Quantum interference as a resource for quantum speedup

TL;DR

This paper investigates how quantum interference contributes to quantum speedup, demonstrating that limited interference operations cannot produce speedup, while large interference can, and provides classical simulation methods for such circuits.

Contribution

It quantifies quantum interference and shows that low-interference operations cannot achieve quantum speedup, offering classical simulation techniques for these cases.

Findings

Low-interference operations cannot generate quantum speedup.

Large interference operations can lead to quantum speedup.

Classical Monte Carlo simulation is feasible for circuits with limited interference.

Abstract

Quantum states can in a sense be thought of as generalizations of classical probability distributions, but are more powerful than probability distributions when used for computation or communication. Quantum speedup therefore requires some feature of quantum states that classical probability distributions lack. One such feature is interference. We quantify interference and show that there can be no quantum speedup due to a small number of operations incapable of generating large amounts of interference (although large numbers of such operations can in fact lead to quantum speedup). Low-interference operations include sparse unitaries, Grover reflections, short time/low energy Hamiltonian evolutions, and the Haar wavelet transform. Circuits built from such operations can be classically simulated via a Monte Carlo technique making use of a convex combination of two Markov chains. Applications to query complexity, communication complexity, and the Wigner representation are discussed.