Hardness of approximation for quantum problems

TL;DR

This paper introduces a quantum generalization of the polynomial hierarchy, establishes the hardness of approximation for certain quantum problems and QCMA, using dispersers inspired by classical complexity results.

Contribution

It defines a quantum polynomial hierarchy, identifies complete problems, and proves their hardness of approximation, extending classical complexity concepts into quantum computing.

Findings

Quantum polynomial hierarchy is defined and studied.

Complete problems for the second level are shown to be hard to approximate.

Hardness results are obtained for quantum versions of classical problems.

Abstract

The polynomial hierarchy plays a central role in classical complexity theory. Here, we define a quantum generalization of the polynomial hierarchy, and initiate its study. We show that not only are there natural complete problems for the second level of this quantum hierarchy, but that these problems are in fact hard to approximate. Using these techniques, we also obtain hardness of approximation for the class QCMA. Our approach is based on the use of dispersers, and is inspired by the classical results of Umans regarding hardness of approximation for the second level of the classical polynomial hierarchy [Umans, FOCS 1999]. The problems for which we prove hardness of approximation for include, among others, a quantum version of the Succinct Set Cover problem, and a variant of the local Hamiltonian problem with hybrid classical-quantum ground states.