  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.
