Quantum algorithms for fermionic simulations

Ortíz, Gerardo; Gubernatis, J. E.; Knill, Emanuel; Laflamme, Raymond

doi:10.1103/physreva.64.022319

Cited by 312 publications

(315 citation statements)

References 39 publications

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“…It was also discovered (17) that such an approach is particularly suitable for simulating single or many-particle systems where the Hamiltonians consist of two terms, namely kinetic energy and potential energy, for example, atomic or molecular systems (18). On the other hand, however, attempt to create quantum algorithms to solve the ground-state (19,20) or thermal-state (21-23) problems of unstructured Hamiltonians failed to show an exponential gain over classical approaches. However, this lack of an exponential advantage does not mean that quantum computers fail to show advantages over classical computers in quantum simulation.…”

Section: Background Of Quantum Simulationmentioning

confidence: 99%

A quantum–quantum Metropolis algorithm

Yung

Aspuru‐Guzik

2012

Proc. Natl. Acad. Sci. U.S.A.

148

120

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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.Metropolis method | statistical physics | quantum computing | quantum simulation | simulated annealing I nteracting many-body problems cover a wide range of applications in many areas in science and technology. These are best exemplified by the Ising spin model, which was originally developed for explaining ferromagnetism and its associated phase transitions (1). The Ising model was later found to be related to many practical applications, for example, error-correcting codes, image restoration, associative memory, and optimization problems (see, e.g., ref. 2). However, there is no efficient method for finding solutions to many-body problems in general. For example, the problem of finding the ground state of the Ising model with arbitrary couplings is known to be a nondeterministic polynomial (NP)-complete problem, meaning that, if an efficient algorithm for solving the Ising model exists, then all of the problems in the class of NP, for example the graph isomorphism problem and some variants of the traveling salesman problem, can be solved efficiently as well (see, e.g., ref.3).The most fundamental challenge for solving many-body problems is that they generally require an exponentially large amount of spatial and temporal computing resources to find the solutions as the system size increases (4). Although no universal method for solving general many-body problems has been found, powerful classical methods such as Markov-chain Monte Carlo (MCMC) or quantum Monte Carlo (QMC) have been invented and proven to be successful in many applications (see, e.g., ref. 5). These methods, however, have certain limitations. For example, the running time of MCMC scales as Oð1∕δÞ (6), where δ is the gap of the transition matrix. For problems such as spin glasses (7) where δ is vani...

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Section: Background Of Quantum Simulationmentioning

confidence: 99%

A quantum–quantum Metropolis algorithm

Yung

Aspuru‐Guzik

2012

Proc. Natl. Acad. Sci. U.S.A.

148

120

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“…30 in elementary gates can be done using the methods described in previous works. 2,3 In particular, in Fig. 3 we show the decomposition of the term exp…”

Section: Evolution Of the Initial Statementioning

confidence: 99%

“…2 The basic idea is to use L extra (ancilla) qubits, then perform unitary evolutions controlled in the state of the ancillas, and finally perform a measurement of the z-component of the spin of the ancillas. In this way, the probability of successful preparation of |ψ is 1/L.…”

Section: Preparation Of the Initial Statementioning

confidence: 99%

Quantum simulations of physics problems

et al. 2003

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If a large Quantum Computer (QC) existed today, what type of physical problems could we efficiently simulate on it that we could not simulate on a classical Turing machine? In this paper we argue that a QC could solve some relevant physical "questions" more efficiently. The existence of one-to-one mappings between different algebras of observables or between different Hilbert spaces allow us to represent and imitate any physical system by any other one (e.g., a bosonic system by a spin-1/2 system). We explain how these mappings can be performed showing quantum networks useful for the efficient evaluation of some physical properties, such as correlation functions and energy spectra.

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“…Armed with these, researchers are able to convert physical questions about complex many-body quantum systems into tractable computational problems. Certain roadblocks faced by these classical approaches, such as the exponential growth of classical resources (Lloyd, 1996) and the dynamical sign problem (Gubernatis et al, 2001), are expected to be overcome by the use of quantum computers.…”

Section: Theoretical Chemistry Tools For Quantum Simulationmentioning

confidence: 99%

Quantum Information and Computation for Chemistry

Kais

2014

Advances in Chemical Physics

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Quantum algorithms for fermionic simulations

Cited by 312 publications

References 39 publications

A quantum–quantum Metropolis algorithm

A quantum–quantum Metropolis algorithm

Quantum simulations of physics problems

Quantum Information and Computation for Chemistry

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