Complexity limitations on quantum computation

TL;DR

This paper investigates the fundamental limitations of quantum computation using counting complexity and oracle techniques, revealing various relativized worlds where quantum and classical complexity classes relate in surprising ways.

Contribution

It provides new insights into the limitations of BQP through relativized worlds and shows how quantum complexity interacts with classical complexity classes.

Findings

BQP is low for PP, i.e., PP^BQP=PP

Existence of relativized worlds where P=BQP and the polynomial hierarchy is infinite

Existence of relativized worlds where BQP lacks complete sets

Abstract

We use the powerful tools of counting complexity and generic oracles to help understand the limitations of the complexity of quantum computation. We show several results for the probabilistic quantum class BQP. 1. BQP is low for PP, i.e., PP^BQP=PP. 2. There exists a relativized world where P=BQP and the polynomial-time hierarchy is infinite. 3. There exists a relativized world where BQP does not have complete sets. 4. There exists a relativized world where P=BQP but P is not equal to UP intersect coUP and one-way functions exist. This gives a relativized answer to an open question of Simon.