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Kovalsky LK, Calderon-Vargas FA, Grace MD, Magann AB, Larsen JB, Baczewski AD, Sarovar M. Self-Healing of Trotter Error in Digital Adiabatic State Preparation. PHYSICAL REVIEW LETTERS 2023; 131:060602. [PMID: 37625062 DOI: 10.1103/physrevlett.131.060602] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/02/2022] [Revised: 04/18/2023] [Accepted: 06/29/2023] [Indexed: 08/27/2023]
Abstract
Adiabatic time evolution can be used to prepare a complicated quantum many-body state from one that is easier to synthesize and Trotterization can be used to implement such an evolution digitally. The complex interplay between nonadiabaticity and digitization influences the infidelity of this process. We prove that the first-order Trotterization of a complete adiabatic evolution has a cumulative infidelity that scales as O(T^{-2}δt^{2}) instead of O(T^{2}δt^{2}) expected from general Trotter error bounds, where δt is the time step and T is the total time. This result suggests a self-healing mechanism and explains why, despite increasing T, infidelities for fixed-δt digitized evolutions still decrease for a wide variety of Hamiltonians. It also establishes a correspondence between the quantum approximate optimization algorithm and digitized quantum annealing.
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Affiliation(s)
- Lucas K Kovalsky
- Quantum Algorithms and Applications Collaboratory, Sandia National Laboratories, Livermore, California 94550, USA
| | - Fernando A Calderon-Vargas
- Quantum Algorithms and Applications Collaboratory, Sandia National Laboratories, Livermore, California 94550, USA
| | - Matthew D Grace
- Quantum Algorithms and Applications Collaboratory, Sandia National Laboratories, Livermore, California 94550, USA
| | - Alicia B Magann
- Quantum Algorithms and Applications Collaboratory, Sandia National Laboratories, Albuquerque, New Mexico 87185, USA
| | - James B Larsen
- Quantum Algorithms and Applications Collaboratory, Sandia National Laboratories, Albuquerque, New Mexico 87185, USA
- Department of Mathematics, Brigham Young University, Provo, Utah 84602, USA
| | - Andrew D Baczewski
- Quantum Algorithms and Applications Collaboratory, Sandia National Laboratories, Albuquerque, New Mexico 87185, USA
| | - Mohan Sarovar
- Quantum Algorithms and Applications Collaboratory, Sandia National Laboratories, Livermore, California 94550, USA
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Chan HHS, Meister R, Jones T, Tew DP, Benjamin SC. Grid-based methods for chemistry simulations on a quantum computer. SCIENCE ADVANCES 2023; 9:eabo7484. [PMID: 36857445 PMCID: PMC9977186 DOI: 10.1126/sciadv.abo7484] [Citation(s) in RCA: 3] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 02/23/2022] [Accepted: 01/30/2023] [Indexed: 06/18/2023]
Abstract
First-quantized, grid-based methods for chemistry modeling are a natural and elegant fit for quantum computers. However, it is infeasible to use today's quantum prototypes to explore the power of this approach because it requires a substantial number of near-perfect qubits. Here, we use exactly emulated quantum computers with up to 36 qubits to execute deep yet resource-frugal algorithms that model 2D and 3D atoms with single and paired particles. A range of tasks is explored, from ground state preparation and energy estimation to the dynamics of scattering and ionization; we evaluate various methods within the split-operator QFT (SO-QFT) Hamiltonian simulation paradigm, including protocols previously described in theoretical papers and our own techniques. While we identify certain restrictions and caveats, generally, the grid-based method is found to perform very well; our results are consistent with the view that first-quantized paradigms will be dominant from the early fault-tolerant quantum computing era onward.
