Quantum error detection .II. Bounds
Ashikhmin, A.E.
Barg, A.M.
Knill, E.
Litsyn, S.N.
Bell Labs., Lucent Technol., Murray Hill, NJ;
This paper appears in: Information Theory, IEEE Transactions on
Publication Date: May 2000
Volume: 46,
Issue: 3
On page(s): 789-800
ISSN: 0018-9448
References Cited: 15
CODEN: IETTAW
INSPEC Accession Number: 6593112
Digital Object Identifier: 10.1109/18.841163
Current Version Published: 2002-08-06
Abstract
For pt.I see ibid., vol.46, no.3, p.778-88 (2000). In Part I of
this paper we formulated the problem of error detection with quantum
codes on the depolarizing channel and gave an expression for the
probability of undetected error via the weight enumerators of the code.
In this part we show that there exist quantum codes whose probability of
undetected error falls exponentially with the length of the code and
derive bounds on this exponent. The lower (existence) bound is proved
for stabilizer codes by a counting argument for classical
self-orthogonal quaternary codes. Upper bounds are proved by linear
programming. First we formulate two linear programming problems that are
convenient for the analysis of specific short codes. Next we give a
relaxed formulation of the problem in terms of optimization on the cone
of polynomials in the Krawtchouk basis. We present two general solutions
of the problem. Together they give an upper bound on the exponent of
undetected error. The upper and lower asymptotic bounds coincide for a
certain interval of code rates close to 1
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