Near-optimal ground state preparation

Lin, Lin; Tong, Yu

doi:10.22331/q-2020-12-14-372

Near-optimal ground state preparation

Journal Article · Mon Dec 14 00:00:00 EST 2020 · Quantum

DOI:https://doi.org/10.22331/q-2020-12-14-372· OSTI ID:1810083

Lin, Lin ^[1]; ^[2]

Univ. of California, Berkeley, CA (United States); Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States)
Univ. of California, Berkeley, CA (United States)

Preparing the ground state of a given Hamiltonian and estimating its ground energy are important but computationally hard tasks. However, given some additional information, these problems can be solved efficiently on a quantum computer. We assume that an initial state with non-trivial overlap with the ground state can be efficiently prepared, and the spectral gap between the ground energy and the first excited energy is bounded from below. With these assumptions we design an algorithm that prepares the ground state when an upper bound of the ground energy is known, whose runtime has a logarithmic dependence on the inverse error. When such an upper bound is not known, we propose a hybrid quantum-classical algorithm to estimate the ground energy, where the dependence of the number of queries to the initial state on the desired precision is exponentially improved compared to the current state-of-the-art algorithm proposed in [Ge et al. 2019]. These two algorithms can then be combined to prepare a ground state without knowing an upper bound of the ground energy. We also prove that our algorithms reach the complexity lower bounds by applying it to the unstructured search problem and the quantum approximate counting problem.

View Accepted Manuscript (DOE)

Research Organization:: Lawrence Berkeley National Laboratory (LBNL), Berkeley, CA (United States)

Sponsoring Organization:: US Air Force Office of Scientific Research (AFOSR); USDOE Office of Science (SC)

Grant/Contract Number:: AC02-05CH11231; SC0017867

OSTI ID:: 1810083

Journal Information:: Quantum, Journal Name: Quantum Vol. 4; ISSN 2521-327X

Publisher:: Quantum Science Open CommunityCopyright Statement

Country of Publication:: United States

Language:: English

References (37)

