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arXiv:1008.4162 · Physical Review Letters

Preparing Thermal States of Quantum Systems by Dimension Reduction

Ersen Bilgin, Sergio Boixo

Published Oct 22, 2010• 65 citations
arXivHTMLPDF

Abstract

We present an algorithm that prepares thermal Gibbs states of one dimensional quantum systems on a quantum computer without any memory overhead, and in a time significantly shorter than other known alternatives. Specifically, the time complexity is dominated by the quantity N(‖h‖/T), where N is the size of the system, ‖h‖ is a bound on the operator norm of the local terms of the Hamiltonian (coupling energy), and T is the temperature. Given other results on the complexity of thermalization, this overall scaling is likely optimal. For higher dimensions, our algorithm lowers the known scaling of the time complexity with the dimension of the system by one.

DOI
10.1103/physrevlett.105.170405

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We present an algorithm that prepares thermal Gibbs states of one dimensional quantum systems on a quantum computer without any memory overhead, and in a time significantly shorter than other known alternatives. Specifically, the time complexity is dominated by the quantity N(‖h‖/T), where N is the size of the system, ‖h‖ is a bound on the operator norm of the local terms of the Hamiltonian (coupling energy), and T is the temperature. Given other results on the complexity of thermalization, this overall scaling is likely optimal. For higher dimensions, our algorithm lowers the known scaling of the time complexity with the dimension of the system by one.
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In a recent work [10], POULIN and one of us presented a quantum algorithm for preparing thermal Gibbs states of interacting quantum systems. This algorithm is based on Grover’s technique for quantum state engineering, and its running time is dominated by the factor (√D/𝒵β), where D and 𝒵βdenote the dimension of the quantum system and its partition function at inverse temperature β, respectively. We present here a modified algorithm and a more detailed analysis of the errors that arise due to imperfect simulation of Hamiltonian time evolutions and limited performance of phase estimation (finite accuracy and nonzero probability of failure). This modification together with the tighter analysis allows us to prove a better running time by the effect of these sources of error on the overall complexity. We think that the ideas underlying of our new analysis could also be used to prove a better performance of quantum Metropolis sampling by TEMME et al. [12].
While recent advances have established efficient quantum algorithms for preparing Gibbs states of finite-dimensional systems, comparable complexity results for bosonic and other infinite-dimensional models remain unexplored. We introduce the first general rigorous Gibbs sampling framework for bosonic many-body systems, showing that physically relevant bosonic models admit gapped dissipative generators, enabling efficient preparation of thermal states, provided that the spectral gap scales favourably with the number of modes. Although our results hold for broad classes of models, we illustrate them using Bose-Hubbard Hamiltonians, both within and beyond the mean-field regime. In both cases, we show that the associated dissipative generators maintain a positive spectral gap, thereby implying exponential convergence to the thermal state. For the Gibbs sampler corresponding to the full Bose-Hubbard model, we obtain a spectral-gap lower bound Ce−cnCe^{-cn}Ce−cn, where nnn is the number of modes. We apply our results to provide a Gibbs-state preparation algorithm on qubit hardware, with runtime polynomial in the number of modes and the inverse spectral gap, and thereby obtain a quantum algorithm to compute thermal properties of the model. This provides the first mathematically controlled route to Gibbs sampling in infinite-dimensional systems, with implications for quantum simulation, thermalization, and many-body complexity, where quantum advantages may arise.
Preparing thermal equilibrium states is an essential task for finite-temperature quantum simulations. In statistical mechanics, microstates in thermal equilibrium can be obtained from statistical ensembles. To date, numerous ensembles have been devised, ranging from Gibbs ensembles such as the canonical and microcanonical ensembles to a variety of generalized ensembles. Since these ensembles yield equivalent thermodynamic predictions, one can freely choose an ensemble for computational convenience. In this paper, we exploit this flexibility to develop an efficient quantum algorithm for preparing thermal equilibrium states. We first present a quantum algorithm for implementing generalized ensembles within the framework of quantum singular value transformation. We then perform a detailed analysis of the computational cost and elucidate its dependence on the choice of the ensemble. Our analysis shows that employing an appropriate ensemble can significantly mitigate ensemble-dependent overhead and yield improved scaling of the computational cost with system size compared to existing methods based on the canonical ensemble. We also numerically demonstrate that our approach achieves a significant reduction in the computational cost even for small finite-size systems. Our algorithm applies to arbitrary thermodynamic systems at any temperature and is thus expected to offer a practical and versatile method for computing finite-temperature properties of quantum many-body systems. These results highlight the potential of ensemble design as a powerful tool for enhancing the efficiency of a broad class of quantum algorithms.
In this work, we show how Gibbs or thermal states appear dynamically in closed quantum many-body systems, building on the program of dynamical typicality. We introduce a novel perturbation theorem for physically relevant weak system-bath couplings that is applicable even in the thermodynamic limit. We identify conditions under which thermalization happens and discuss the underlying physics. Based on these results, we also present a fully general quantum algorithm for preparing Gibbs states on a quantum computer with a certified runtime and error bound. This complements quantum Metropolis algorithms, which are expected to be efficient but have no known runtime estimates and only work for local Hamiltonians.