Introducing Structure to Expedite Quantum Search

TL;DR

This paper introduces a new quantum algorithm that reduces the number of quantum gates needed for unstructured search problems, making it more suitable for NISQ devices and applicable to complex combinatorial problems.

Contribution

The paper presents a quantum algorithm with fewer gates than Grover's, proves its optimality, and introduces the partial uncompute technique to exploit oracle structure for gate reduction.

Findings

Uses asymptotically fewer gates than Grover's algorithm.

Proves the optimality of the proposed algorithm in gate count.

Demonstrates applicability to problems like Unique k-SAT and multi-marked elements.

Abstract

We present a novel quantum algorithm for solving the unstructured search problem with one marked element. Our algorithm allows generating quantum circuits that use asymptotically fewer additional quantum gates than the famous Grover's algorithm and may be successfully executed on NISQ devices. We prove that our algorithm is optimal in the total number of elementary gates up to a multiplicative constant. As many NP-hard problems are not in fact unstructured, we also describe the \emph{partial uncompute} technique which exploits the oracle structure and allows a significant reduction in the number of elementary gates required to find the solution. Combining these results allows us to use asymptotically smaller number of elementary gates than the Grover's algorithm in various applications, keeping the number of queries to the oracle essentially the same. We show how the results can be applied to solve hard combinatorial problems, for example Unique k-SAT. Additionally, we show how to asymptotically reduce the number of elementary gates required to solve the unstructured search problem with multiple marked elements.