Quantum Dynamic Programming

TL;DR

This paper presents a quantum extension of dynamic programming that reduces circuit depth exponentially for certain recursive problems, with trade-offs in circuit width, and demonstrates applications in quantum search, evolution, and state preparation.

Contribution

It introduces a novel quantum dynamic programming framework that coherently generates recursion unitaries using memorized states, enabling depth reduction and flexible resource trade-offs.

Findings

Achieves exponential circuit depth reduction for fixed-point quantum recursions.

Provides a hybrid approach balancing circuit depth and width.

Demonstrates applications in Grover's search, quantum evolution, and state preparation.

Abstract

We introduce a quantum extension of dynamic programming, a fundamental computational method that efficiently solves recursive problems using memory. Our innovation lies in showing how to coherently generate recursion step unitaries by using memorized intermediate quantum states. Quantum dynamic programming achieves an exponential reduction in circuit depth for a broad class of fixed-point quantum recursions, though this comes at the cost of increased circuit width. Interestingly, the trade-off becomes more favourable when the initial state is pure. By hybridizing our approach with a conventional memoryless one, we can flexibly balance circuit depth and width to optimize performance on quantum devices with fixed hardware constraints. Finally, we showcase applications of quantum dynamic programming to several quantum recursions, including a variant of Grover's search, quantum imaginary-time evolution, and a new protocol for obliviously preparing a quantum state in its Schmidt basis.