Beyond stabilizer codes .I. Nice error bases

Klappenecker, Andreas; Rötteler, Martin

doi:10.1109/tit.2002.800471

Cited by 71 publications

(79 citation statements)

References 19 publications

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“…Furthermore, they generate an error group G 1 of size pq 2 with center ζ(G 1 ) = ωI (see [18,19]). Any element of G 1 can uniquely be written as ω γ X α Z β where γ ∈ {0, .…”

Section: Unitary Error Basesmentioning

confidence: 99%

Efficient Quantum Circuits for Non-Qubit Quantum Error-Correcting Codes

Grassl

Rötteler

Beth

2003

Int. J. Found. Comput. Sci.

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We present two methods for the construction of quantum circuits for quantum errorcorrecting codes (QECC). The underlying quantum systems are tensor products of subsystems (qudits) of equal dimension which is a prime power. For a QECC encoding k qudits into n qudits, the resulting quantum circuit has O(n(n − k)) gates. The running time of the classical algorithm to compute the quantum circuit is O(n(n − k) 2 ).

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“…Furthermore, they generate an error group G 1 of size pq 2 with center ζ(G 1 ) = ωI (see [18,19]). Any element of G 1 can uniquely be written as ω γ X α Z β where γ ∈ {0, .…”

Section: Unitary Error Basesmentioning

confidence: 99%

Efficient Quantum Circuits for Non-Qubit Quantum Error-Correcting Codes

Grassl

Rötteler

Beth

2003

Int. J. Found. Comput. Sci.

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“…There are also a series of interesting extensions to non-stabilizer codes [14], [15] with the aim of increasing the coding capabilities of quantum codes. For instance, a type of non-additive codes can beat the ratio 1:5 of perfect linear codes.…”

Section: Introductionmentioning

confidence: 99%

Homological error correction: Classical and quantum codes

Bombin

Martín-Delgado

2007

Journal of Mathematical Physics

102

125

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We prove several theorems characterizing the existence of homological error correction codes both classically and quantumly. Not every classical code is homological, but we find a family of classical homological codes saturating the Hamming bound. In the quantum case, we show that for nonorientable surfaces it is impossible to construct homological codes based on qudits of dimension D > 2, while for orientable surfaces with boundaries it is possible to construct them for arbitrary dimension D. We give a method to obtain planar homological codes based on the construction of quantum codes on compact surfaces without boundaries. We show how the original Shor's 9-qubit code can be visualized as a homological quantum code. We study the problem of constructing quantum codes with optimal encoding rate. In the particular case of toric codes we construct an optimal family and give an explicit proof of its optimality. For homological quantum codes on surfaces of arbitrary genus we also construct a family of codes asymptotically attaining the maximum possible encoding rate. We provide the tools of homology group theory for graphs embedded on surfaces in a self-contained manner.

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“…Our codes differ from the "Clifford codes" associated with "nice error bases" as proposed by Knill [38] and developed by Klappenecker and Rötteler [32,34,35]. The most significant difference is that the KKR approach does not yield new binary codes.…”

Section: Quantum Error Correctionmentioning

confidence: 92%

Noisy Quantum Communication and Computation

Ruskai¹

2006

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show abstract

Beyond stabilizer codes .I. Nice error bases

Cited by 71 publications

References 19 publications

Efficient Quantum Circuits for Non-Qubit Quantum Error-Correcting Codes

Efficient Quantum Circuits for Non-Qubit Quantum Error-Correcting Codes

Homological error correction: Classical and quantum codes

Noisy Quantum Communication and Computation

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