On the Complexity of Computing Zero-Error and Holevo Capacity of Quantum Channels
Abstract
One of the main problems in quantum complexity theory is that our understanding of the theory of QMA-completeness is not as rich as its classical analogue, the NP- completeness. In this paper we consider the clique problem in graphs, which is NP- complete, and try to find its quantum analogue. We show that, quantum clique problem can be defined as follows; Given a quantum channel, decide whether there are k states that are distinguishable, with no error, after passing through channel. This definition comes from reconsidering the clique problem in terms of the zero-error capacity of graphs, and then redefining it in quantum information theory. We prove that, quantum clique problem is QMA-complete. In the second part of paper, we consider the same problem for the Holevo capacity. We prove that computing the Holevo capacity as well as the minimum entropy of a quantum channel is NP-complete. Also, we show these results hold even if the set of quantum channels is restricted to entanglement breaking ones.
Cite
@article{arxiv.0709.2090,
title = {On the Complexity of Computing Zero-Error and Holevo Capacity of Quantum Channels},
author = {Salman Beigi and Peter W. Shor},
journal= {arXiv preprint arXiv:0709.2090},
year = {2008}
}
Comments
19 pages, no figure, minor error fixed