Quantum algorithm for approximating the expected value of a random-exist quantified oracle

TL;DR

This paper introduces a quantum algorithm that achieves a quadratic speedup in estimating the probability of a successful reaction in a game involving a random oracle, outperforming classical Monte-Carlo methods under certain conditions.

Contribution

It demonstrates how quantum amplitude amplification and estimation can be combined to efficiently approximate probabilities in a random-exist quantified oracle setting, extending quantum speedups to this class of problems.

Findings

Quantum algorithm achieves quadratic speedup over classical methods.

Performance depends on specific problem parameters.

A regime exists where the quadratic speedup is optimal.

Abstract

Quantum amplitude amplification and estimation have shown quadratic speedups to unstructured search and estimation tasks. We show that a coherent combination of these quantum algorithms also provides a quadratic speedup to calculating the expectation value of a random-exist quantified oracle. In this problem, Nature makes a decision randomly, i.e. chooses a bitstring according to some probability distribution, and a player has a chance to react by finding a complementary bitstring such that an black-box oracle evaluates to $1$ (or True). Our task is to approximate the probability that the player has a valid reaction to Nature's initial decision. We compare the quantum algorithm to the average-case performance of Monte-Carlo integration over brute-force search, which is, under reasonable assumptions, the best performing classical algorithm. We find the performance separation depends on some problem parameters, and show a regime where the canonical quadratic speedup exists.