Nonbinary quantum stabilizer codes

Ashikhmin, Alexei; Knill, Emanuel

doi:10.1109/18.959288

Cited by 500 publications

(492 citation statements)

References 21 publications

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“…, 0 in this notation. That this collection forms an orthonormal basis follows from the fact that the Z p operators commute with the C lm operators, so can be moved through the unitary U on the right side of (6). As the states Z ν |+ , 0 ≤ ν ≤ D − 1, are an orthonormal basis for a single qudit, their products form an orthonormal basis for n qudits.…”

Section: B Graph Statesmentioning

confidence: 99%

“…It has been extended to cases where D is prime or a prime power in [6,24,30]. In [8] stabilizer codes were extended in a very general fashion to arbitrary D from a point of view that includes encoding.…”

Section: G-additive Codes As Stabilizer Codesmentioning

confidence: 99%

“…To apply these results to a general graph G on n qubits, note that the unitary U defined in (7) provides, through (6) and (10), a one-to-one map of the graph basis states of the trivial G 0 onto the graph basis states of G. At the same time the one-to-one map UP U † carries the S satisfying C1 and C2 (and possibly C3) for the G 0 code to the corresponding S, satisfying the same conditions for the G code. (The reverse maps are obtained by interchanging U † and U.)…”

Section: C3mentioning

confidence: 99%

“…10 of [4]. Most of the early work dealt with codes for qubits, with a Hilbert space of dimension D = 2, but qudit codes with D > 2 have also been studied [5,6,7,8,9,10,11]. They are of intrinsic interest and could turn out to be of some practical value.…”

Section: Introductionmentioning

confidence: 99%

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Quantum-error-correcting codes using qudit graph states

Looi

Gheorghiu

et al. 2008

Phys. Rev. A

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Graph states are generalized from qubits to collections of n qudits of arbitrary dimension D, and simple graphical methods are used to construct both additive and nonadditive, as well as degenerate and nondegenerate, quantum error correcting codes. Codes of distance 2 saturating the quantum Singleton bound for arbitrarily large n and D are constructed using simple graphs, except when n is odd and D is even. Computer searches have produced a number of codes with distances 3 and 4, some previously known and some new. The concept of a stabilizer is extended to general D, and shown to provide a dual representation of an additive graph code.

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Section: B Graph Statesmentioning

confidence: 99%

Section: G-additive Codes As Stabilizer Codesmentioning

confidence: 99%

Section: C3mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Quantum-error-correcting codes using qudit graph states

Looi

Gheorghiu

et al. 2008

Phys. Rev. A

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“…32,33,34. The argument in this subsection can be extended to linear l-adic stabilizer codes for a prime l with the following exception: In the proof of Lemma 11, there does not always exist σ ∈ GU n (F l 2 ) such that σ x = y for a pair of vectors x, y ∈ F n l 2 with τ ( x, x) = 0 and τ ( y, y) = 0.…”

Section: Remark 13 From Lemma 11 We Can Show That There Exists An [[Nmentioning

confidence: 99%

Lower bound for the quantum capacity of a discrete memoryless quantum channel

Matsumoto

Uyematsu

2002

Journal of Mathematical Physics

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We generalize the random coding argument of stabilizer codes and derive a lower bound on the quantum capacity of an arbitrary discrete memoryless quantum channel. For the depolarizing channel, our lower bound coincides with that obtained by Bennett et al. We also slightly improve the quantum Gilbert-Varshamov bound for general stabilizer codes, and establish an analogue of the quantum GilbertVarshamov bound for linear stabilizer codes. Our proof is restricted to the binary quantum channels, but its extension of to l-adic channels is straightforward.

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