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Mori Y, Kawabata S, Matsuzaki Y. How to experimentally evaluate the adiabatic condition for quantum annealing. Sci Rep 2024; 14:8177. [PMID: 38589470 PMCID: PMC11001971 DOI: 10.1038/s41598-024-58286-2] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/06/2023] [Accepted: 03/27/2024] [Indexed: 04/10/2024] Open
Abstract
We propose an experimental method for evaluating the adiabatic condition during quantum annealing (QA), which will be essential for solving practical problems. The adiabatic condition consists of the transition matrix element and the energy gap, and our method simultaneously provides information about these components without diagonalizing the Hamiltonian. The key idea is to measure the power spectrum of a time domain signal by adding an oscillating field during QA, and we can estimate the values of the transition matrix element and energy gap from the measurement output. Our results provides a powerful experimental basis for analyzing the performance of QA.
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Affiliation(s)
- Yuichiro Mori
- Global Research and Development Center for Business by Quantum-AI Technology (G-QuAT), National Institute of Advanced Industrial Science and Technology (AIST), 1-1-1, Umezono, Tsukuba, Ibaraki, 305-8568, Japan.
| | - Shiro Kawabata
- Global Research and Development Center for Business by Quantum-AI Technology (G-QuAT), National Institute of Advanced Industrial Science and Technology (AIST), 1-1-1, Umezono, Tsukuba, Ibaraki, 305-8568, Japan.
- NEC-AIST Quantum Technology Cooperative Research Laboratory, National Institute of Advanced Industrial Science and Technology (AIST), Tsukuba, Ibaraki, 305-8568, Japan.
| | - Yuichiro Matsuzaki
- Global Research and Development Center for Business by Quantum-AI Technology (G-QuAT), National Institute of Advanced Industrial Science and Technology (AIST), 1-1-1, Umezono, Tsukuba, Ibaraki, 305-8568, Japan.
- NEC-AIST Quantum Technology Cooperative Research Laboratory, National Institute of Advanced Industrial Science and Technology (AIST), Tsukuba, Ibaraki, 305-8568, Japan.
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2
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Misiewicz J, Evangelista FA. Implementation of the Projective Quantum Eigensolver on a Quantum Computer. J Phys Chem A 2024; 128:2220-2235. [PMID: 38452262 PMCID: PMC10961848 DOI: 10.1021/acs.jpca.3c07429] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/09/2023] [Revised: 01/29/2024] [Accepted: 02/20/2024] [Indexed: 03/09/2024]
Abstract
We study the performance of our previously proposed projective quantum eigensolver (PQE) on IBM's quantum hardware in conjunction with error mitigation techniques. For a single qubit model of H2, we find that we are able to obtain energies within 4 millihartree (2.5 kcal/mol) of the exact energy along the entire potential energy curve, with the accuracy limited by both the stochastic error and the inconsistent performance of the IBM devices. We find that an optimization algorithm using direct inversion of the iterative subspace can converge swiftly, even to excited states, but stochastic noise can prompt large parameter updates. For the 4-site transverse-field Ising model at its critical point, PQE with an appropriate application of qubit tapering can recover 99% of the correlation energy, even after discarding several parameters. The large number of CNOT gates needed for the additional parameters introduces a concomitant error that, on the IBM devices, results in a loss of accuracy despite the increased expressivity of the trial state. Error extrapolation techniques and tapering or postselection are recommended to mitigate errors in PQE hardware experiments.
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Affiliation(s)
| | - Francesco A. Evangelista
- Department of Chemistry and
Cherry Emerson Center for Scientific Computation, Emory University, Atlanta, Georgia 30322, United States
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3
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Molecular dynamics on quantum annealers. Sci Rep 2022; 12:16824. [PMID: 36207401 PMCID: PMC9547079 DOI: 10.1038/s41598-022-21163-x] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/05/2022] [Accepted: 09/23/2022] [Indexed: 11/08/2022] Open
Abstract
In this work we demonstrate a practical prospect of using quantum annealers for simulation of molecular dynamics. A methodology developed for this goal, dubbed Quantum Differential Equations (QDE), is applied to propagate classical trajectories for the vibration of the hydrogen molecule in several regimes: nearly harmonic, highly anharmonic, and dissociative motion. The results obtained using the D-Wave 2000Q quantum annealer are all consistent and quickly converge to the analytical reference solution. Several alternative strategies for such calculations are explored and it was found that the most accurate results and the best efficiency are obtained by combining the quantum annealer with classical post-processing (greedy algorithm). Importantly, the QDE framework developed here is entirely general and can be applied to solve any system of first-order ordinary nonlinear differential equations using a quantum annealer.
