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New Quantum Burst-Error Correcting Codes from Interleaving Technique

Cibele Cristina Trinca, Clarice Dias de Albuquerque, Reginaldo Palazzo, J. Carmelo Interlando, Antonio Aparecido de Andrade, Ricardo Augusto Watanabe

Published Dec 4, 2022• 4 citations
Publisher

Abstract

Quantum memory is essential for the development of many devices in quantum information processing, including a synchronization tool that can match the various processes in a quantum computer, a quantum gate that maintains the identity of any state, and a mechanism for converting predetermined photons into on-demand photons. Quantum memory can be used in many aspects, such as quantum computing and quantum communication. Continuous research and experiments have enabled quantum memory to realize the storage of qubits. On the other hand, interleaving techniques are used to deal with burst of errors. Many works from literature provide effective interleaving techniques for combating burst of errors by using classical error-correcting codes. However, to the best of our knowledge, little is known regarding interleaving techniques for combating cluster of errors in topological quantum error-correcting codes. Thenceforward, it is proposed in this work a new interleaving technique by using good toric quantum codes, namely Bombin and Martin-Delgado toric codes and another class of toric codes from literature which is described in this work. As a consequence, by applying this new interleaving method to such toric quantum codes, new quantum burst-error correcting codes are obtained which improve the coding gain. In addition to the proposed quantum interleaving technique improves such parameter, it can be used for burst-error correction in errors which are located, quantum data stored and quantum channels with memory.

DOI
10.1109/globecom48099.2022.10000761

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Quantum memory is essential for the development of many devices in quantum information processing, including a synchronization tool that can match the various processes in a quantum computer, a quantum gate that maintains the identity of any state, and a mechanism for converting predetermined photons into on-demand photons. Quantum memory can be used in many aspects, such as quantum computing and quantum communication. Continuous research and experiments have enabled quantum memory to realize the storage of qubits. On the other hand, interleaving techniques are used to deal with burst of errors. Many works from literature provide effective interleaving techniques for combating burst of errors by using classical error-correcting codes. However, to the best of our knowledge, little is known regarding interleaving techniques for combating cluster of errors in topological quantum error-correcting codes. Thenceforward, it is proposed in this work a new interleaving technique by using good toric quantum codes, namely Bombin and Martin-Delgado toric codes and another class of toric codes from literature which is described in this work. As a consequence, by applying this new interleaving method to such toric quantum codes, new quantum burst-error correcting codes are obtained which improve the coding gain. In addition to the proposed quantum interleaving technique improves such parameter, it can be used for burst-error correction in errors which are located, quantum data stored and quantum channels with memory.
Characterization of nnn-Dimensional Toric and Burst-Error-Correcting Quantum Codes from Lattice Codes
Quantum error correction is essential for the development of any scalable quantum computer. In this work we introduce a generalization of a quantum interleaving method for combating clusters of errors in toric quantum error-correcting codes. We present new nnn-dimensional toric quantum codes, where n≥5n\geq 5n≥5, which are featured by lattice codes and apply the proposed quantum interleaving method to such new nnn-dimensional toric quantum codes. Through the application of this method to these novel nnn-dimensional toric quantum codes we derive new nnn-dimensional quantum burst-error-correcting codes. Consequently, nnn-dimensional toric quantum codes and burst-error-correcting quantum codes are provided offering both a good code rate and a significant coding gain when it comes to toric quantum codes. Another important consequence from the presented nnn-dimensional toric quantum codes is that if the Golomb and Welch conjecture in \cite{perfcodes} regarding the Lee sphere in nnn dimensions for the respective close packings holds true, then it follows that these nnn-dimensional toric quantum codes are the only possible ones to be obtained from lattice codes. Moreover, such a methodology can be applied for burst error correction in cases involving localized errors, quantum data storage and quantum channels with memory.
To construct a quantum code with much error-correcting ability, the philosophy of the quantum interleaving is illustrated by introducing the block interleaving techniques. Quantum interleaving is a process to rearrange the qubits of several states of the same code so as to spread the same quantum errors over multiple states. Utilizing the interleaving technique, the concatenated k n -qubit code is constructed in this paper with more error-correcting ability than utilizing only one n -qubit quantum error-correcting code (QECC) without introducing the redundant qubits. It hence suggests an important way to design an quantum convolution code as the classical convolutional code has been constructed. For the application, the proposed quantum interleaving technique becomes an effective means to combat quantum bursts of errors by converting quantum bursts of errors into quantum random errors.
In a noisy quantum channel, burst errors usually causes multiple bit-phase flips that are adjacent to each other (Vatan, Raichowdry and Anantram, 1999; Grassl and Beth, 2000). This effectively makes some quantum error-correcting codes to be unusable. In this paper, the concept of interleaving qubit transmission is proposed. By using the interleaving method in a quantum information transmission process, burst errors in a quantum channel are re-arranged into different time slots. This relocation makes the error bits non-adjacent and hence increases the probability of information recovery. Note that although this scheme enhances the error-correcting capability of the encoded quantum information, it does not increase the redundancy of the code, because only the qubit sequences are relocated. In addition to the interleaving scheme, implementation of the proposed scheme at the circuit level is also discussed in this paper.
Quantum information theory is concerned with identifying how quantum mechanical resources, such as entangled quantum states, can be utilized for a number of information processing tasks, including data storage, computation, communication, and cryptography. Efficient quantum algorithms and protocols have been developed for performing some tasks (e.g., factoring large numbers, securely communicating over a public channel, and simulating quantum mechanical systems) that appear to be very difficult with just classical resources. In addition to identifying the separation between classical and quantum computational power, much of the theoretical focus in this field over the last decade has been concerned with finding novel ways of encoding quantum information that are robust against errors, which is an important step toward building practical quantum information processing devices. In this thesis I present some results on the quantum error-correcting properties of oscillator codes (also described as symplectic lattice codes) and toric codes. Any harmonic oscillator system, such as a mode of light, can be encoded with quantum information via symplectic lattice codes that are robust against shifts in the system's continuous quantum variables. I show the existence of lattice codes whose achievable rates match the one-shot coherent information over the Gaussian quantum channel. Also, I construct a family of symplectic self-dual lattices and search for optimal encodings of quantum information distributed between several oscillators. Toric codes provide encodings of quantum information into two-dimensional spin lattices that are robust against local clusters of errors and which require only local quantum operations for error correction. Numerical simulations of this system under realistic error models provide a calculation of the accuracy threshold for quantum memory using toric codes, which can be related to phase transitions in particular condensed matter models. I also present a local classical processing scheme for correcting errors on toric codes, which demonstrates that quantum information can be maintained in two dimensions by purely local quantum and classical resources.
In practical communication and computation systems, errors occur predominantly in adjacent positions rather than in a random manner. In this paper, we develop a stabilizer formalism for quantum burst error correction codes (QBECC) to combat such error patterns in the quantum regime. Our contributions are as follows. Firstly, we derive an upper bound for the correctable burst errors of QBECCs, the quantum Reiger bound (QRB). Secondly, we propose two constructions of QBECCs: one by heuristic computer search and the other by concatenating two quantum tensor product codes (QTPCs). We obtain several new QBECCs with better parameters than existing codes with the same coding length. Moreover, some of the constructed codes can saturate the quantum Reiger bounds. Finally, we perform numerical experiments for our constructed codes over Markovian correlated depolarizing quantum memory channels, and show that QBECCs indeed outperform standard QECCs in this scenario.