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EXPONENTIAL IMPROVEMENT IN PRECISION FOR SIMULATING SPARSE HAMILTONIANS

Part of: Theory of computing

Published online by Cambridge University Press:  02 March 2017

DOMINIC W. BERRY ,
ANDREW M. CHILDS ,
RICHARD CLEVE ,
ROBIN KOTHARI  and
ROLANDO D. SOMMA
DOMINIC W. BERRY
Affiliation:
Department of Physics and Astronomy, Macquarie University, Sydney, NSW 2109, Australia; dominic.berry@mq.edu.au
ANDREW M. CHILDS
Affiliation:
Department of Combinatorics & Optimization and Institute for Quantum Computing, University of Waterloo, Waterloo, ON N2L 3G1, Canada Department of Computer Science, Institute for Advanced Computer Studies, and Joint Center for Quantum Information and Computer Science, University of Maryland, College Park, MD 20742, USA; amchilds@umd.edu
RICHARD CLEVE
Affiliation:
David R. Cheriton School of Computer Science and Institute for Quantum Computing, University of Waterloo, Waterloo, ON N2L 3G1, Canada; cleve@uwaterloo.ca Canadian Institute for Advanced Research, Toronto, ON M5G 1Z8, Canada
ROBIN KOTHARI
Affiliation:
David R. Cheriton School of Computer Science and Institute for Quantum Computing, University of Waterloo, Waterloo, ON N2L 3G1, Canada; cleve@uwaterloo.ca Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA; rkothari@mit.edu
ROLANDO D. SOMMA
Affiliation:
Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA; somma@lanl.gov

Abstract

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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.

MSC classification

Primary: 68Q12: Quantum algorithms and complexity
Secondary: 81P68: Quantum computation
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 5 , 2017 , e8
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2017

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