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Journal ArticleDOI

Simulating physical phenomena by quantum networks

09 Apr 2002-Physical Review A (American Physical Society)-Vol. 65, Iss: 4, pp 042323
TL;DR: In this paper, the authors exploit the Jordan-Wigner isomorphism between the two algebras and give quantum circuits useful for the efficient evaluation of the physical properties (e.g., spectrum of observables or relevant correlation functions) of an arbitrary system with Hamiltonian H.
Abstract: Physical systems, characterized by an ensemble of interacting constituents, can be represented and studied by different algebras of operators (observables). For example, a fully polarized electronic system can be studied by means of the algebra generated by the usual fermionic creation and annihilation operators or by the algebra of Pauli (spin-1/2) operators. The Jordan-Wigner isomorphism gives the correspondence between the two algebras. As we previously noted, similar isomorphisms enable one to represent any physical system in a quantum computer. In this paper we evolve and exploit this fundamental observation to simulate generic physical phenomena by quantum networks. We give quantum circuits useful for the efficient evaluation of the physical properties (e.g., the spectrum of observables or relevant correlation functions) of an arbitrary system with Hamiltonian H.
Citations
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Journal ArticleDOI
TL;DR: The main theoretical and experimental aspects of quantum simulation have been discussed in this article, and some of the challenges and promises of this fast-growing field have also been highlighted in this review.
Abstract: Simulating quantum mechanics is known to be a difficult computational problem, especially when dealing with large systems However, this difficulty may be overcome by using some controllable quantum system to study another less controllable or accessible quantum system, ie, quantum simulation Quantum simulation promises to have applications in the study of many problems in, eg, condensed-matter physics, high-energy physics, atomic physics, quantum chemistry and cosmology Quantum simulation could be implemented using quantum computers, but also with simpler, analog devices that would require less control, and therefore, would be easier to construct A number of quantum systems such as neutral atoms, ions, polar molecules, electrons in semiconductors, superconducting circuits, nuclear spins and photons have been proposed as quantum simulators This review outlines the main theoretical and experimental aspects of quantum simulation and emphasizes some of the challenges and promises of this fast-growing field

1,941 citations

Journal ArticleDOI
TL;DR: Peruzzo et al. as mentioned in this paper developed a variational adiabatic ansatz and explored unitary coupled cluster where they established a connection from second order unitary cluster to universal gate sets through a relaxation of exponential operator splitting.
Abstract: Many quantum algorithms have daunting resource requirements when compared to what is available today. To address this discrepancy, a quantum-classical hybrid optimization scheme known as 'the quantum variational eigensolver' was developed (Peruzzo et al 2014 Nat. Commun. 5 4213) with the philosophy that even minimal quantum resources could be made useful when used in conjunction with classical routines. In this work we extend the general theory of this algorithm and suggest algorithmic improvements for practical implementations. Specifically, we develop a variational adiabatic ansatz and explore unitary coupled cluster where we establish a connection from second order unitary coupled cluster to universal gate sets through a relaxation of exponential operator splitting. We introduce the concept of quantum variational error suppression that allows some errors to be suppressed naturally in this algorithm on a pre-threshold quantum device. Additionally, we analyze truncation and correlated sampling in Hamiltonian averaging as ways to reduce the cost of this procedure. Finally, we show how the use of modern derivative free optimization techniques can offer dramatic computational savings of up to three orders of magnitude over previously used optimization techniques.

1,490 citations

Journal ArticleDOI
TL;DR: This review presents strategies employed to construct quantum algorithms for quantum chemistry, with the goal that quantum computers will eventually answer presently inaccessible questions, for example, in transition metal catalysis or important biochemical reactions.
Abstract: One of the most promising suggested applications of quantum computing is solving classically intractable chemistry problems. This may help to answer unresolved questions about phenomena such as high temperature superconductivity, solid-state physics, transition metal catalysis, and certain biochemical reactions. In turn, this increased understanding may help us to refine, and perhaps even one day design, new compounds of scientific and industrial importance. However, building a sufficiently large quantum computer will be a difficult scientific challenge. As a result, developments that enable these problems to be tackled with fewer quantum resources should be considered important. Driven by this potential utility, quantum computational chemistry is rapidly emerging as an interdisciplinary field requiring knowledge of both quantum computing and computational chemistry. This review provides a comprehensive introduction to both computational chemistry and quantum computing, bridging the current knowledge gap. Major developments in this area are reviewed, with a particular focus on near-term quantum computation. Illustrations of key methods are provided, explicitly demonstrating how to map chemical problems onto a quantum computer, and how to solve them. The review concludes with an outlook on this nascent field.

