Quantum simulation of real-space dynamics

by   Andrew M. Childs, et al.

Quantum simulation is a prominent application of quantum computers. While there is extensive previous work on simulating finite-dimensional systems, less is known about quantum algorithms for real-space dynamics. We conduct a systematic study of such algorithms. In particular, we show that the dynamics of a d-dimensional Schrödinger equation with η particles can be simulated with gate complexity Õ(η d F poly(log(g'/ϵ))), where ϵ is the discretization error, g' controls the higher-order derivatives of the wave function, and F measures the time-integrated strength of the potential. Compared to the best previous results, this exponentially improves the dependence on ϵ and g' from poly(g'/ϵ) to poly(log(g'/ϵ)) and polynomially improves the dependence on T and d, while maintaining best known performance with respect to η. For the case of Coulomb interactions, we give an algorithm using η^3(d+η)Tpoly(log(η dTg'/(Δϵ)))/Δ one- and two-qubit gates, and another using η^3(4d)^d/2Tpoly(log(η dTg'/(Δϵ)))/Δ one- and two-qubit gates and QRAM operations, where T is the evolution time and the parameter Δ regulates the unbounded Coulomb interaction. We give applications to several computational problems, including faster real-space simulation of quantum chemistry, rigorous analysis of discretization error for simulation of a uniform electron gas, and a quadratic improvement to a quantum algorithm for escaping saddle points in nonconvex optimization.


page 1

page 2

page 3

page 4


On the complexity of implementing Trotter steps

Quantum dynamics can be simulated on a quantum computer by exponentiatin...

Eliminating Intermediate Measurements using Pseudorandom Generators

We show that quantum algorithms of time T and space S≥log T with unitary...

Basic quantum subroutines: finding multiple marked elements and summing numbers

We show how to find all k marked elements in a list of size N using the ...

Practical implementation of a quantum backtracking algorithm

In previous work, Montanaro presented a method to obtain quantum speedup...

A Theory of Trotter Error

The Lie-Trotter formula, together with its higher-order generalizations,...

Sensitivity of a Chaotic Logic Gate

Chaotic logic gates or `chaogates' are a promising mixed-signal approach...

Nearly tight Trotterization of interacting electrons

We consider simulating quantum systems on digital quantum computers. We ...

Please sign up or login with your details

Forgot password? Click here to reset