Measures of quantum computing speedup

TL;DR

This paper introduces the concept of strong quantum speedup, demonstrating that certain eigenvalue problems, like approximating ground state energies, can be solved exponentially faster on quantum computers, challenging existing beliefs.

Contribution

It defines strong quantum speedup and proves exponential speedup for a class of eigenvalue problems, clarifying misconceptions in quantum complexity theory.

Findings

Strong quantum speedup for ground state energy approximation

Exponential quantum speedup demonstrated for Schrödinger equation problems

Challenges the belief that quantum computation is ineffective for eigenvalue problems

Abstract

We introduce the concept of strong quantum speedup. We prove that approximating the ground state energy of an instance of the time-independent Schr\"odinger equation, with $d$ degrees of freedom, $d$ large, enjoys strong exponential quantum speedup. It can be easily solved on a quantum computer. Some researchers in discrete complexity theory believe that quantum computation is not effective for eigenvalue problems. One of our goals in this paper is to explain this dissonance.