954 citations

Journal ArticleDOI
TL;DR: In this paper, the first electronic structure calculation performed on a quantum computer without exponentially costly precompilation is reported, where a programmable array of superconducting qubits is used to compute the energy surface of molecular hydrogen using two distinct quantum algorithms.
Abstract: We report the first electronic structure calculation performed on a quantum computer without exponentially costly precompilation. We use a programmable array of superconducting qubits to compute the energy surface of molecular hydrogen using two distinct quantum algorithms. First, we experimentally execute the unitary coupled cluster method using the variational quantum eigensolver. Our efficient implementation predicts the correct dissociation energy to within chemical accuracy of the numerically exact result. Second, we experimentally demonstrate the canonical quantum algorithm for chemistry, which consists of Trotterization and quantum phase estimation. We compare the experimental performance of these approaches to show clear evidence that the variational quantum eigensolver is robust to certain errors. This error tolerance inspires hope that variational quantum simulations of classically intractable molecules may be viable in the near future.

925 citations

Journal ArticleDOI
TL;DR: The time is ripe for describing some of the recent development of superconducting devices, systems and applications as well as practical applications of QIP, such as computation and simulation in Physics and Chemistry.
Abstract: During the last ten years, superconducting circuits have passed from being interesting physical devices to becoming contenders for near-future useful and scalable quantum information processing (QIP). Advanced quantum simulation experiments have been shown with up to nine qubits, while a demonstration of quantum supremacy with fifty qubits is anticipated in just a few years. Quantum supremacy means that the quantum system can no longer be simulated by the most powerful classical supercomputers. Integrated classical-quantum computing systems are already emerging that can be used for software development and experimentation, even via web interfaces. Therefore, the time is ripe for describing some of the recent development of superconducting devices, systems and applications. As such, the discussion of superconducting qubits and circuits is limited to devices that are proven useful for current or near future applications. Consequently, the centre of interest is the practical applications of QIP, such as computation and simulation in Physics and Chemistry.

