Hamiltonian Simulation with Optimal Sample Complexity

TL;DR

This paper establishes the optimal sample complexity for Hamiltonian simulation from unknown quantum states, extends the method to multiple states, and demonstrates its applications in quantum algorithms and universal computation.

Contribution

It proves the optimality of a known Hamiltonian simulation method, extends it to multiple states, and introduces new applications in quantum algorithms and computation.

Findings

Proves the optimality of Lloyd et al.'s Hamiltonian simulation method.

Extends simulation to multiple input states and Hermitian polynomials.

Provides applications in commutator simulation, orthogonality testing, and universal quantum computation.

Abstract

We investigate the sample complexity of Hamiltonian simulation: how many copies of an unknown quantum state are required to simulate a Hamiltonian encoded by the density matrix of that state? We show that the procedure proposed by Lloyd, Mohseni, and Rebentrost [Nat. Phys., 10(9):631--633, 2014] is optimal for this task. We further extend their method to the case of multiple input states, showing how to simulate any Hermitian polynomial of the states provided. As applications, we derive optimal algorithms for commutator simulation and orthogonality testing, and we give a protocol for creating a coherent superposition of pure states, when given sample access to those states. We also show that this sample-based Hamiltonian simulation can be used as the basis of a universal model of quantum computation that requires only partial swap operations and simple single-qubit states.