New Developments in Quantum Algorithms

TL;DR

This survey highlights recent advances in quantum algorithms, including a near-optimal quantum method for evaluating Boolean formulas and a logarithmic-time quantum algorithm for solving linear systems, showcasing significant speedups over classical approaches.

Contribution

The paper introduces two new quantum algorithms: one for Boolean formula evaluation with O(√N) complexity and another for linear systems with logarithmic complexity, advancing quantum computational capabilities.

Findings

Quantum algorithm for Boolean formulas runs in O(√N) time.

Quantum linear system solver operates in O(log^c N) time.

Provides optimal quantum algorithms in the black-box query model.

Abstract

In this survey, we describe two recent developments in quantum algorithms. The first new development is a quantum algorithm for evaluating a Boolean formula consisting of AND and OR gates of size N in time O(\sqrt{N}). This provides quantum speedups for any problem that can be expressed via Boolean formulas. This result can be also extended to span problems, a generalization of Boolean formulas. This provides an optimal quantum algorithm for any Boolean function in the black-box query model. The second new development is a quantum algorithm for solving systems of linear equations. In contrast with traditional algorithms that run in time O(N^{2.37...}) where N is the size of the system, the quantum algorithm runs in time O(\log^c N). It outputs a quantum state describing the solution of the system.