809 citations

References
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Book
27 Aug 1986
TL;DR: In this paper, the authors define the notion of generalized coherent states and define a generalization of the Coherent State Representation T?(g) of the Heisenberg-Weyl Group.
Abstract: I Generalized Coherent States for the Simplest Lie Groups.- 1. Standard System of Coherent States Related to the Heisenberg-Weyl Group: One Degree of Freedom.- 1.1 The Heisenberg-Weyl Group and Its Representations.- 1.1.1 The Heisenberg-Weyl Group.- 1.1.2 Representations of the Heisenberg-Weyl Group.- 1.1.3 Concrete Realization of the Representation T?(g).- 1.2 Coherent States.- 1.3 The Fock-Bargmann Representation.- 1.4 Completeness of Coherent-State Subsystems.- 1.5 Coherent States and Theta Functions.- 1.6 Operators and Their Symbols.- 1.7 Characteristic Functions.- 2. Coherent States for Arbitrary Lie Groups.- 2.1 Definition of the Generalized Coherent State.- 2.2 General Properties of Coherent-State Systems.- 2.3 Completeness and Expansion in States of the CS System.- 2.4 Selection of Generalized CS Systems with States Closest to Classical.- 3. The Standard System of Coherent States Several Degrees of Freedom.- 3.1 General Properties.- 3.2 Coherent States and Theta Functions for Several Degrees of Freedom.- 4. Coherent States for the Rotation Group of Three-Dimensional Space.- 4.1 Structure of the Groups SO(3) and SU(2).- 4.2 Representations of SU(2).- 4.3 Coherent States.- 5. The Most Elementary Noneompact, Non-Abelian Simple Lie Group: SU(1,1).- 5.1 Group SU(1,1) and Its Representations.- 5.1.1 Fundamental Properties ofU(1,1) 67.- 5.1.2 Discrete Series.- 5.1.3 Principal (Continuous) Series.- 5.2 Coherent States.- 5.2.1 Discrete Series.- 5.2.2 Principal (Continuous) Series.- 6. The Lorentz Group: SO(3,1).- 6.1 Representations of the Lorentz Group.- 6.2 Coherent States.- 7. Coherent States for the SO(n, 1) Group: Class-1 Representations of the Principal Series.- 7.1 Class-I Representations of SO(n,1).- 7.2 Coherent States.- 8. Coherent States for a Bosonic System with Finite Number of Degrees of Freedom.- 8.1 Canonical Transformations.- 8.2 Coherent States.- 8.3 Operators in the Space ?B(+).- 9. Coherent States for a Fermionic System with Finite Number of Degrees of Freedom.- 9.1 Canonical Transformations.- 9.2 Coherent States.- 9.3 Operators in the Space ?F(+).- II General Case.- 10. Coherent States for Nilpotent Lie Groups.- 10.1 Structure of Nilpotent Lie Groups.- 10.2 Orbits of Coadjoint Representation.- 10.3 Orbits of Nilpotent Lie Groups.- 10.4 Representations of Nilpotent Lie Groups.- 10.5 Coherent States.- 11. Coherent States for Compact Semisimple Lie Groups.- 11.1 Elements of the Theory of Compact Semisimple Lie Groups..- 11.2 Representations of Compact Simple Lie Groups.- 11.3 Coherent States.- 12. Discrete Series of Representations: The General Case.- 12.1 Discrete Series.- 12.2 Bounded Domains.- 12.3 Coherent States.- 13. Coherent States for Real Semisimple Lie Groups: Class-I Representations of Principal Series.- 13.1 Class-I Representations.- 13.2 Coherent States.- 13.3 Horocycles in Symmetric Space.- 13.4 Rank-1 Symmetric Spaces.- 13.5 Properties of Rank-1 CS Systems.- 13.6 Complex Homogeneous Bounded Domains.- 13.6.1 Type-I Tube Domains.- 13.6.2 Type-II Tube Domains.- 13.6.3 Type-III Tube Domains.- 13.6.4 Type-IV Domains.- 13.6.5 The Exceptional Domain Dv.- 13.7 Properties of the Coherent States.- 14. Coherent States and Discrete Subgroups: The Case of SU(1,1).- 14.1 Preliminaries.- 14.2 Incompleteness Criterion for CS Subsystems Related to Discrete Subgroups.- 14.3 Growth of a Function Analytical in a Disk Related to the Distribution of Its Zeros.- 14.4 Completeness Criterion for CS Subsystems.- 14.5 Discrete Subgroups of SU(1,1) and Automorphic Forms.- 15. Coherent States for Discrete Series and Discrete Subgroups: General Case.- 15.1 Automorphic Forms.- 15.2 Completeness of Some CS Subsystems.- 16. Coherent States and Berezin's Quantization.- 16.1 Classical Mechanics.- 16.2 Quantization.- 16.3 Quantization on the Lobachevsky Plane.- 16.3.1 Description of Operators.- 16.3.2 The Correspondence Principle.- 16.3.3 Operator Th in Terms of a Laplacian.- 16.3.4 Representation of Group of Motions of the Lobachevsky Plane in Space ?h.- 16.3.5 Quantization by Inversions Analog to Weyl Quantization.- 16.4 Quantization on a Sphere.- 16.5 Quantization on Homogeneous Kahler Manifolds.- III Physical Applications.- 17. Preliminaries.- 18. Quantum Oscillators.- 18.1 Quantum Oscillator Acted on by a Variable External Force..- 18.2 Parametric Excitation of a Quantum Oscillator.- 18.3 Quantum Singular Oscillator.- 18.3.1 The Stationary Case.- 18.3.2 The Nonstationary Case.- 18.3.3 The Case of N Interacting Particles.- 18.4 Oscillator with Variable Frequency Acted on by an External Force.- 19. Particles in External Electromagnetic Fields.- 19.1 Spin Motion in a Variable Magnetic Field.- 19.2 Boson Pair Production in a Variable Homogeneous External Field.- 19.2.1 Dynamical Symmetry for Scalar Particles.- 19.2.2 The Multidimensional Case: Coherent States.- 19.2.3 The Multidimensional Case: Nonstationary Problem..- 19.3 Fermion Pair Production in a Variable Homogeneous External Field.- 19.3.1 Dynamical Symmetry for Spin-1/2 particles.- 19.3.2 Heisenberg Representation.- 19.3.3 The Multidimensional Case: Coherent States.- 20. Generating Function for Clebsch-Gordan Coefficients of the SU(2) group.- 21. Coherent States and the Quasiclassical Limit.- 22. 1/N Expansion for Gross-Neveu Models.- 22.1 Description of the Model.- 22.2 Dimensionality of Space ?N= ?O in the Fermion Case.- 22.3 Quasiclassical Limit.- 23. Relaxation to Thermodynamic Equilibrium.- 23.1 Relaxation of Quantum Oscillator to Thermodynamic Equilibrium.- 23.1.1 Kinetic Equation.- 23.1.2 Characteristic Functions and Quasiprobability Distributions.- 23.1.3 Use of Operator Symbols.- 23.2 Relaxation of a Spinning Particle to Thermodynamic Equilibrium in the Presence of a Magnetic Field.- 24. Landau Diamagnetism.- 25. The Heisenberg-Euler Lagrangian.- 26. Synchrotron Radiation.- 27. Classical and Quantal Entropy.- Appendix A. Proof of Completeness for Certain CS Subsystems.- Appendix B. Matrix Elements of the Operator D(y).- Appendix C. Jacobians of Group Transformations for Classical Domains.- Further Applications of the CS Method.- References.- Subject-Index.- Addendum. Further Applications of the CS Method.- References.- References to Addendum.- Subject-Index.

