Entanglement-assisted quantum MDS codes from negacyclic codes

The entanglement-assisted formalism generalizes the standard stabilizer formalism, which can transform arbitrary classical linear codes into entanglement-assisted quantum error-correcting codes (EAQECCs) by using pre-shared entanglement between the sender and the receiver. In this work, we construct six classes of q-ary entanglement-assisted quantum MDS (EAQMDS) codes based on classical negacyclic MDS codes by exploiting two or more pre-shared maximally entangled states. We show that two of these six classes q-ary EAQMDS have minimum distance more larger than \(q+1\). Most of these q-ary EAQMDS codes are new in the sense that their parameters are not covered by the codes available in the literature.     

Entanglement-assisted quantum MDS codes constructed from negacyclic codes

Recently, entanglement-assisted quantum codes have been constructed from cyclic codes by some scholars. However, how to determine the number of shared pairs required to construct entanglement-assisted quantum codes is not an easy work. In this paper, we propose a decomposition of the defining set of negacyclic codes. Based on this method, four families of entanglement-assisted quantum codes constructed in this paper satisfy the entanglement-assisted quantum Singleton bound, where the minimum distance satisfies \(q+1 \le d\le \frac{n+2}{2}\). Furthermore, we construct two families of entanglement-assisted quantum codes with maximal entanglement.     

Entanglement-assisted quantum MDS codes constructed from constacyclic codes

Recently, entanglement-assisted quantum error-correcting codes (EAQECCs) have been constructed by cyclic codes and negacyclic codes. In this paper, by decomposing the defining set of constacyclic codes, we construct four classes of new EAQECCs, which satisfy the entanglement-assisted quantum Singleton bound.     

Entanglement-assisted quantum feedback control

The main advantage of quantum metrology relies on the effective use of entanglement, which indeed allows us to achieve strictly better estimation performance over the standard quantum limit. In this paper, we propose an analogous method utilizing entanglement for the purpose of feedback control. The system considered is a general linear dynamical quantum system, where the control goal can be systematically formulated as a linear quadratic Gaussian control problem based on the quantum Kalman filtering method; in this setting, an entangled input probe field is effectively used to reduce the estimation error and accordingly the control cost function. In particular, we show that, in the problem of cooling an opto-mechanical oscillator, the entanglement-assisted feedback control can lower the stationary occupation number of the oscillator below the limit attainable by the controller with a coherent probe field and furthermore beats the controller with an optimized squeezed probe field.     

Constructing quantum codes

Quantum error correcting codes are indispensable for quantum information processing and quantum computation. In 1995 and 1996, Shor and Steane gave first several examples of quantum codes from classical error correcting codes. The construction of efficient quantum codes is now an active multi-discipline research field. In this paper we review the known several constructions of quantum codes and present some examples.     

Constructing quantum codes

Quantum error correcting codes are indispensable for quantum information processing and quantum computation. In 1995 and 1996, Shor and Steane gave first several examples of quantum codes from classical error correcting codes. The construction of efficient quantum codes is now an active multi-discipline research field. In this paper we review the known several constructions of quantum codes and present some examples.     

Some negacyclic BCH codes and quantum codes

Quantum Information Processing - Any quantum communication task requires a common reference frame (i.e., phase, coordinate system). In particular, quantum key distribution requires different bases...     

Asymmetric quantum codes: new codes from old

In this paper we extend to asymmetric quantum error-correcting codes the construction methods, namely: puncturing, extending, expanding, direct sum and the $({ \mathbf u}| \mathbf{u}+{ \mathbf v})$ construction. By applying these methods, several families of asymmetric quantum codes can be constructed. Consequently, as an example of application of quantum code expansion developed here, new families of asymmetric quantum codes derived from generalized Reed-Muller codes, quadratic residue, Bose-Chaudhuri-Hocquenghem, character codes and affine-invariant codes are constructed.     

New quantum codes constructed from quaternary BCH codes

In this paper, we firstly study construction of new quantum error-correcting codes (QECCs) from three classes of quaternary imprimitive BCH codes. As a result, the improved maximal designed distance of these narrow-sense imprimitive Hermitian dual-containing quaternary BCH codes are determined to be much larger than the result given according to Aly et al. (IEEE Trans Inf Theory 53:1183–1188, 2007) for each different code length. Thus, families of new QECCs are newly obtained, and the constructed QECCs have larger distance than those in the previous literature. Secondly, we apply a combinatorial construction to the imprimitive BCH codes with their corresponding primitive counterpart and construct many new linear quantum codes with good parameters, some of which have parameters exceeding the finite Gilbert–Varshamov bound for linear quantum codes.     

Two families of BCH codes and new quantum codes

In this paper, two families of non-narrow-sense (NNS) BCH codes of lengths \(n=\frac{q^{2m}-1}{q^2-1}\) and \(n=\frac{q^{2m}-1}{q+1}\) (\(m\ge 3)\) over the finite field \(\mathbf {F}_{q^2}\) are studied. The maximum designed distances \(\delta ^\mathrm{new}_\mathrm{max}\) of these dual-containing BCH codes are determined by a careful analysis of properties of the cyclotomic cosets. NNS BCH codes which achieve these maximum designed distances are presented, and a sequence of nested NNS BCH codes that contain these BCH codes with maximum designed distances are constructed and their parameters are computed. Consequently, new nonbinary quantum BCH codes are derived from these NNS BCH codes. The new quantum codes presented here include many classes of good quantum codes, which have parameters better than those constructed from narrow-sense BCH codes, negacyclic and constacyclic BCH codes in the literature.     

