Inapproximability of Positive Semidefinite Permanents and Quantum State Tomography

TL;DR

This paper proves that approximating the permanents of positive semidefinite matrices and certain quantum state tomography tasks are NP-hard, indicating their computational intractability even with limited quantum resources.

Contribution

It establishes the NP-hardness of approximating PSD matrix permanents and related quantum state tomography tasks, linking computational complexity with quantum information processing.

Findings

PSD permanents are NP-hard to approximate within a constant factor

Quantum state tomography tasks are NP-hard in the Hilbert space dimension

Hardness persists even with logarithmic number of qubits

Abstract

Matrix 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 FPTAS (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.