3,565 citations

Book
21 Jan 1988
TL;DR: The Landay Theory of Germi Liquids has been used in this paper to describe the second quantization and coherent states of the first quantization of a function at finite temperature.
Abstract: * Second Quantization and Coherent States * General Formalism at Finite Temperature * Perturbation Theory at Zero Temperature * Order Parameters and Broken Symmetry * Greens Functions * The Landay Theory of Germi Liquids * Further Development of Functional Integrals * Stochastic Methods

1,619 citations

Book
19 Dec 1985
TL;DR: In this paper, the authors provide a comprehensive and pedagogical account of the various methods used in the quantum theory of finite systems, including molecular, atomic, nuclear, and particle phenomena.
Abstract: This book provides a comprehensive and pedagogical account of the various methods used in the quantum theory of finite systems, including molecular, atomic, nuclear, and particle phenomena. Covering both background material and advanced topics and including nearly 200 problems, "Quantum Theory of Finite Systems" has been designed to serve primarily as a text and will also prove useful as a reference in research.The first of the book's four parts introduces the basic mathematical apparatus: second quantization, canonical transformations, Wick theorems and the resulting diagram expansions, and oscillator models. The second part presents mean field approximations and the recently developed path integral methods for the quantization of collective modes. Part three develops perturbation theory in terms of both time-dependent Feynman diagrams and time-independent Goldstone diagrams. A fourth part discusses variational methods based on correlated wavefunctions, including spin correlations. The approximation schemes are formulated for fermions and bosons at eigher zero or non-zero temperature.Although the formalism developed applies to both finite and infinite systems, the book stresses those aspects of the theory that are specific to the description of finite systems. Thus special attention is given to mean field approximations, the ensuing broken symmetries, and the associated collective motions such as rotations. Conversely, some specific features of systems with infinite numbers of degrees of freedom (such as the thermodynamic limit, critical phenomena, and the elimination of ultraviolet divergencies) are deliberately omitted.Jean-Paul Blaizot and Georges Ripka are associated with the Centre d'Etudes Nucleaires de Saclay.

952 citations


"Simulating physical phenomena by qu..." refers background or methods in this paper

  • ...Simplification occurs, however, by recalling the Thouless’ s theorem [16] which says that if |φ〉 = Ne ∏...

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  • ...In Appendix A we illustrate procedures and recipes to prepare one such cl ass of correlated (entangled) states, the so-called Jastrow states [16]....

    [...]

  • ...The classic form of a Jastrow-type wave function for fermion s is[16] |Ψ0〉 = e|φ〉 , (A9)...

    [...]

  • ...In quantum many-body physics, considerable experience and interest exists in developing simple approaches for generating several specific classes of corre lated wave functions [16]....

    [...]

Book
01 Jan 1974
TL;DR: In this article, a phenomenological theory of magnetic order impurities and alloys external fields electrons, phonons, and transport aspects of the electron-electron interaction is presented.
Abstract: Electronic structure lattice dynamics symmetry and its consequences phenomenological theories of magnetic order impurities and alloys external fields electrons, phonons, and transport aspects of the electron-electron interaction.

832 citations