A Quantum Time-Space Lower Bound for the Counting Hierarchy

TL;DR

This paper establishes the first nontrivial time-space lower bounds for quantum algorithms solving problems in the counting hierarchy, showing limitations on their efficiency for certain satisfiability problems.

Contribution

It introduces new lower bounds for quantum algorithms on MajSAT and MajMajSAT, and develops a simulation technique for quantum computations with intermediate measurements.

Findings

MajMajSAT cannot be solved in near-linear time and sublinear space by quantum algorithms.

Quantum algorithms for MajSAT and MajMajSAT require superpolynomial resources under certain bounds.

A new simulation method for quantum computations with intermediate measurements was developed.

Abstract

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.