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Birkhäuser Basel eBooks

Imprimitivity Systems and Quantum Codes

Kalyanapuram Rangachari Parthasarathy

Published Jan 1, 2003• 0 citations
Publisher

Abstract

It is shown how classical error correcting codes can be quantized to yield quantum error correcting codes as defined by E. Knill and R. Laflamme [2] by using the notion of an imprimitivity system for a group action in the sense of Frobenius and Mackey [4], [8].
DOI
10.1007/978-3-0348-8018-3_9

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We introduce a generalisation of quantum error correction, relaxing the requirement that a code should identify and correct a set of physical errors on the Hilbert space of a quantum computer exactly, instead allowing recovery up to a pre-specified admissible set of errors on the code space. We call these quantum error transmuting codes. They are of particular interest for the simulation of noisy quantum systems, and for use in algorithms inherently robust to errors of a particular character. Necessary and sufficient algebraic conditions on the set of physical and admissible errors for error transmutation are derived, generalising the Knill-Laflamme quantum error correction conditions. We demonstrate how some existing codes, including fermionic encodings, have error transmuting properties to interesting classes of admissible errors. Additionally, we report on the existence of some new codes, including low-qubit and translation invariant examples.
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Abstract. The hereby paper presents an overview of the current interpretation of decoherence in quantum theory, at the same time firmly stating that decoherence can be successfully fought against by means of the quantum error-correcting codes developed in the recent years. While decoherence itself is but a natural transition from quantum to classical, explaining the so disputed quantum origins of the classical world, its effects have to be combated or severely diminished if revolutionary techniques such as quantum communication or quantum computation are to be ever implemented. We will argue and try to prove by reference to existing leading research papers that quantum error-correction codes are a viable solution for the decoherence problem in particular settings. As a clear-cut example we tackle the five-qubit quantum circuit described by Shor, DiVincenzo and Terhal. Full generalization is in our view a simple problem of time as the theory describing it exists already and is continuously revised by top scientists. In this sense the general QECC framework developed by Knill and Laflamme is thoroughly analyzed and commented upon. 1. Scene-Setting: Decoherence as Transition Concept between Quantum and Classical If we were to consider decoherence in its full generality