01705nas a2200121 4500008004100000245005700041210005500098260001500153520133500168100002001503700002401523856003601547 2016 eng d00aA Complete Characterization of Unitary Quantum Space0 aComplete Characterization of Unitary Quantum Space c2016/04/053 aWe give two complete characterizations of unitary quantum space-bounded classes. The first is based on the Matrix Inversion problem for well-conditioned matrices. We show that given the size-n efficient encoding of a 2O(k(n))×2O(k(n)) well-conditioned matrix H, approximating a particular entry of H−1 is complete for the class of problems solvable by a quantum algorithm that uses O(k(n)) space and performs all quantum measurements at the end of the computation. In particular, the problem of computing entries of H−1 for an explicit well-conditioned n×n matrix H is complete for unitary quantum logspace.
We then show that the problem of approximating to high precision the least eigenvalue of a positive semidefinite matrix H, encoded as a circuit, gives a second characterization of unitary quantum space complexity. In the process we also establish an equivalence between unitary quantum space-bounded classes and certain QMA proof systems. As consequences, we establish that QMA with exponentially small completeness-soundness gap is equal to PSPACE, that determining whether a local Hamiltonian is frustration-free is PSPACE-complete, and give a provable setting in which the ability to prepare PEPS states gives less computational power than the ability to prepare the ground state of a generic local Hamiltonian.1 aFefferman, Bill1 aLin, Cedric, Yen-Yu uhttp://arxiv.org/abs/1604.01384