Inapproximability of Positive Semidefinite Permanents and Quantum State Tomography

Author:

Meiburg AlexanderORCID

Abstract

AbstractMatrix permanents are hard to compute or even estimate in general. It had been previously suggested that the permanents of Positive Semidefinite (PSD) matrices may have efficient approximations. By relating PSD permanents to a task in quantum state tomography, we show that PSD permanents are NP-hard to approximate within a constant factor, and so admit no polynomial-time approximation scheme (unless P = NP). We also establish that several natural tasks in quantum state tomography, even approximately, are NP-hard in the dimension of the Hilbert space. These state tomography tasks therefore remain hard even with only logarithmically few qubits.

Publisher

Springer Science and Business Media LLC

Subject

Applied Mathematics,Computer Science Applications,General Computer Science

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