EXPONENTIAL IMPROVEMENT IN PRECISION FOR SIMULATING SPARSE HAMILTONIANS

Dominic W Berry,Rolando D Somma,Andrew M Childs,Richard Cleve,Robin Kothari

doi:10.1017/fms.2017.2

Abstract

We provide a quantum algorithm for simulating the dynamics of sparse Hamiltonians with complexity sublogarithmic in the inverse error, an exponential improvement over previous methods. Specifically, we show that a $d$-sparse Hamiltonian $H$ acting on $n$ qubits can be simulated for time $t$ with precision $\unicode[STIX]{x1D716}$ using $O(\unicode[STIX]{x1D70F}(\log (\unicode[STIX]{x1D70F}/\unicode[STIX]{x1D716})/\log \log (\unicode[STIX]{x1D70F}/\unicode[STIX]{x1D716})))$ queries and $O(\unicode[STIX]{x1D70F}(\log ^{2}(\unicode[STIX]{x1D70F}/\unicode[STIX]{x1D716})/\log \log (\unicode[STIX]{x1D70F}/\unicode[STIX]{x1D716}))n)$ additional 2-qubit gates, where $\unicode[STIX]{x1D70F}=d^{2}\Vert H\Vert _{\max }t$. Unlike previous approaches based on product formulas, the query complexity is independent of the number of qubits acted on, and for time-varying Hamiltonians, the gate complexity is logarithmic in the norm of the derivative of the Hamiltonian. Our algorithm is based on a significantly improved simulation of the continuous- and fractional-query models using discrete quantum queries, showing that the former models are not much more powerful than the discrete model even for very small error. We also simplify the analysis of this conversion, avoiding the need for a complex fault-correction procedure. Our simplification relies on a new form of ‘oblivious amplitude amplification’ that can be applied even though the reflection about the input state is unavailable. Finally, we prove new lower bounds showing that our algorithms are optimal as a function of the error.

Highlights

Simulation of quantum mechanical systems is a major potential application of quantum computers
A d-sparse Hamiltonian H acting on n qubits can be simulated for time t within error with O(τ (log(τ/ )/ log log(τ/ ))) queries and O(τ (log2(τ/ )/ log log(τ/ ))n) additional 2-qubit gates, where τ := d2 H maxt
Our new approach to Hamiltonian simulation strictly improves all previous approaches based on product formulas

Summary

Introduction

Simulation of quantum mechanical systems is a major potential application of quantum computers. Reference [11] does not address the more general problem of simulating the fractional-query model, as we do here Another algorithm [12] uses a quantum walk to simulate time-independent sparse Hamiltonians. For the notable special case of local Hamiltonians (or more generally, those with a given decomposition into 1-sparse terms), the present work already has complexity nearly linear in the sparsity, so [12] does not give a significant improvement. It is unclear whether ideas from [12] can be applied to simulate timedependent Hamiltonians. The concept of oblivious amplitude amplification introduced in this paper has been applied to quantum circuit synthesis [37, 44]

Summary of results

From continuous to discrete queries

Findings

Hamiltonian simulation