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Negre CFA, Lopez-Bezanilla A, Zhang Y, Akrobotu PD, Mniszewski SM, Tretiak S, Dub PA. Toward a QUBO-Based Density Matrix Electronic Structure Method. J Chem Theory Comput 2022; 18:4177-4185. [PMID: 35658437 DOI: 10.1021/acs.jctc.2c00090] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
Abstract
Density matrix electronic structure theory is used in many quantum chemistry methods to "alleviate" the computational cost that arises from directly using wave functions. Although density matrix based methods are computationally more efficient than wave function based methods, significant computational effort is involved. Because the Schrödinger equation needs to be solved as an eigenvalue problem, the time-to-solution scales cubically with the system size in mean-field type approaches such as Hartree-Fock and density functional theory and is solved as many times in order to reach charge or field self-consistency. We hereby propose and study a method to compute the density matrix by using a quadratic unconstrained binary optimization (QUBO) solver. This method could be useful to solve the problem with quantum computers and, more specifically, quantum annealers. Our proposed approach is based on a direct construction of the density matrix using a QUBO eigensolver. We explore the main parameters of the algorithm focusing on precision and efficiency. We show that, while direct construction of the density matrix using a QUBO formulation is possible, the efficiency and precision have room for improvement. Moreover, calculations performed with quantum annealing on D-Wave's new Advantage quantum computer are compared with results obtained with classical simulated annealing, further highlighting some problems of the proposed method. We also suggest alternative methods that could lead to a more efficient QUBO-based density matrix construction.
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Affiliation(s)
| | | | | | - Prosper D Akrobotu
- Department of Mathematical Sciences, University of Texas at Dallas, Richardson, Texas 75080, United States
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Krakoff B, Mniszewski SM, Negre CFA. Controlled precision QUBO-based algorithm to compute eigenvectors of symmetric matrices. PLoS One 2022; 17:e0267954. [PMID: 35533179 PMCID: PMC9084526 DOI: 10.1371/journal.pone.0267954] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/22/2021] [Accepted: 04/19/2022] [Indexed: 11/18/2022] Open
Abstract
We describe an algorithm to compute the extremal eigenvalues and corresponding eigenvectors of a symmetric matrix which is based on solving a sequence of Quadratic Binary Optimization problems. This algorithm is robust across many different classes of symmetric matrices; It can compute the eigenvector/eigenvalue pair to essentially any arbitrary precision, and with minor modifications, can also solve the generalized eigenvalue problem. Performance is analyzed on small random matrices and selected larger matrices from practical applications.
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Affiliation(s)
- Benjamin Krakoff
- Department of Mathematics, University of Michigan, Ann Arbor, Michigan, United States of America
| | - Susan M. Mniszewski
- Computer, Computational, and Statistical Sciences Division, Los Alamos National Laboratory, Los Alamos, New Mexico, United States of America
| | - Christian F. A. Negre
- Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico, United States of America
- * E-mail:
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6
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Oshiyama H, Suzuki S, Shibata N. Classical Simulation and Theory of Quantum Annealing in a Thermal Environment. PHYSICAL REVIEW LETTERS 2022; 128:170502. [PMID: 35570457 DOI: 10.1103/physrevlett.128.170502] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/01/2021] [Revised: 02/21/2022] [Accepted: 03/22/2022] [Indexed: 06/15/2023]
Abstract
We study quantum annealing in the quantum Ising model coupled to a thermal environment. When the speed of quantum annealing is sufficiently slow, the system evolves following the instantaneous thermal equilibrium. This quasistatic and isothermal evolution, however, fails near the end of annealing because the relaxation time grows infinitely, therefore yielding excess energy from the thermal equilibrium. We develop a phenomenological theory based on this picture and derive a scaling relation of the excess energy after annealing. The theoretical results are numerically confirmed using a novel non-Markovian method that we recently proposed based on a path-integral representation of the reduced density matrix and the infinite time evolving block decimation. In addition, we discuss crossovers from weak to strong coupling as well as from the adiabatic to quasistatic regime, and propose experiments on the D-Wave quantum annealer.
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Affiliation(s)
- Hiroki Oshiyama
- Department of Physics, Tohoku University, Sendai 980-8578, Japan
| | - Sei Suzuki
- Department of Liberal Arts, Saitama Medical University, Moroyama, Saitama 350-0495, Japan
| | - Naokazu Shibata
- Department of Physics, Tohoku University, Sendai 980-8578, Japan
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Teplukhin A, Kendrick BK, Mniszewski SM, Tretiak S, Dub PA. Sampling electronic structure quadratic unconstrained binary optimization problems (QUBOs) with Ocean and Mukai solvers. PLoS One 2022; 17:e0263849. [PMID: 35148343 PMCID: PMC8836298 DOI: 10.1371/journal.pone.0263849] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/09/2021] [Accepted: 01/27/2022] [Indexed: 11/18/2022] Open
Abstract
The most advanced D-Wave Advantage quantum annealer has 5000+ qubits, however, every qubit is connected to a small number of neighbors. As such, implementation of a fully-connected graph results in an order of magnitude reduction in qubit count. To compensate for the reduced number of qubits, one has to rely on special heuristic software such as qbsolv, the purpose of which is to decompose a large quadratic unconstrained binary optimization (QUBO) problem into smaller pieces that fit onto a quantum annealer. In this work, we compare the performance of the open-source qbsolv which is a part of the D-Wave Ocean tools and a new Mukai QUBO solver from Quantum Computing Inc. (QCI). The comparison is done for solving the electronic structure problem and is implemented in a classical mode (Tabu search techniques). The Quantum Annealer Eigensolver is used to map the electronic structure eigenvalue-eigenvector equation to a QUBO problem, solvable on a D-Wave annealer. We find that the Mukai QUBO solver outperforms the Ocean qbsolv with one to two orders of magnitude more accurate energies for all calculations done in the present work, both the ground and excited state calculations. This work stimulates the further development of software to assist in the utilization of modern quantum annealers.
