HighDist Framework: Algorithms and Applications

TL;DR

This paper introduces the HighDist framework with quantum algorithms for distribution mode and amplitude thresholds, improving query complexities for several problems and enabling efficient implementation on current quantum hardware.

Contribution

It presents novel quantum algorithms for HighDist and HighAmp problems with logarithmic space complexity and constant query complexity, enhancing solutions for multiple distribution estimation tasks.

Findings

Reduced query complexity for distribution problems

Algorithms feasible on current quantum hardware

Improved bounds for distribution and Boolean function analysis

Abstract

We introduce the problem of determining if the mode of the output distribution of a quantum circuit (given as a black-box) is larger than a given threshold, named HighDist, and a similar problem based on the absolute values of the amplitudes, named HighAmp. We design quantum algorithms for promised versions of these problems whose space complexities are logarithmic in the size of the domain of the distribution, but the query complexities are independent. Using these, we further design algorithms to estimate the largest probability and the largest amplitude among the output distribution of a quantum black-box. All of these allow us to improve the query complexity of a few recently studied problems, namely, $k$-distinctness and its gapped version, estimating the largest frequency in an array, estimating the min-entropy of a distribution, and the non-linearity of a Boolean function, in the $\tilde{O}(1)$-qubits scenario. The time-complexities of almost all of our algorithms have a small overhead over their query complexities making them efficiently implementable on currently available quantum backends.