Quantum Convolutional Codes Research Articles

Transforming collections of Pauli operators into equivalent collections of Pauli operators over minimal registers

Transformations which convert between fermionic modes and qubit operations have become a ubiquitous tool in quantum algorithms for simulating systems. Similarly, collections of Pauli operators might be obtained from solutions of nonlocal games and satisfiability problems. Drawing on ideas from entanglement-assisted quantum error-correcting codes and quantum convolutional codes, we prove the obtainable lower bound for the number of qubits needed to represent such Pauli operations which are equivalent to the initial ones and provide a procedure for determining such a set of minimal register Pauli operations.

Using a Deep Quantum Neural Network to Enhance the Fidelity of Quantum Convolutional Codes

The fidelity of quantum states is an important concept in quantum information. Improving quantum fidelity is very important for both quantum communication and quantum computation. In this paper, we use a quantum neural network (QNN) to enhance the fidelity of [6, 2, 2] quantum convolutional codes. Towards the circuit of quantum convolutional codes, the target quantum state |0⟩ or |1⟩ is turned into entangled quantum states, which can defend against quantum noise more effectively. As the quantum neural network works better for quantum states with low dimension, we divide the quantum circuits into two parts. Then we apply the quantum neural network to each part of the circuit. The results of the simulation show that the network performs well in enhancing the fidelity of the quantum states. Through the quantum neural network, the fidelity of the first part is enhanced from 95.2% to 99.99%, and the fidelity of the second part is enhanced from 93.88% to 94.57%.

Quantum convolutional codes concatenated with the GKP code for correcting continuous errors

Quantum convolutional codes concatenated with the GKP code for correcting continuous errors

New optimal asymmetric quantum codes and quantum convolutional codes derived from constacyclic codes

In this paper, some families of asymmetric quantum codes and quantum convolutional codes that satisfy the quantum Singleton bound are constructed by utilizing constacyclic codes with length $$n=\frac{q^2+1}{10h}$$ , where q is an odd prime power with the form $$q=10hm+t$$ or $$q=10hm+10h-t$$ , where m is a positive integer, and both h and t are odd with $$10h=t^2+1$$ and $$t\ge 3$$ . Compared with those codes constructed in the literature, the parameters of these constructed quantum codes in this paper are more general. Moreover, the distance $$d_z$$ of optimal asymmetric quantum codes $$[[n,k,d_z/d_x]]_{q^2}$$ here is larger than most of the ones given in the literature.

Classical and Quantum Convolutional Codes Derived From Algebraic Geometry Codes

In this paper, we construct new families of convolutional codes. Such codes are obtained by means of algebraic geometry codes. Additionally, more families of convolutional codes are constructed by means of puncturing, extending, expanding and by the direct product code construction applied to algebraic geometry codes. The parameters of the new convolutional codes are better than or comparable to the ones available in literature. In particular, a family of almost near MDS codes is presented.

Some families of optimal quantum codes derived from constacyclic codes

ABSTRACTQuantum maximum-distance-separable (MDS) codes form an important family of quantum codes. Quantum MDS codes are optimal, which can attain the quantum Singleton bound. In this paper, using some classes of constacyclic codes and Hermitian constructions, we propose the constructions of optimal quantum convolutional codes and optimal asymmetric quantum codes in the sense of the quantum Singleton bound. Compared with the codes obtained in the literature, the parameters of the constructed quantum codes are new as well as the distance of optimal asymmetric quantum codes constructed in this paper is larger.

A construction of quantum turbo product codes based on CSS-type quantum convolutional codes

As in classical coding theory, turbo product codes (TPCs) through serially concatenated block codes can achieve approximatively Shannon capacity limit and have low decoding complexity. However, special requirements in the quantum setting severely limit the structures of turbo product codes (QTPCs). To design a good structure for QTPCs, we present a new construction of QTPCs with the interleaved serial concatenation of [Formula: see text]-type quantum convolutional codes (QCCs). First, [Formula: see text]-type QCCs are proposed by exploiting the theory of CSS-type quantum stabilizer codes and QCCs, and the description and the analysis of encoder circuit are greatly simplified in the form of Hadamard gates and C-NOT gates. Second, the interleaved coded matrix of QTPCs is derived by quantum permutation SWAP gate definition. Finally, we prove the corresponding relation on the minimum Hamming distance of QTPCs associated with classical TPCs, and describe the state diagram of encoder and decoder of QTPCs that have a highly regular structure and simple design idea.

