Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:0706.4094 · Physical Review B

Quantum belief propagation: An algorithm for thermal quantum systems

M. B. Hastings

Published Nov 13, 2007• 102 citations
arXivHTMLPDF

Abstract

We present an accurate numerical algorithm, called quantum belief propagation, for simulation of one-dimensional quantum systems at nonzero temperature. The algorithm exploits the fact that quantum effects are short-range in these systems at nonzero temperature, decaying on a length scale inversely proportional to the temperature. We compare to exact results on a spin-$1∕2$ Heisenberg chain. Even a very modest calculation, requiring diagonalizing only ten-by-ten matrices, reproduces the peak susceptibility with a relative error of less than ${10}^{\ensuremath{-}5}$, while more elaborate calculations further reduce the error.

DOI
10.1103/physrevb.76.201102

Related papers

Coarse-grained belief propagation for simulation of interacting quantum systems at all temperatures

Ersen Bilgin, David Poulin

We continue our numerical study of quantum belief propagation initiated in [Phys. Rev. A 77, 052318 (2008)]. We demonstrate how the method can be expressed in terms of an effective thermal potential that materializes when the system presents quantum correlations, but is insensitive to classical correlations. The thermal potential provides an efficient means to assess the precision of belief propagation on graphs with no loops. We illustrate these concepts using the one-dimensional quantum Ising model and compare our results with exact solutions. We also use the method to study the transverse field quantum Ising spin glass for which we obtain a phase diagram that is largely in agreement with the one obtained in [Phys. Rev. B 78, 134424 (2008).] using a different approach. Finally, we introduce the coarse-grained belief propagation (CGBP) algorithm to improve belief propagation at low temperatures. This method combines the reliability of belief propagation at high temperatures with the ability of entanglement renormalization to efficiently describe low-energy subspaces of quantum systems with local interactions. With CGBP, thermodynamic properties of quantum systems can be calculated with a high degree of accuracy at all temperatures.

arXiv:0910.2299•Physical Review B•Feb 9, 2010•19 citations

Quantum Belief Propagation Algorithm versus Suzuki-Trotter approach in the one-dimensional Heisenberg chains

Farzad Ghafari, Mohammad Babazadeh, Davoud Salami +1 more

Quantum systems are the future candidates for computers and information processing devices. Information about quantum states and processes may be incomplete and scattered in these systems. We use a quantum version of Belief Propagation(BP) Algorithm to integrate the distributed information. In this algorithm the distributed information, which is in the form of density matrix, can be approximated to local structures. The validity of this algorithm is measured in comparison with Suzuki-Trotter(ST) method, using simulated information. ST in 3-body Heisenberg example gives a more accurate answer, however Quantum Belief Propagation (QBP) runs faster based on complexity. In order to develop it in the future, we should be looking for ways to increase the accuracy of QBP.

arXiv:1409.2048•Sep 6, 2014•0 citations

Belief propagation algorithm for computing correlation functions in finite-temperature quantum many-body systems on loopy graphs

David Poulin, Ersen Bilgin

Belief propagation---a powerful heuristic method to solve inference problems involving a large number of random variables---was recently generalized to quantum theory. Like its classical counterpart, this algorithm is exact on trees when the appropriate independence conditions are met and is expected to provide reliable approximations when operated on loopy graphs. In this paper, we benchmark the performances of loopy quantum belief propagation (QBP) in the context of finite-temperature quantum many-body physics. Our results indicate that QBP provides reliable estimates of the high-temperature correlation function when the typical loop size in the graph is large. As such, it is suitable, e.g., for the study of quantum spin glasses on Bethe lattices and the decoding of sparse quantum error correction codes.

arXiv:0710.4304•Physical Review A•May 16, 2008•24 citations

Computation of ESR spectra from the time evolution of the magnetization: Comparison of autocorrelation and Wiener-Khinchin-relation-based methods

Hiroki Ikeuchi, Hans A. De Raedt, Sylvain Bertaina +1 more

The calculation of finite temperature electron spin resonance (ESR) spectra for concrete specified crystal configurations is a very important issue in the study of quantum spin systems. Although direct evaluation of the Kubo formula by means of numerical diagonalization yields exact results, memory and CPU time restrictions limit the applicability of this approach to small system sizes. Methods based on the time evolution of a single pure quantum state can be used to study larger systems. One such method exploits the property that the expectation value of the autocorrelation function obtained for a few samples of so-called thermal typical states yields a good estimate of the thermal equilibrium value. In this paper, we propose a new method based on a Wiener-Khinchin-like theorem for quantum system. By comparison with exact diagonalization results, it is shown that both methods yield correct results. As the Wiener-Khinchin-based method involves sampling over thermal typical states, we study the statistical properties of the sampling distribution. Effects due to finite observation time are investigated and found to differ for the two methods but it is also found that, for both methods, the effects vanish as the system size increases. We present ESR spectra of the one-dimensional XXZ Heisenberg chain of up to 28 spins and discuss the dependence of separation of double peaks on the chain length.

arXiv:1503.06111•Physical Review B•Dec 21, 2015•7 citations

Non-Markov quantum belief propagation

Jack Ceroni, Ian MacCormack, Guillaume Verdon

We provide a rigorous proof of the approximate convergence of sliding-window quantum belief-propagation as outlined heuristically in the work of Bilgin and Poulin [Phys. Rev. B 81, 054106 (2010)], in the absence of the quantum Markov property. In particular, we confirm the hypothesis outlined in this work that the approximation error of each step in the belief-propagation algorithm decreases exponentially with the sliding-window size, under the assumption that the underlying state on which belief-propagation is being performed possesses a so-called thermal boundedness property: a relaxation of the Markov property required for exact convergence.

