## A Quantum Time-Space Lower Bound for the Counting Hierarchy

##### Date

##### Authors

van Melkebeek, Dieter

Watson, Thomas

##### Journal Title

##### Journal ISSN

##### Volume Title

##### Publisher

##### Abstract

##### Description

We obtain the first nontrivial time-space lower bound for 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 epsilon there exists a real c>1 such that either: MajMajSAT does
not have a quantum algorithm with bounded two-sided error that runs in time
n^c, or MajSAT does not have a quantum algorithm with bounded two-sided error
that runs in time n^d and space n^{1-\epsilon}. In particular, MajMajSAT cannot
be solved by a quantum algorithm with bounded two-sided error running in time
n^{1+o(1)} and space n^{1-\epsilon} for any epsilon>0. The key technical
novelty is a time- and space-efficient simulation of quantum computations with
intermediate measurements by probabilistic machines with unbounded error. We
also develop a model that is particularly suitable for the study of general
quantum computations with simultaneous time and space bounds. However, our
arguments hold for any reasonable uniform model of quantum computation.

Comment: 25 pages

Comment: 25 pages

##### Keywords

Quantum Physics