On the MacWilliams Identity for Classical and Quantum Convolutional Codes

The weight generating functions associated with convolutional codes (CCs) are based on state space realizations or the weight adjacency matrices (WAMs). The MacWilliams identity for CCs on the WAMs was first established by Gluesing- Luerssen and Schneider in the case of minimal encoders, and generalized by Forney. We consider this problem in the viewpoint of constraint codes and obtain a simple and direct proof of this MacWilliams identity in the case of minimal encoders. For our purpose, we choose a different representation for the exact weight generating function (EWGF) of a block code, by defining it as a linear combination of orthonormal vectors in Dirac bra-ket notation. This representation provides great flexibility so that general split weight generating functions and their MacWilliams identities can be easily obtained from the MacWilliams identity for EWGFs. As a result, we also obtain the MacWilliams identity for the input-parity weight adjacency matrices of a systematic convolutional code and its dual. Finally, paralleling the development of the classical case, we establish the MacWilliams identity for quantum convolutional codes.

New quantum MDS-convolutional codes derived from constacyclic codes

In this paper, we utilize a family of Hermitian dual-containing constacyclic codes to construct classical and quantum MDS convolutional codes. Our classical and quantum convolutional codes are optimal in the sense that they attain the classical (quantum) generalized Singleton bound.

Asymmetric quantum convolutional codes

In this paper, we construct the first families of asymmetric quantum convolutional codes (AQCCs). These new AQCCs are constructed by means of the CSS-type construction applied to suitable families of classical convolutional codes, which are also constructed here. The new codes have non-catastrophic generator matrices, and they have great asymmetry. Since our constructions are performed algebraically, i.e. we develop general algebraic methods and properties to perform the constructions, it is possible to derive several families of such codes and not only codes with specific parameters. Additionally, several different types of such codes are obtained.

New optimal quantum convolutional codes

One of the most challenges to prove the feasibility of quantum computers is to protect the quantum nature of information. Quantum convolutional codes are aimed at protecting a stream of quantum information in a long distance communication, which are the correct generalization to the quantum domain of their classical analogs. In this paper, we construct some classes of quantum convolutional codes by employing classical constacyclic codes. These codes are optimal in the sense that they attain the Singleton bound for pure convolutional stabilizer codes.

Some families of asymmetric quantum codes and quantum convolutional codes from constacyclic codes

Quantum maximal-distance-separable (MDS) codes are an important class of quantum codes. Recently, many scholars utilize constacyclic codes to construct some quantum MDS codes. In this paper, several new families of optimal asymmetric quantum codes and optimal quantum convolutional codes are constructed by using constacyclic codes. Moreover, these quantum codes constructed in this paper are different from the ones in the literature.

A Construction of MDS Quantum Convolutional Codes

In this paper, two new families of MDS quantum convolutional codes are constructed. The first one can be regarded as a generalization of [36, Theorem 6.5], in the sense that we do not assume that q≡1 (mod 4). More specifically, we obtain two classes of MDS quantum convolutional codes with parameters: (i) [(q 2+1,q 2−4i+3,1;2,2i+2)] q , where q≥5 is an odd prime power and 2≤i≤(q−1)/2; (ii) $[(\frac {q^{2}+1}{10},\frac {q^{2}+1}{10}-4i,1;2,2i+3)]_{q}$ , where q is an odd prime power with the form q=10m+3 or 10m+7 (m≥2), and 2≤i≤2m−1.