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Affiliation(s)
| | - Richard Meister
- Department of Materials, University of Oxford, Oxford OX1 3PH, UK
| | - Tyson Jones
- Department of Materials, University of Oxford, Oxford OX1 3PH, UK
| | - David P. Tew
- Department of Chemistry, University of Oxford, Oxford OX1 3TA, UK
- Duality Quantum Photonics, 6 Lower Park Row, Bristol BS1 5BJ, UK
| | - Simon C. Benjamin
- Department of Materials, University of Oxford, Oxford OX1 3PH, UK
- Quantum Motion, 9 Sterling Way, London N7 9HJ, UK
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Russo AE, Rudinger KM, Morrison BCA, Baczewski AD. Evaluating Energy Differences on a Quantum Computer with Robust Phase Estimation. PHYSICAL REVIEW LETTERS 2021; 126:210501. [PMID: 34114854 DOI: 10.1103/physrevlett.126.210501] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/20/2020] [Revised: 03/17/2021] [Accepted: 03/22/2021] [Indexed: 06/12/2023]
Abstract
We adapt the robust phase estimation algorithm to the evaluation of energy differences between two eigenstates using a quantum computer. This approach does not require controlled unitaries between auxiliary and system registers or even a single auxiliary qubit. As a proof of concept, we calculate the energies of the ground state and low-lying electronic excitations of a hydrogen molecule in a minimal basis on a cloud quantum computer. The denominative robustness of our approach is then quantified in terms of a high tolerance to coherent errors in the state preparation and measurement. Conceptually, we note that all quantum phase estimation algorithms ultimately evaluate eigenvalue differences.
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Affiliation(s)
- A E Russo
- Center for Computing Research, Sandia National Laboratories, Albuquerque, New Mexico 87185, USA
| | - K M Rudinger
- Center for Computing Research, Sandia National Laboratories, Albuquerque, New Mexico 87185, USA
| | - B C A Morrison
- Center for Computing Research, Sandia National Laboratories, Albuquerque, New Mexico 87185, USA
- Center for Quantum Information and Control (CQuIC), Department of Physics and Astronomy, University of New Mexico, Albuquerque, New Mexico 87131, USA
| | - A D Baczewski
- Center for Computing Research, Sandia National Laboratories, Albuquerque, New Mexico 87185, USA
- Center for Quantum Information and Control (CQuIC), Department of Physics and Astronomy, University of New Mexico, Albuquerque, New Mexico 87131, USA
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Poulin D, Kitaev A, Steiger DS, Hastings MB, Troyer M. Quantum Algorithm for Spectral Measurement with a Lower Gate Count. PHYSICAL REVIEW LETTERS 2018; 121:010501. [PMID: 30028152 DOI: 10.1103/physrevlett.121.010501] [Citation(s) in RCA: 12] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/22/2017] [Indexed: 06/08/2023]
Abstract
We present two techniques that can greatly reduce the number of gates required to realize an energy measurement, with application to ground state preparation in quantum simulations. The first technique realizes that to prepare the ground state of some Hamiltonian, it is not necessary to implement the time-evolution operator: any unitary operator which is a function of the Hamiltonian will do. We propose one such unitary operator which can be implemented exactly, circumventing any Taylor or Trotter approximation errors. The second technique is tailored to lattice models, and is targeted at reducing the use of generic single-qubit rotations, which are very expensive to produce by standard fault tolerant techniques. In particular, the number of generic single-qubit rotations used by our method scales with the number of parameters in the Hamiltonian, which contrasts with a growth proportional to the lattice size required by other techniques.
