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arXiv:quant-ph/9709049

Upper Bounds on the Size of Quantum Codes

Alexei Ashikhmin, Simon N. Litsyn

Published Sep 23, 1997• 2 citations
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Abstract

Several upper bounds on the size of quantum codes are derived using the linear programming approach. These bounds are strengthened for the linear quantum codes.
DOI
10.48550/arxiv.quant-ph/9709049

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This paper is concerned with bounds for quantum error-correcting codes. Using the quantum MacWilliams (1972, 1977) identities, we generalize the linear programming approach from classical coding theory to the quantum case. Using this approach, we obtain Singleton-type, Hamming-type, and the first linear-programming-type bounds for quantum codes. Using the special structure of linear quantum codes, we derive an upper bound that is better than both Hamming and the first linear programming bounds on some subinterval of rates.
The most powerful technique known at present for bounding the size of quantum codes of prescribed minimum distance is the quantum linear programming bound. Unlike the classical linear programming bound, it is not immediately obvious that if the quantum linear programming constraints are satisfiable for dimension K, then the constraints can be satisfied for all lower dimensions. We show that the quantum linear programming bound is monotonic in this sense, and give an explicitly monotonic reformulation.
A fundamental problem in quantum coding theory is to determine the maximum size of quantum codes of given block length and distance. A recent work introduced bounds based on semidefinite programming, strengthening the well-known quantum linear programming bounds. However, floating-point inaccuracies prevent the extraction of rigorous non-existence proofs from the numerical methods. Here, we address this by providing rational infeasibility certificates for a range of quantum codes. Using a clustered low-rank solver with heuristic rounding to algebraic expressions, we can improve upon 181818 upper bounds on the maximum size of nnn-qubit codes with 6≤n≤196 \leq n \leq 196≤n≤19. Our work highlights the practicality and scalability of semidefinite programming for quantum coding bounds.