Optimal binary codes and binary construction of quantum codes

This paper discusses optimal binary codes and pure binary quantum codes created using Steane construction. First, a local search algorithm for a special subclass of quasi-cyclic codes is proposed, then five binary quasi-cyclic codes are built. Second, three classical construction methods are generalized for new codes from old such that they are suitable for constructing binary self-orthogonal codes, and 62 binary codes and six subcode chains of obtained self-orthogonal codes are designed. Third, six pure binary quantum codes are constructed from the code pairs obtained through Steane construction. There are 66 good binary codes that include 12 optimal linear codes, 45 known optimal linear codes, and nine known optimal self-orthogonal codes. The six pure binary quantum codes all achieve the performance of their additive counterparts constructed by quaternary construction and thus are known optimal codes.     

On quantum SPC product codes

Quantum Information Processing - One central theme in quantum error correction is to construct quantum codes that have large minimum distances. It has been a great challenge to construct new...     

Non-binary entanglement-assisted quantum stabilizer codes

In this paper, we present the p^m-ary entanglement-assisted (EA) stabilizer formalism, where p is a prime and m is a positive integer. Given an arbitrary non-abelian “stabilizer”, the problem of code construction and encoding is settled perfectly in the case of m = 1. The optimal number of required maximally entangled pairs is discussed and an algorithm to determine the encoding and decoding circuits is proposed. We also generalize several bounds on p-ary EA stabilizer codes, such as the BCH bound, the G-V bound and the linear programming bound. However, the issue becomes tricky when it comes to m > 1, in which case, the former construction method applies only when the non-commuting “stabilizer” satisfies a sophisticated limitation.     

New quantum codes from dual-containing cyclic codes over finite rings

Let \(R=\mathbb {F}_{2^{m}}+u\mathbb {F}_{2^{m}}+\cdots +u^{k}\mathbb {F}_{2^{m}}\), where \(\mathbb {F}_{2^{m}}\) is the finite field with \(2^{m}\) elements, m is a positive integer, and u is an indeterminate with \(u^{k+1}=0.\) In this paper, we propose the constructions of two new families of quantum codes obtained from dual-containing cyclic codes of odd length over R. A new Gray map over R is defined, and a sufficient and necessary condition for the existence of dual-containing cyclic codes over R is given. A new family of \(2^{m}\)-ary quantum codes is obtained via the Gray map and the Calderbank–Shor–Steane construction from dual-containing cyclic codes over R. In particular, a new family of binary quantum codes is obtained via the Gray map, the trace map and the Calderbank–Shor–Steane construction from dual-containing cyclic codes over R.     

基于三个自对偶码的S-链和量子码构造

用随机搜索算法和典型群理论,研究了双循环形自对偶码D3,D4和D5的对偶距离d⊥满足3≤d⊥≤7的子码,确立了这些子码构成的自正交子码链及它们的对偶构成的S-链。利用得到的S-链,由Steane构造法构造出新的量子纠错码。     

Application of constacyclic codes to entanglement-assisted quantum maximum distance separable codes

The entanglement-assisted stabilizer formalism overcomes the dual-containing constraint of standard stabilizer formalism for constructing quantum codes. This allows ones to construct entanglement-assisted quantum error-correcting codes (EAQECCs) from arbitrary linear codes by pre-shared entanglement between the sender and the receiver. However, it is not easy to determine the number c of pre-shared entanglement pairs required to construct an EAQECC from arbitrary linear codes. In this paper, let q be a prime power, we aim to construct new q-ary EAQECCs from constacyclic codes. Firstly, we define the decomposition of the defining set of constacyclic codes, which transforms the problem of determining the number c into determining a subset of the defining set of underlying constacyclic codes. Secondly, five families of non-Hermitian dual-containing constacyclic codes are discussed. Hence, many entanglement-assisted quantum maximum distance separable codes with \(c\le 7\) are constructed from them, including ones with minimum distance \(d\ge q+1\). Most of these codes are new, and some of them have better performance than ones obtained in the literature.     

Dualities and identities for entanglement-assisted quantum codes

The dual of an entanglement-assisted quantum error-correcting (EAQEC) code is the code resulting from exchanging the original code’s information qubits with its ebits. To introduce this notion, we show how entanglement-assisted repetition codes and accumulator codes are dual to each other, much like their classical counterparts, and we give an explicit, general quantum shift-register circuit that encodes both classes of codes. We later show that our constructions are optimal, and this result completes our understanding of these dual classes of codes. We also establish the Gilbert–Varshamov bound and the Plotkin bound for EAQEC codes, and we use these to examine the existence of some EAQEC codes. Finally, we provide upper bounds on the block error probability when transmitting maximal-entanglement EAQEC codes over the depolarizing channel, and we derive variations of the hashing bound for EAQEC codes, which is a lower bound on the maximum rate at which reliable communication over Pauli channels is possible with the use of pre-shared entanglement.     

Optimal correction of concatenated fault-tolerant quantum codes

We present a method of concatenated quantum error correction in which improved classical processing is used with existing quantum codes and fault-tolerant circuits to more reliably correct errors. Rather than correcting each level of a concatenated code independently, our method uses information about the likelihood of errors having occurred at lower levels to maximize the probability of correctly interpreting error syndromes. Results of simulations of our method applied to the [[4,1,2]] subsystem code indicate that it can correct a number of discrete errors up to half of the distance of the concatenated code, which is optimal.