Efficient discrete-time simulations of continuous-time quantum query algorithms

TL;DR

This paper establishes a method to efficiently simulate continuous-time quantum query algorithms within the discrete query model, providing bounds that connect the complexities of both models.

Contribution

It introduces the first general upper bound for converting continuous-time quantum algorithms into discrete-time algorithms, independent of the driving Hamiltonian's norm.

Findings

Simulation of continuous-time algorithms with O[T log(T) / log(log(T))] discrete queries.

Lower bounds in the discrete model imply similar bounds in the continuous model.

The result is independent of the size of the driving Hamiltonian.

Abstract

The continuous-time query model is a variant of the discrete query model in which queries can be interleaved with known operations (called "driving operations") continuously in time. Interesting algorithms have been discovered in this model, such as an algorithm for evaluating nand trees more efficiently than any classical algorithm. Subsequent work has shown that there also exists an efficient algorithm for nand trees in the discrete query model; however, there is no efficient conversion known for continuous-time query algorithms for arbitrary problems. We show that any quantum algorithm in the continuous-time query model whose total query time is T can be simulated by a quantum algorithm in the discrete query model that makes O[T log(T) / log(log(T))] queries. This is the first upper bound that is independent of the driving operations (i.e., it holds even if the norm of the driving Hamiltonian is very large). A corollary is that any lower bound of T queries for a problem in the discrete-time query model immediately carries over to a lower bound of \Omega[T log(log(T))/log (T)] in the continuous-time query model.