Asymmetric Quantum Codes and Quantum Convolutional Codes Derived from Nonprimitive Non-Narrow-Sense BCH Codes

Nonprimitive non-narrow-sense BCH codes have been studied by many scholars. In this paper, we utilize nonprimitive non-narrow-sense BCH codes to construct a family of asymmetric quantum codes and two families of quantum convolutional codes. Most quantum codes constructed in this paper are different from the ones in the literature. Moreover, some quantum codes constructed in this paper have good parameters compared with the ones in the literature.

Quantum convolutional codes derived from constacyclic codes

In this paper, three families of quantum convolutional codes are constructed. The first one and the second one can be regarded as a generalization of Theorems 3, 4, 7 and 8 [J. Chen, J. Li, F. Yang and Y. Huang, Int. J. Theor. Phys., doi:10.1007/s10773-014-2214-6 (2014)], in the sense that we drop the constraint q ≡ 1 ( mod 4). Furthermore, the second one and the third one attain the quantum generalized Singleton bound.

Nonbinary Quantum Convolutional Codes Derived from Negacyclic Codes

In this paper, some families of nonbinary quantum convolutional codes are constructed by using negacyclic codes. These nonbinary quantum convolutional codes are different from quantum convolutional codes in the literature. Moreover, we construct a family of optimal quantum convolutional codes.

On Classical and Quantum MDS-Convolutional BCH Codes

Several new families of multi-memory classical convolutional Bose-Chaudhuri-Hocquenghem (BCH) codes as well as families of unit-memory quantum convolutional codes are constructed in this paper. Our unit-memory classical and quantum convolutional codes are optimal in the sense that they attain the classical (quantum) generalized Singleton bound. The constructions presented in this paper are performed algebraically and not by computational search.

Recursive Quantum Convolutional Encoders are Catastrophic: A Simple Proof

Poulin, Tillich, and Ollivier discovered an important separation between the classical and quantum theories of convolutional coding, by proving that a quantum convolutional encoder cannot be both non-catastrophic and recursive. Non-catastrophicity is desirable so that an iterative decoding algorithm converges when decoding a quantum turbo code whose constituents are quantum convolutional codes, and recursiveness is as well so that a quantum turbo code has a minimum distance growing nearly linearly with the length of the code, respectively. Their proof of the aforementioned theorem was admittedly "rather involved," and as such, it has been desirable since their result to find a simpler proof. In this paper, we furnish a proof that is arguably simpler. Our approach is group-theoretic---we show that the subgroup of memory states that are part of a zero physical-weight cycle of a quantum convolutional encoder is equivalent to the centralizer of its "finite-memory" subgroup (the subgroup of memory states which eventually reach the identity memory state by identity operator inputs for the information qubits and identity or Pauli-Z operator inputs for the ancilla qubits). After proving that this symmetry holds for any quantum convolutional encoder, it easily follows that an encoder is non-recursive if it is non-catastrophic. Our proof also illuminates why this no-go theorem does not apply to entanglement-assisted quantum convolutional encoders---the introduction of shared entanglement as a resource allows the above symmetry to be broken.

Degenerate Viterbi Decoding

We present a decoding algorithm for quantum convolutional codes that finds the class of degenerate errors with the largest probability conditioned on a given error syndrome. The algorithm runs in time linear with the number of qubits. Previous decoding algorithms for quantum convolutional codes optimized the probability over individual errors instead of classes of degenerate errors. Using Monte Carlo simulations, we show that this modification to the decoding algorithm results in a significantly lower block error rate.

Minimal-Memory, Noncatastrophic, Polynomial-Depth Quantum Convolutional Encoders

Quantum convolutional coding is a technique for encoding a stream of quantum information before transmitting it over a noisy quantum channel. Two important goals in the design of quantum convolutional encoders are to minimize the memory required by them and to avoid the catastrophic propagation of errors. In a previous paper, we determined minimal-memory, non-catastrophic, polynomial-depth encoders for a few exemplary quantum convolutional codes. In this paper, we elucidate a general technique for finding an encoder of an arbitrary quantum convolutional code such that the encoder possesses these desirable properties. We also provide an elementary proof that these encoders are non-recursive. Finally, we apply our technique to many quantum convolutional codes from the literature.