Quantum and classical probability distributions for arbitrary Hamiltonians

Semay, Claude; Ducobu, Ludovic

doi:10.1088/0143-0807/37/4/045403

Cited by 11 publications

(15 citation statements)

References 25 publications

(59 reference statements)

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“…It is not clear how accurately one needs to sample from Boltzmann distributions for machine learning, or even that Boltzmann distributions are optimal for this purpose. A tantalizing research direction is the use of distributions that have a large quantum component [2], particularly given that certain distributions generated by quantum Hamiltonians are believed to have no efficient classical sampling mechanism [27,28]. A deeper understanding of these processes will have profound implications for the design of future annealers and the prospects of utilizing quantum annealers as efficient Boltzmann samplers for machine learning and beyond.…”

Section: Discussionmentioning

confidence: 99%

Thermalization, Freeze-out, and Noise: Deciphering Experimental Quantum Annealers

2017

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By contrasting the performance of two quantum annealers operating at different temperatures, we address recent questions related to the role of temperature in these devices and their function as 'Boltzmann samplers'. Using a method to reliably calculate the degeneracies of the energy levels of large-scale spin-glass instances, we are able to estimate the instance-dependent effective temperature from the output of annealing runs. Our results corroborate the 'freeze-out' picture which posits two regimes, one in which the final state corresponds to a Boltzmann distribution of the final Hamiltonian with a well-defined 'effective temperature' determined at a freeze-out point late in the anneal, and another regime in which such a distribution is not necessarily expected. We find that the output distributions of the annealers do not in general correspond to a classical Boltzmann distribution for the final Hamiltonian. We also find that the effective temperatures at different programming cycles fluctuate greatly, with the effect worsening with problem size. We discuss the implications of our results for the design of future quantum annealers to act as more effective Boltzmann samplers and for the programming of such annealers.

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Section: Discussionmentioning

confidence: 99%

Thermalization, Freeze-out, and Noise: Deciphering Experimental Quantum Annealers

2017

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“…One might also argue that our results are merely a consequence of the fact that, even for a single energy eigenstate, there is typically a close correspondence between the quantum position distribution and a classically defined probability distribution associated with the given energy [23][24][25][26]. However, this fact cannot fully explain our observations.…”

Section: Discussionmentioning

confidence: 80%

The Boltzmann distribution and the quantum-classical correspondence

Alterman

Choi

Durst

et al. 2018

J. Phys. A: Math. Theor.

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In this paper we explore the following question: can the probabilities constituting the quantum Boltzmann distribution, P B n ∝ e −En/kT , be derived from a requirement that the quantum configuration-space distribution for a system in thermal equilibrium be very similar to the corresponding classical distribution? It is certainly to be expected that the quantum distribution in configuration space will approach the classical distribution as the temperature approaches infinity, and a well-known equation derived from the Boltzmann distribution shows that this is generically the case. Here we ask whether one can reason in the opposite direction, that is, from quantumclassical agreement to the Boltzmann probabilities. For two of the simple examples we consider-a particle in a one-dimensional box and a simple harmonic oscillator-this approach leads to probability distributions that provably approach the Boltzmann probabilities at high temperature, in the sense that the Kullback-Leibler divergence between the distributions approaches zero. arXiv:1710.06051v5 [quant-ph]

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“…To the author's best knowledge, the first generalization of the WKB method to relativistic Hamiltonian was presented in ref. [26], where the WKB probability distributions for Hamiltonians involving arbitrary kinetic energy are derived. In particular the kinetic term √ m 2 + p 2 and its m → 0 limit were discussed in [26], in the context of the HO.…”

Section: Approximate Wkb Results and Exact Analytical Results For M →mentioning

confidence: 99%

“…[26], where the WKB probability distributions for Hamiltonians involving arbitrary kinetic energy are derived. In particular the kinetic term √ m 2 + p 2 and its m → 0 limit were discussed in [26], in the context of the HO.…”

Section: Approximate Wkb Results and Exact Analytical Results For M →mentioning

confidence: 99%

A Non-relativistic Approach to Relativistic Quantum Mechanics: The Case of the Harmonic Oscillator

et al. 2022

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A recently proposed approach to relativistic quantum mechanics (Grave de Peralta, Poveda, Poirier in Eur J Phys 42:055404, 2021) is applied to the problem of a particle in a quadratic potential. The methods, both exact and approximate, allow one to obtain eigenstate energy levels and wavefunctions, using conventional numerical eigensolvers applied to Schrödinger-like equations. Results are obtained over a nine-order-of-magnitude variation of system parameters, ranging from the non-relativistic to the ultrarelativistic limits. Various trends are analyzed and discussed—some of which might have been easily predicted, others which may be a bit more surprising.

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Quantum and classical probability distributions for arbitrary Hamiltonians

Cited by 11 publications

References 25 publications

Thermalization, Freeze-out, and Noise: Deciphering Experimental Quantum Annealers

Thermalization, Freeze-out, and Noise: Deciphering Experimental Quantum Annealers

The Boltzmann distribution and the quantum-classical correspondence

A Non-relativistic Approach to Relativistic Quantum Mechanics: The Case of the Harmonic Oscillator

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