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Affiliation(s)
- Alexander Teplukhin
- Theoretical Division (T-1, MS B221), Los Alamos National Laboratory, Los Alamos, New Mexico, United States of America
| | - Brian K. Kendrick
- Theoretical Division (T-1, MS B221), Los Alamos National Laboratory, Los Alamos, New Mexico, United States of America
| | - Susan M. Mniszewski
- Computer, Computational and Statistical Sciences Division (CCS, MS B214), Los Alamos National Laboratory, Los Alamos, New Mexico, United States of America
| | - Sergei Tretiak
- Theoretical Division (T-1, MS B221), Los Alamos National Laboratory, Los Alamos, New Mexico, United States of America
| | - Pavel A. Dub
- Chemistry Division (C-IIAC, MS K558), Los Alamos National Laboratory, Los Alamos, New Mexico, United States of America
- * E-mail:
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GPS: A New TSP Formulation for Its Generalizations Type QUBO. MATHEMATICS 2022. [DOI: 10.3390/math10030416] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 12/10/2022]
Abstract
We propose a new Quadratic Unconstrained Binary Optimization (QUBO) formulation of the Travelling Salesman Problem (TSP), with which we overcame the best formulation of the Vehicle Routing Problem (VRP) in terms of the minimum number of necessary variables. After, we will present a detailed study of the constraints subject to the new TSP model and benchmark it with MTZ and native formulations. Finally, we will test whether the correctness of the formulation by entering it into a QUBO problem solver. The solver chosen is a D-Wave_2000Q6 quantum computer simulator due to the connection between Quantum Annealing and QUBO formulations.
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Teplukhin A, Kendrick BK, Mniszewski SM, Zhang Y, Kumar A, Negre CFA, Anisimov PM, Tretiak S, Dub PA. Computing molecular excited states on a D-Wave quantum annealer. Sci Rep 2021; 11:18796. [PMID: 34552136 PMCID: PMC8458378 DOI: 10.1038/s41598-021-98331-y] [Citation(s) in RCA: 7] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/08/2021] [Accepted: 08/31/2021] [Indexed: 11/29/2022] Open
Abstract
The possibility of using quantum computers for electronic structure calculations has opened up a promising avenue for computational chemistry. Towards this direction, numerous algorithmic advances have been made in the last five years. The potential of quantum annealers, which are the prototypes of adiabatic quantum computers, is yet to be fully explored. In this work, we demonstrate the use of a D-Wave quantum annealer for the calculation of excited electronic states of molecular systems. These simulations play an important role in a number of areas, such as photovoltaics, semiconductor technology and nanoscience. The excited states are treated using two methods, time-dependent Hartree–Fock (TDHF) and time-dependent density-functional theory (TDDFT), both within a commonly used Tamm–Dancoff approximation (TDA). The resulting TDA eigenvalue equations are solved on a D-Wave quantum annealer using the Quantum Annealer Eigensolver (QAE), developed previously. The method is shown to reproduce a typical basis set convergence on the example \documentclass[12pt]{minimal}
\usepackage{amsmath}
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\begin{document}$$\hbox {H}_2$$\end{document}H2 molecule and is also applied to several other molecular species. Characteristic properties such as transition dipole moments and oscillator strengths are computed as well. Three potential energy profiles for excited states are computed for \documentclass[12pt]{minimal}
\usepackage{amsmath}
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\begin{document}$$\hbox {NH}_3$$\end{document}NH3 as a function of the molecular geometry. Similar to previous studies, the accuracy of the method is dependent on the accuracy of the intermediate meta-heuristic software called qbsolv.
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Affiliation(s)
- Alexander Teplukhin
- Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM, 87545, USA
| | - Brian K Kendrick
- Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM, 87545, USA
| | - Susan M Mniszewski
- Computer, Computational and Statistical Sciences Division, Los Alamos National Laboratory, Los Alamos, NM, 87545, USA
| | - Yu Zhang
- Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM, 87545, USA
| | - Ashutosh Kumar
- Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM, 87545, USA
| | - Christian F A Negre
- Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM, 87545, USA
| | - Petr M Anisimov
- Accelerator Operations and Technology Division, Los Alamos National Laboratory, Los Alamos, NM, 87545, USA
| | - Sergei Tretiak
- Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM, 87545, USA.
| | - Pavel A Dub
- Chemistry Division, Los Alamos National Laboratory, Los Alamos, NM, 87545, USA.
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