# An algorithm for solving unconstrained unitary quantum brachistochrone problems

## TL;DR

This paper presents an iterative algorithm to find time-optimal Hamiltonians for driving quantum systems between states, with rapid convergence and applications in quantum state preparation and gate design.

## Contribution

The paper introduces a novel iterative method for solving quantum brachistochrone problems that converges quickly and saturates minimal evolution time bounds.

## Key findings

- Method converges rapidly for large systems
- Solutions reach the minimal evolution time bound
- Provides a geometric interpretation linked to quantum phases

## Abstract

We introduce an iterative method to search for time-optimal Hamiltonians that drive a quantum system between two arbitrary, and in general mixed, quantum states. The method is based on the idea of progressively improving the efficiency of an initial, randomly chosen, Hamiltonian, by reducing its components that do not actively contribute to driving the system. We show that our method converges rapidly even for large dimensional systems, and that its solutions saturate any attainable bound for the minimal time of evolution. We provide a rigorous geometric interpretation of the iterative method by exploiting an isomorphism between geometric phases acquired by the system along a path and the Hamiltonian that generates it, and discuss resulting similarities with Grover's quantum search algorithm. Our method is directly applicable as a powerful tool for state preparation and gate design problems.