Application of Pontryagin's Minimum Principle to Grover's Quantum Search Problem

TL;DR

This paper maps Grover's quantum search to a time-optimal control problem and uses Pontryagin's Minimum Principle to derive a bang-singular-bang optimal control structure, offering new insights into quantum algorithm design.

Contribution

It introduces a novel application of Pontryagin's Minimum Principle to quantum search, revealing the optimal control structure and connecting control theory with quantum computation.

Findings

Optimal control structure is bang-singular-bang.

Quantum search can be approximated by bang-bang protocols.

Provides a geometric control perspective on quantum algorithms.

Abstract

Grover's algorithm is one of the most famous algorithms which explicitly demonstrates how the quantum nature can be utilized to accelerate the searching process. In this work, Grover's quantum search problem is mapped to a time-optimal control problem. Resorting to Pontryagin's Minimum Principle we find that the time-optimal solution has the bang-singular-bang structure. This structure can be derived naturally, without integrating the differential equations, using the geometric control technique where Hamiltonians in the Schr\"odinger's equation are represented as vector fields. In view of optimal control, Grover's algorithm uses the bang-bang protocol to approximate the optimal protocol with a minimized number of bang-to-bang switchings to reduce the query complexity. Our work provides a concrete example how Pontryagin's Minimum Principle is connected to quantum computation, and offers insight into how a quantum algorithm can be designed.