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Affiliation(s)
- David Poulin
- Département de Physique & Institut Quantique, Université de Sherbrooke, J1K 2R1, Canada
- Canadian Institute for Advanced Research, Toronto, Ontario M5G 1Z8, Canada
| | - Alexei Kitaev
- California Institute of Technology, Pasadena, California 91125, USA
| | | | - Matthew B Hastings
- Station Q, Microsoft Research, Santa Barbara, California 93106-6105, USA
- Quantum Architecture and Computation Group, Microsoft Research, Redmond, Washington 98052, USA
| | - Matthias Troyer
- Theoretische Physik, ETH Zurich, 8093 Zurich, Switzerland
- Quantum Architecture and Computation Group, Microsoft Research, Redmond, Washington 98052, USA
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Yung MH, Whitfield JD, Boixo S, Tempel DG, Aspuru-Guzik A. Introduction to Quantum Algorithms for Physics and Chemistry. ADVANCES IN CHEMICAL PHYSICS 2014. [DOI: 10.1002/9781118742631.ch03] [Citation(s) in RCA: 16] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 12/04/2022]
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Yung MH, Casanova J, Mezzacapo A, McClean J, Lamata L, Aspuru-Guzik A, Solano E. From transistor to trapped-ion computers for quantum chemistry. Sci Rep 2014; 4:3589. [PMID: 24395054 PMCID: PMC5378044 DOI: 10.1038/srep03589] [Citation(s) in RCA: 58] [Impact Index Per Article: 5.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/02/2013] [Accepted: 12/06/2013] [Indexed: 12/02/2022] Open
Abstract
Over the last few decades, quantum chemistry has progressed through the development of computational methods based on modern digital computers. However, these methods can hardly fulfill the exponentially-growing resource requirements when applied to large quantum systems. As pointed out by Feynman, this restriction is intrinsic to all computational models based on classical physics. Recently, the rapid advancement of trapped-ion technologies has opened new possibilities for quantum control and quantum simulations. Here, we present an efficient toolkit that exploits both the internal and motional degrees of freedom of trapped ions for solving problems in quantum chemistry, including molecular electronic structure, molecular dynamics, and vibronic coupling. We focus on applications that go beyond the capacity of classical computers, but may be realizable on state-of-the-art trapped-ion systems. These results allow us to envision a new paradigm of quantum chemistry that shifts from the current transistor to a near-future trapped-ion-based technology.
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Affiliation(s)
- M.-H. Yung
- Center for Quantum Information, Institute for Interdisciplinary Information Sciences, Tsinghua University, Beijing, 100084, P. R. China
- Department of Chemistry and Chemical Biology, Harvard University, Cambridge MA, 02138, USA
- These authors contributed equally to this work
| | - J. Casanova
- Department of Physical Chemistry, University of the Basque Country UPV/EHU, Apartado 644, 48080 Bilbao, Spain
- These authors contributed equally to this work
| | - A. Mezzacapo
- Department of Physical Chemistry, University of the Basque Country UPV/EHU, Apartado 644, 48080 Bilbao, Spain
| | - J. McClean
- Department of Chemistry and Chemical Biology, Harvard University, Cambridge MA, 02138, USA
| | - L. Lamata
- Department of Physical Chemistry, University of the Basque Country UPV/EHU, Apartado 644, 48080 Bilbao, Spain
| | - A. Aspuru-Guzik
- Department of Chemistry and Chemical Biology, Harvard University, Cambridge MA, 02138, USA
| | - E. Solano
- Department of Physical Chemistry, University of the Basque Country UPV/EHU, Apartado 644, 48080 Bilbao, Spain
- IKERBASQUE, Basque Foundation for Science, Alameda Urquijo 36, 48011 Bilbao, Spain
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7
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Solving quantum ground-state problems with nuclear magnetic resonance. Sci Rep 2012; 1:88. [PMID: 22355607 PMCID: PMC3216574 DOI: 10.1038/srep00088] [Citation(s) in RCA: 19] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/20/2011] [Accepted: 08/17/2011] [Indexed: 11/27/2022] Open
Abstract
Quantum ground-state problems are computationally hard problems for general many-body Hamiltonians; there is no classical or quantum algorithm known to be able to solve them efficiently. Nevertheless, if a trial wavefunction approximating the ground state is available, as often happens for many problems in physics and chemistry, a quantum computer could employ this trial wavefunction to project the ground state by means of the phase estimation algorithm (PEA). We performed an experimental realization of this idea by implementing a variational-wavefunction approach to solve the ground-state problem of the Heisenberg spin model with an NMR quantum simulator. Our iterative phase estimation procedure yields a high accuracy for the eigenenergies (to the 10−5 decimal digit). The ground-state fidelity was distilled to be more than 80%, and the singlet-to-triplet switching near the critical field is reliably captured. This result shows that quantum simulators can better leverage classical trial wave functions than classical computers
