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arXiv:quant-ph/9910061 · Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences

Cyclic quantum error–correcting codes and quantum shift registers

Markus Grassl, Thomas Beth

Published Nov 8, 2000• 45 citations
arXivHTMLPDF

Abstract

We transfer the concept of linear feedback shift registers to quantum circuits. It is shown how to use these quantum linear shift registers for encoding and decoding cyclic quantum error–correcting codes.
DOI
10.1098/rspa.2000.0633

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A quantum-shift-register circuit acts on a set of input qubits and memory qubits, outputs a set of output qubits and updated memory qubits, and feeds the memory back into the device for the next cycle (similar to the operation of a classical shift register). Such a device finds application as an encoding and decoding circuit for a particular type of quantum error-correcting code called a quantum convolutional code. Building on the Ollivier-Tillich and Grassl-R\"otteler encoding algorithms for quantum convolutional codes, I present a method to determine a quantum-shift-register encoding circuit for a quantum convolutional code. I also determine a formula for the amount of memory that a Calderbank-Shor-Steane (CSS) quantum convolutional code requires. I then detail primitive quantum-shift-register circuits that realize all of the finite- and infinite-depth transformations in the shift-invariant Clifford group (the class of transformations important for encoding and decoding quantum convolutional codes). The memory formula for a CSS quantum convolutional code then immediately leads to a formula for the memory required by a CSS entanglement-assisted quantum convolutional code.
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.
Introduction Many of the quantum codes discussed in this book are quantum block codes. Quantum block codes are useful both in quantum computing and in quantum communication, but one of the drawbacks for quantum communication is that, in general, the sender must have her block of qubits ready before encoding takes place. This preparation may be a heavy demand on the sender when the size of the block code is large. Quantum convolutional coding theory [OT03, OT04, FGG07] offers a paradigm different from quantum block coding and has numerous benefits for quantum communication. The convolutional structure is useful when a sender possesses a stream of qubits to transmit to a receiver. It is possible to encode and decode quantum convolutional codes with quantum shift-register circuits [W09b], the natural “quantization” of a classical shift register circuit. These quantum circuits ensure a low complexity for encoding and decoding while also providing higher performance than a block code with equivalent encoding complexity [FGG07]. Quantum shift-register circuits have the property that the sender Alice and the receiver Bob can respectively send and receive qubits in an “online” fashion. Alice can encode an arbitrary number of information qubits without worrying beforehand how many she may want to send over the quantum communication channel.