Time-Space Efficient Simulations of Quantum Computations Download as PDF


We give two time- and space-efficient simulations of quantum computations with intermediate measurements, one by classical randomized computations with unbounded error and the other by quantum computations that use an arbitrary fixed universal set of gates. Specifically, our simulations show that every language solvable by a bounded-error quantum algorithm running in time t and space s is also solvable by an unbounded-error randomized algorithm running in time O(t log t) and space O(s + log t), as well as by a bounded-error quantum algorithm restricted to use an arbitrary universal set and running in time O(t polylog t) and space O(s + log t), provided the universal set is closed under adjoint.

We also develop a quantum model that is particularly suitable for the study of general computations with simultaneous time and space bounds.

As an application of our randomized simulation we obtain the first nontrivial lower bound for general quantum algorithms solving problems related to satisfiability. Our bound applies to MajSAT and MajMajSAT, which are complete problems for the first and second levels of the counting hierarchy, respectively. We prove that for every real d and every positive real ε there exists a real c > 1 such that either:

In particular, MajMajSAT cannot be solved by a quantum algorithm with bounded two-sided error running in time n1+o(1) and space n1-ε for any positive real ε. Our lower bounds hold for any reasonable uniform model of quantum computation, in particular for the model we develop.