On Physical Problems that are Slightly More Difficult than QMA Ambainis, Andris 2014 IEEE Conference on Computational Complexity (CCC), 2014 IEEE 29th Conference on Computational Complexity (CCC) https://doi.org/10.1109/ccc.2014.12	conference	June 2014
Variable time amplitude amplification and quantum algorithms for linear algebra problems Ambainis, Andris Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik GmbH, Wadern/Saarbruecken, Germany https://doi.org/10.4230/lipics.stacs.2012.636	text	January 2012
The complexity of quantum spin systems on a two-dimensional square lattice Oliveira, Roberto; Terhal, Barbara M. arXiv https://doi.org/10.48550/arxiv.quant-ph/0504050	text	January 2005
A fast quantum mechanical algorithm for database search Grover, Lov K. arXiv https://doi.org/10.48550/arxiv.quant-ph/9605043	preprint	January 1996
The Complexity of the Local Hamiltonian Problem Kempe, Julia; Kitaev, Alexei; Regev, Oded FSTTCS 2004: Foundations of Software Technology and Theoretical Computer Science https://doi.org/10.1007/978-3-540-30538-5_31	book	January 2004
The Power of Quantum Systems on a Line Aharonov, Dorit; Gottesman, Daniel; Irani, Sandy Communications in Mathematical Physics, Vol. 287, Issue 1 https://doi.org/10.1007/s00220-008-0710-3	journal	January 2009
Uniform approximation of sgn x by polynomials and entire functions Eremenko, Alexandre; Yuditskii, Peter Journal d'Analyse Mathématique, Vol. 101, Issue 1 https://doi.org/10.1007/s11854-007-0011-3	journal	March 2007
A Multireference Quantum Krylov Algorithm for Strongly Correlated Electrons Stair, Nicholas H.; Huang, Renke; Evangelista, Francesco A. Journal of Chemical Theory and Computation, Vol. 16, Issue 4 https://doi.org/10.1021/acs.jctc.9b01125	journal	February 2020
Undecidability of the spectral gap Cubitt, Toby S.; Perez-Garcia, David; Wolf, Michael M. Nature, Vol. 528, Issue 7581 https://doi.org/10.1038/nature16059	journal	December 2015
A variational eigenvalue solver on a photonic quantum processor Peruzzo, Alberto; McClean, Jarrod; Shadbolt, Peter Nature Communications, Vol. 5, Issue 1 https://doi.org/10.1038/ncomms5213	journal	July 2014
Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution Motta, Mario; Sun, Chong; Tan, Adrian T. K. Nature Physics, Vol. 16, Issue 2 https://doi.org/10.1038/s41567-019-0704-4	journal	November 2019
Faster ground state preparation and high-precision ground energy estimation with fewer qubits Ge, Yimin; Tura, Jordi; Cirac, J. Ignacio Journal of Mathematical Physics, Vol. 60, Issue 2 https://doi.org/10.1063/1.5027484	journal	February 2019
Toward the first quantum simulation with quantum speedup Childs, Andrew M.; Maslov, Dmitri; Nam, Yunseong Proceedings of the National Academy of Sciences, Vol. 115, Issue 38 https://doi.org/10.1073/pnas.1801723115	journal	September 2018
Quantum amplitude amplification and estimation Brassard, Gilles; Høyer, Peter; Mosca, Michele Quantum Computation and Information https://doi.org/10.1090/conm/305/05215	book	January 2002
Classical and Quantum Computation Kitaev, A.; Shen, A.; Vyalyi, M. https://doi.org/10.1090/gsm/047	book	January 2002
Quantum search by measurement Childs, Andrew M.; Deotto, Enrico; Farhi, Edward Physical Review A, Vol. 66, Issue 3 https://doi.org/10.1103/PhysRevA.66.032314	journal	September 2002
Lower bound for quantum phase estimation Bessen, Arvid J. Physical Review A, Vol. 71, Issue 4 https://doi.org/10.1103/PhysRevA.71.042313	journal	April 2005
Quantum Algorithms to Simulate Many-Body Physics of Correlated Fermions Jiang, Zhang; Sung, Kevin J.; Kechedzhi, Kostyantyn Physical Review Applied, Vol. 9, Issue 4 https://doi.org/10.1103/PhysRevApplied.9.044036	journal	April 2018
Preparing Ground States of Quantum Many-Body Systems on a Quantum Computer Poulin, David; Wocjan, Pawel Physical Review Letters, Vol. 102, Issue 13 https://doi.org/10.1103/PhysRevLett.102.130503	journal	April 2009
Simulating Hamiltonian Dynamics with a Truncated Taylor Series Berry, Dominic W.; Childs, Andrew M.; Cleve, Richard Physical Review Letters, Vol. 114, Issue 9 https://doi.org/10.1103/PhysRevLett.114.090502	journal	March 2015
Optimal Hamiltonian Simulation by Quantum Signal Processing Low, Guang Hao; Chuang, Isaac L. Physical Review Letters, Vol. 118, Issue 1 https://doi.org/10.1103/PhysRevLett.118.010501	journal	January 2017
Quantum Simulation of Electronic Structure with Linear Depth and Connectivity Kivlichan, Ian D.; McClean, Jarrod; Wiebe, Nathan Physical Review Letters, Vol. 120, Issue 11 https://doi.org/10.1103/PhysRevLett.120.110501	journal	March 2018
Undecidability of the Spectral Gap in One Dimension Bausch, Johannes; Cubitt, Toby S.; Lucia, Angelo Physical Review X, Vol. 10, Issue 3 https://doi.org/10.1103/PhysRevX.10.031038	journal	August 2020
Methodology of Resonant Equiangular Composite Quantum Gates Low, Guang Hao; Yoder, Theodore J.; Chuang, Isaac L. Physical Review X, Vol. 6, Issue 4 https://doi.org/10.1103/PhysRevX.6.041067	journal	December 2016
Low-Depth Quantum Simulation of Materials Babbush, Ryan; Wiebe, Nathan; McClean, Jarrod Physical Review X, Vol. 8, Issue 1 https://doi.org/10.1103/PhysRevX.8.011044	journal	March 2018
On Physical Problems that are Slightly More Difficult than QMA Ambainis, Andris 2014 IEEE Conference on Computational Complexity (CCC), 2014 IEEE 29th Conference on Computational Complexity (CCC) https://doi.org/10.1109/CCC.2014.12	conference	June 2014
Hamiltonian Simulation with Nearly Optimal Dependence on all Parameters Berry, Dominic W.; Childs, Andrew M.; Kothari, Robin 2015 IEEE 56th Annual Symposium on Foundations of Computer Science (FOCS) https://doi.org/10.1109/FOCS.2015.54	conference	October 2015
Universal Quantum Simulators Lloyd, S. Science, Vol. 273, Issue 5278 https://doi.org/10.1126/science.273.5278.1073	journal	August 1996
Quantum Algorithm for Systems of Linear Equations with Exponentially Improved Dependence on Precision Childs, Andrew M.; Kothari, Robin; Somma, Rolando D. SIAM Journal on Computing, Vol. 46, Issue 6 https://doi.org/10.1137/16M1087072	journal	January 2017
Strengths and Weaknesses of Quantum Computing Bennett, Charles H.; Bernstein, Ethan; Brassard, Gilles SIAM Journal on Computing, Vol. 26, Issue 5 https://doi.org/10.1137/S0097539796300933	journal	October 1997
A fast quantum mechanical algorithm for database search Grover, Lov K. Proceedings of the twenty-eighth annual ACM symposium on Theory of computing - STOC '96 https://doi.org/10.1145/237814.237866	conference	January 1996
The quantum query complexity of approximating the median and related statistics Nayak, Ashwin; Wu, Felix Proceedings of the thirty-first annual ACM symposium on Theory of computing - STOC '99 https://doi.org/10.1145/301250.301349	conference	January 1999
Quantum singular value transformation and beyond: exponential improvements for quantum matrix arithmetics Gilyén, András; Su, Yuan; Low, Guang Hao Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing - STOC 2019 https://doi.org/10.1145/3313276.3316366	conference	January 2019
Hamiltonian Simulation by Qubitization Low, Guang Hao; Chuang, Isaac L. Quantum, Vol. 3 https://doi.org/10.22331/q-2019-07-12-163	journal	July 2019
Product Decomposition of Periodic Functions in Quantum Signal Processing Haah, Jeongwan Quantum, Vol. 3 https://doi.org/10.22331/q-2019-10-07-190	journal	October 2019
Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems Lin, Lin; Tong, Yu Quantum, Vol. 4 https://doi.org/10.22331/q-2020-11-11-361	journal	November 2020
Eigenpath traversal by phase randomization Boixo, S.; Knill, E.; Somma, R. Quantum Information and Computation, Vol. 9, Issue 9&10 https://doi.org/10.26421/QIC9.9-10-7	journal	September 2009

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