On the Complexity of Computing Zero-Error and Holevo Capacity of Quantum Channels

TL;DR

This paper investigates the computational complexity of quantum channel capacities, establishing that certain problems like quantum clique and Holevo capacity computation are QMA-complete or NP-complete, highlighting their computational difficulty.

Contribution

It introduces the quantum clique problem, proves its QMA-completeness, and shows that computing Holevo capacity and minimum entropy are NP-complete, even for entanglement-breaking channels.

Findings

Quantum clique problem is QMA-complete.

Computing Holevo capacity is NP-complete.

Minimum entropy computation is NP-complete.

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.