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Cyclic Quantum Error-Correcting Codes and Quantum Shift Registers

Markus Grassl, Thomas Beth

Published Jan 1, 1999• 36 citations
OpenAlex

Abstract

We transfer the concept of linear feed-back 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.

Related papers

Cyclic quantum error–correcting codes and quantum shift registers

Markus Grassl, Thomas Beth

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.

arXiv:quant-ph/9910061•Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences•Nov 8, 2000•45 citations

Quantum-shift-register circuits

Mark M. Wilde

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.

arXiv:0903.3894•Physical Review A•Jun 30, 2009•27 citations

The Quantum Hamming and Hexacodes

T. Beth, Markus Grassl

In this paper we present the relations between algebra and physics in the context of quantum error-correcting codes, showing the connection between Pauli matrices, finite fields, classical and quantum error-correcting codes. We show how to encode and especially decode (i.e., correct errors) quantum error-correcting codes. This is exemplarily demonstrated for the cyclic quantum Hamming code of length five and its extension, the quantum Hexacode.

Fortschritte der Physik•Jun 1, 1998•21 citations

Bounds on Quantum Error Correcting Codes

Frank Gaitan

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.

CRC Press eBooks•Feb 7, 2008•0 citations

Quantum convolutional codes

Mark M. Wilde

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.

Cambridge University Press eBooks•Sep 5, 2013•1 citation

Good additive cyclic quantum error-correcting codes

Ai Luo

This thesis is about the quantum error-correcting codes. Although there are many methods to construct them, finding a good quantum code is still a complicated and difficult task, for we do not know which methods to use. To solve this problem, we did a thorough research on a special class of quantum codes--additive cyclic quantum codes. A new search algorithm has been designed and a lot of good additive cyclic quantum codes found by this algorithm are presented in this thesis. By showing the success of this algorithm and great value of additive cyclic quantum codes, we have greatly reduced the complexity of finding good quantum codes. Quantum error-correction theory is key part of quantum information theory. As quantum information theory is a quite new field which has been developing fast during the last decade, we spent much time explaining the primary concepts of it. We introduced linear algebra, quantum mechanics, quantum operations, quantum error-correction theory and quantum error-correcting codes. In a word, this thesis serves two functions: (1) an exploration of quantum error-correcting codes and (2) an introduction to quantum information theory.

•Jan 1, 2004•1 citation
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.
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.
This thesis is about the quantum error-correcting codes. Although there are many methods to construct them, finding a good quantum code is still a complicated and difficult task, for we do not know which methods to use. To solve this problem, we did a thorough research on a special class of quantum codes--additive cyclic quantum codes. A new search algorithm has been designed and a lot of good additive cyclic quantum codes found by this algorithm are presented in this thesis. By showing the success of this algorithm and great value of additive cyclic quantum codes, we have greatly reduced the complexity of finding good quantum codes. Quantum error-correction theory is key part of quantum information theory. As quantum information theory is a quite new field which has been developing fast during the last decade, we spent much time explaining the primary concepts of it. We introduced linear algebra, quantum mechanics, quantum operations, quantum error-correction theory and quantum error-correcting codes. In a word, this thesis serves two functions: (1) an exploration of quantum error-correcting codes and (2) an introduction to quantum information theory.