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Abstract
The classical Metropolis sampling method is a cornerstone of many statistical modeling applications that range from physics, chemistry, and biology to economics. This method is particularly suitable for sampling the thermal distributions of classical systems. The challenge of extending this method to the simulation of arbitrary quantum systems is that, in general, eigenstates of quantum Hamiltonians cannot be obtained efficiently with a classical computer. However, this challenge can be overcome by quantum computers. Here, we present a quantum algorithm which fully generalizes the classical Metropolis algorithm to the quantum domain. The meaning of quantum generalization is twofold: The proposed algorithm is not only applicable to both classical and quantum systems, but also offers a quantum speedup relative to the classical counterpart. Furthermore, unlike the classical method of quantum Monte Carlo, this quantum algorithm does not suffer from the negative-sign problem associated with fermionic systems. Applications of this algorithm include the study of low-temperature properties of quantum systems, such as the Hubbard model, and preparing the thermal states of sizable molecules to simulate, for example, chemical reactions at an arbitrary temperature.
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Kassal I, Whitfield JD, Perdomo-Ortiz A, Yung MH, Aspuru-Guzik A. Simulating Chemistry Using Quantum Computers. Annu Rev Phys Chem 2011; 62:185-207. [PMID: 21166541 DOI: 10.1146/annurev-physchem-032210-103512] [Citation(s) in RCA: 193] [Impact Index Per Article: 14.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/09/2022]
Affiliation(s)
| | | | - Alejandro Perdomo-Ortiz
- Department of Chemistry and Chemical Biology, Harvard University, Cambridge, Massachusetts 02138;
| | - Man-Hong Yung
- Department of Chemistry and Chemical Biology, Harvard University, Cambridge, Massachusetts 02138;
| | - Alán Aspuru-Guzik
- Department of Chemistry and Chemical Biology, Harvard University, Cambridge, Massachusetts 02138;
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Farhi E, Gosset D, Hassidim A, Lutomirski A, Nagaj D, Shor P. Quantum state restoration and single-copy tomography for ground states of Hamiltonians. PHYSICAL REVIEW LETTERS 2010; 105:190503. [PMID: 21231156 DOI: 10.1103/physrevlett.105.190503] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/08/2010] [Indexed: 05/30/2023]
Abstract
Given a single copy of an unknown quantum state, the no-cloning theorem limits the amount of information that can be extracted from it. Given a gapped Hamiltonian, in most situations it is impractical to compute properties of its ground state, even though in principle all the information about the ground state is encoded in the Hamiltonian. We show in this Letter that if you know the Hamiltonian of a system and have a single copy of its ground state, you can use a quantum computer to efficiently compute its local properties. Specifically, in this scenario, we give efficient algorithms that copy small subsystems of the state and estimate the full statistics of any local measurement.
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Affiliation(s)
- Edward Farhi
- Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
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Poulin D, Wocjan P. Sampling from the thermal quantum Gibbs state and evaluating partition functions with a quantum computer. PHYSICAL REVIEW LETTERS 2009; 103:220502. [PMID: 20366078 DOI: 10.1103/physrevlett.103.220502] [Citation(s) in RCA: 14] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/15/2009] [Indexed: 05/29/2023]
Abstract
We present a quantum algorithm to prepare the thermal Gibbs state of interacting quantum systems. This algorithm sets a universal upper bound D(alpha) on the thermalization time of a quantum system, where D is the system's Hilbert space dimension and alpha < or = 1/2 is proportional to the Helmholtz free energy density. We also derive an algorithm to evaluate the partition function of a quantum system in a time proportional to the system's thermalization time and inversely proportional to the targeted accuracy squared.
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Affiliation(s)
- David Poulin
- Département de Physique, Université de Sherbrooke, Québec, Canada, J1K 2R1
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