Speedup via quantum sampling

Wocjan, Paweł; Abeyesinghe, Anura

doi:10.1103/physreva.78.042336

Cited by 103 publications

(134 citation statements)

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“…1 is one of the eigenstates of W , and (ii) the eigenvalue gap scales Oð ffiffi ffi δ p Þ. These two properties are important for generating classical thermal states through annealing (31,32).…”

Section: Summary Of Szegedy's Methodsmentioning

confidence: 99%

“…Motivation, Results, and Outline of the Quantum-Quantum Metropolis Algorithm (Q 2 MA) To summarize, before this work, the existing quantum algorithms were capable of either (i) preparing thermal states of classical systems with quantum speedup (31,32), or (ii) preparing thermal states of quantum systems without quantum speedup (33,34); this suggests that the potential of quantum computers may not have been fully exploited. This problem motivates us to further explore the power of quantum computation by providing a quantum algorithm that allows us to prepare thermal states of quantum systems with quantum speedup.…”

Section: Background Of Quantum Simulationmentioning

confidence: 99%

“…The key result is that a quadratic speedup Oð1∕ ffiffi ffi δ p Þ in the gap δ of the transition matrix is possible. This idea was later adapted to other algorithms (31,32) for preparing thermal states of classical Hamiltonians, based on the idea of quantum simulated annealing (QSA) (31).…”

Section: Background Of Quantum Simulationmentioning

confidence: 99%

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A quantum–quantum Metropolis algorithm

Yung

Aspuru‐Guzik

2012

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

145

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: Summary Of Szegedy's Methodsmentioning

confidence: 99%

Section: Background Of Quantum Simulationmentioning

confidence: 99%

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A quantum–quantum Metropolis algorithm

Yung

Aspuru‐Guzik

2012

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

145

120

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“…Somma et al [6] combined quantum walk and quantum Zeno effect to achieve quantum speedup. Wocjan and Abeyesinghe [7] improved it by using fixed point quantum search. Generally, these quantum algorithms allow the running time to scale as τ ∼ O(1/ √ ∆), a quadratic speedup compared with the classical counterparts.…”

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confidence: 99%

Simulation of classical thermal states on a quantum computer: A transfer-matrix approach

et al. 2010

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We present a hybrid quantum-classical algorithm to simulate thermal states of classical Hamiltonians on a quantum computer. Our scheme employs a sequence of locally controlled rotations, building up the desired state by adding qubits one at a time. We identify a class of classical models for which our method is efficient and avoids potential exponential overheads encountered by Groverlike or quantum Metropolis schemes. Our algorithm also gives an exponential advantage for 2D Ising models with magnetic field on a square lattice, compared with the previously known Zalka's algorithm.PACS numbers: 03.67.Ac, 05.10.CcSimulation of a finite-temperature physical system with a controllable quantum device is one of the most important goals of quantum simulation [1, 2]. Classical Markov-Chain Monte Carlo (MCMC) algorithms are powerful tools for sampling Gibbs distributions. They are efficient provided that the gap ∆ of the transition matrix is non-vanishing; the running time typically scales as τ ∼ O (1/∆). A quantum generalization [3] of MCMC has recently been explored by the quantum information community due to the connection to quantum walks [4]. Richter [5] developed a method for sampling from the Gibbs distribution for periodic lattices. Somma et al.[6] combined quantum walk and quantum Zeno effect to achieve quantum speedup. Wocjan and Abeyesinghe [7] improved it by using fixed point quantum search. Generally, these quantum algorithms allow the running time to scale as τ ∼ O(1/ √ ∆), a quadratic speedup compared with the classical counterparts. However, for many problems of practical interest, such as optimization problems and spin glasses, the gap ∆ may become exponentially small when the system size increases, making it unpractical to use MCMC algorithms for solving them (see Fig. 1). Therefore, gap-independent methods are more desirable for solving these problems.A class of gap-independent methods is called belief propagation [8], which generalizes the transfer matrix methods in statistical physics. For problems involving a regular geometry, it can be very efficient, e.g. onedimensional spin chains. This property will be exploited in this letter, where a different way for obtaining samples from the thermal state is discussed. This approach is a generalization of the state preparation method by Lidar and Biham [9], and Zalka [10]. We show that in some cases, the structure of the system under investigation allows for large speedups over the general methods. This is because the cost of our method is independent of the temperature and the gap size. Our proposed strategy is to construct a coherent encodings of a thermal state (CETS) |ψ CET S directly, rather than sampling from the thermal probability distribution:where s = {0, 1} N , β = 1/k B T is the inverse temperature, H(s) is the eigen-energy of some classical spin Hamiltonian for the N -spin configuration s = s 1 s 2 · · · s N and Z is the partition function. This CETS can be transformed into the corresponding thermal state ρ = e −βH /Tr e −βH by including a set...

