d+2 + c,d;c]]_q[[aq2−1​,aq2−1​−2d+2+c,d;c]]q​, where qqq is a prime power with the form q=am±ℓq = am \pm \ellq=am±ℓ, a=ℓ2−13a = \frac{{\ell^2} - 1}{3}a=3ℓ2−1​ is an odd integer, ℓ≡2 ( mod  6)\ell \equiv 2\ (\bmod\ 6)ℓ≡2 (mod 6) or ℓ≡4 ( mod  6)\ell \equiv 4\ (\bmod\ 6)ℓ≡4 (mod 6), and mmm is a positive integer. Most of these codes are new in the sense that their parameters are not covered by the codes available in the literature.
We generalize the stabilizer formalism for entanglement-assisted quantum error-correcting codes with noisy ebits (EAQECCs-Ne) from the binary case to the general qqq-ary case, where qqq is a prime power. By leveraging the structure of the generalized Pauli group over Fq\mathbb{F}_qFq​ and symplectic geometry over Fq2n\mathbb{F}_q^{2n}Fq2n​, we establish a unified framework for constructing EAQECCs-Ne for qudit systems. Equivalent formulations in terms of symplectic geometry over Fq\mathbb{F}_qFq​ and additive codes over Fq2n\mathbb{F}_q^{2n}Fq2n​ are derived. We further construct several families of qqq-ary EAQECCs with noise ebits and analyze their performance compared to optimal stabilizer codes. Our results demonstrate that under certain noise conditions, the proposed EAQECCs-Ne can outperform standard stabilizer codes with equivalent error-correcting capability, offering a promising approach for fault-tolerant quantum computation in high-dimensional quantum systems.
We develop the theory of entanglement-assisted quantum error-correcting (EAQEC) codes, a generalization of the stabilizer formalism to the setting in which the sender and receiver have access to preshared entanglement. Conventional stabilizer codes are equivalent to self-orthogonal symplectic codes. In contrast, EAQEC codes do not require self-orthogonality, which greatly simplifies their construction. We show how any classical binary or quaternary block code can be made into an EAQEC code. We provide a table of best known EAQEC codes with code length up to 10. With the self-orthogonality constraint removed, we see that the distance of an EAQEC code can be better than any standard quantum error-correcting code with the same fixed net yield. In a quantum computation setting, EAQEC codes give rise to catalytic quantum codes, which assume a subset of the qubits are noiseless. We also give an alternative construction of EAQEC codes by making classical entanglement-assisted codes coherent.