  Motivated by understanding the power of quantum computation with restricted
number of qubits, we give two complete characterizations of unitary quantum
space bounded computation. First we show that approximating an element of the
inverse of a well-conditioned efficiently encoded $2^{k(n)}\times 2^{k(n)}$
matrix is complete for the class of problems solvable by quantum circuits
acting on $\mathcal{O}(k(n))$ qubits with all measurements at the end of the
computation. Similarly, estimating the minimum eigenvalue of an efficiently
encoded Hermitian $2^{k(n)}\times 2^{k(n)}$ matrix is also complete for this
class. In the logspace case, our results improve on previous results of Ta-Shma
[STOC '13] by giving new space-efficient quantum algorithms that avoid
intermediate measurements, as well as showing matching hardness results.
  Additionally, as a consequence we show that PreciseQMA, the version of QMA
with exponentially small completeness-soundess gap, is equal to PSPACE. Thus,
the problem of estimating the minimum eigenvalue of a local Hamiltonian to
inverse exponential precision is PSPACE-complete, which we show holds even in
the frustration-free case. Finally, we can use this characterization to give a
provable setting in which the ability to prepare the ground state of a local
Hamiltonian is more powerful than the ability to prepare PEPS states.
  Interestingly, by suitably changing the parameterization of either of these
problems we can completely characterize the power of quantum computation with
simultaneously bounded time and space.