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“…Some quantum algorithms have been proposed to approximate the partition functions of certain statistical physics models [5], or even obtain it exactly in very specific cases [6]. In the very recent past, many works have focused on quantum algorithms implementing classical Markov Chain Monte Carlo (MCMC) methods through quantum walks [7][8][9][10]. In general, these methods give a quadratic gain compared to classical simulations.…”

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confidence: 99%

Quantum Algorithm for Exact Monte Carlo Sampling

Destainville¹,

Georgeot²,

Giraud³

2010

Phys. Rev. Lett.

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We build a quantum algorithm which uses the Grover quantum search procedure in order to sample the exact equilibrium distribution of a wide range of classical statistical mechanics systems. The algorithm is based on recently developed exact Monte Carlo sampling methods, and yields a polynomial gain compared to classical procedures.PACS numbers: 03.67.Ac, 05.10.LnThe possibility of using quantum mechanics to treat information and perform computation has attracted great interest in the recent past (see e.g. [1] for a review). Quantum algorithms have been devised, which solve computational problems faster than their classical counterpart, such as the factorization algorithm of Shor [2]. However, relatively few problems have been identified which are amenable to quantum speedup. While many works have proposed methods to simulate quantum systems using a quantum processor (see e.g.[1] and references therein), fewer have tried to build quantum algorithms to speed up classical physical problems [3]. In particular, statistical physics is the source of many computational problems which have led to great efforts to develop efficient classical algorithms. For example, the goal of many Monte Carlo algorithms is to sample a configuration set Ω from an equilibrium probability distribution π [4]. It is therefore important to explore the possibilities to use quantum computers in order to speed up such problems. Some quantum algorithms have been proposed to approximate the partition functions of certain statistical physics models [5], or even obtain it exactly in very specific cases [6]. In the very recent past, many works have focused on quantum algorithms implementing classical Markov Chain Monte Carlo (MCMC) methods through quantum walks [7][8][9][10]. In general, these methods give a quadratic gain compared to classical simulations.Here we consider another type of MCMC algorithm recently developed, the "coupling from the past" (CFTP) procedure of Propp and Wilson, which leads in finite time to the exact equilibrium distribution [11]. We propose a quantum algorithm combining this CFTP procedure and the quantum search procedure of Grover [12], enabling a quadratic speedup over the classical algorithm, without using quantum walks. Our method enables to sample the exact equilibrium distribution in finite time for a wide class of systems, while previous algorithms either provide an approximate version whose error has to be controlled [5,[7][8][9][10], or are restricted to specific models [6]. Our algorithm is also rather simple compared to these other methods, while yielding a comparable polynomial gain.One of the key issues in classical MCMC algorithms is that they must be iterated sufficiently many times so that the final state is a "typical" configuration, in other words has a probability distribution close to the stationary one, π, independently of the initial state. In order to get close to the correct distribution π, one should be able to know when sufficient convergence is achieved. In a few particular cases, it is possible...

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Speedup via quantum sampling

Cited by 103 publications

References 16 publications

A quantum–quantum Metropolis algorithm

A quantum–quantum Metropolis algorithm

Simulation of classical thermal states on a quantum computer: A transfer-matrix approach

Quantum Algorithm for Exact Monte Carlo Sampling

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