arXiv:2406.06360•Physical Review A•Jul 12, 2024•0 citations

Quantum decimation for spin-(1/2) chains in a magnetic field

Chen Xi-Yao, G. F. Tuthill

We have extended the range of application of the decimation renormalization-group (RG) method, using a combination of analytic and numerical techniques in calculating thermal and magnetic properties of a wide variety of one-dimensional quantum spin-(1/2) systems. Efforts to improve the accuracy of the approach include increasing the spatial rescaling factor and investigating the effect of free versus periodic boundary conditions for each renormalization cluster. We have used the RG method to calculate the specific heat and susceptibility for chains with uniaxial exchange anisotropy (ranging from the Ising to the XY limits) in both longitudinal and transverse magnetic fields, as well as for the Heisenberg antiferromagnet with alternating couplings, and have compared our results, when possible, with either exact solutions or with numerical extrapolations from finite-chain calculations. Decimation is a high-temperature approximation, but its range of reliability is found in some cases (e.g., transverse susceptibility for the Ising chain, or susceptibility for the Heisenberg ferromagnet in a field) to extend well into the strong-coupling regime. We note that the decimation results for magnetic properties are, in general, more accurate than those for the thermal response.

Physical review. B, Condensed matter•Dec 1, 1985•3 citations
We present an accurate numerical algorithm, called quantum belief propagation, for simulation of one-dimensional quantum systems at nonzero temperature. The algorithm exploits the fact that quantum effects are short-range in these systems at nonzero temperature, decaying on a length scale inversely proportional to the temperature. We compare to exact results on a spin-1∕21∕21∕2 Heisenberg chain. Even a very modest calculation, requiring diagonalizing only ten-by-ten matrices, reproduces the peak susceptibility with a relative error of less than 10\ensuremath−5{10}^{\ensuremath{-}5}10\ensuremath−5, while more elaborate calculations further reduce the error.
We continue our numerical study of quantum belief propagation initiated in [Phys. Rev. A 77, 052318 (2008)]. We demonstrate how the method can be expressed in terms of an effective thermal potential that materializes when the system presents quantum correlations, but is insensitive to classical correlations. The thermal potential provides an efficient means to assess the precision of belief propagation on graphs with no loops. We illustrate these concepts using the one-dimensional quantum Ising model and compare our results with exact solutions. We also use the method to study the transverse field quantum Ising spin glass for which we obtain a phase diagram that is largely in agreement with the one obtained in [Phys. Rev. B 78, 134424 (2008).] using a different approach. Finally, we introduce the coarse-grained belief propagation (CGBP) algorithm to improve belief propagation at low temperatures. This method combines the reliability of belief propagation at high temperatures with the ability of entanglement renormalization to efficiently describe low-energy subspaces of quantum systems with local interactions. With CGBP, thermodynamic properties of quantum systems can be calculated with a high degree of accuracy at all temperatures.
Quantum systems are the future candidates for computers and information processing devices. Information about quantum states and processes may be incomplete and scattered in these systems. We use a quantum version of Belief Propagation(BP) Algorithm to integrate the distributed information. In this algorithm the distributed information, which is in the form of density matrix, can be approximated to local structures. The validity of this algorithm is measured in comparison with Suzuki-Trotter(ST) method, using simulated information. ST in 3-body Heisenberg example gives a more accurate answer, however Quantum Belief Propagation (QBP) runs faster based on complexity. In order to develop it in the future, we should be looking for ways to increase the accuracy of QBP.
Belief propagation---a powerful heuristic method to solve inference problems involving a large number of random variables---was recently generalized to quantum theory. Like its classical counterpart, this algorithm is exact on trees when the appropriate independence conditions are met and is expected to provide reliable approximations when operated on loopy graphs. In this paper, we benchmark the performances of loopy quantum belief propagation (QBP) in the context of finite-temperature quantum many-body physics. Our results indicate that QBP provides reliable estimates of the high-temperature correlation function when the typical loop size in the graph is large. As such, it is suitable, e.g., for the study of quantum spin glasses on Bethe lattices and the decoding of sparse quantum error correction codes.
The calculation of finite temperature electron spin resonance (ESR) spectra for concrete specified crystal configurations is a very important issue in the study of quantum spin systems. Although direct evaluation of the Kubo formula by means of numerical diagonalization yields exact results, memory and CPU time restrictions limit the applicability of this approach to small system sizes. Methods based on the time evolution of a single pure quantum state can be used to study larger systems. One such method exploits the property that the expectation value of the autocorrelation function obtained for a few samples of so-called thermal typical states yields a good estimate of the thermal equilibrium value. In this paper, we propose a new method based on a Wiener-Khinchin-like theorem for quantum system. By comparison with exact diagonalization results, it is shown that both methods yield correct results. As the Wiener-Khinchin-based method involves sampling over thermal typical states, we study the statistical properties of the sampling distribution. Effects due to finite observation time are investigated and found to differ for the two methods but it is also found that, for both methods, the effects vanish as the system size increases. We present ESR spectra of the one-dimensional XXZ Heisenberg chain of up to 28 spins and discuss the dependence of separation of double peaks on the chain length.
We provide a rigorous proof of the approximate convergence of sliding-window quantum belief-propagation as outlined heuristically in the work of Bilgin and Poulin [Phys. Rev. B 81, 054106 (2010)], in the absence of the quantum Markov property. In particular, we confirm the hypothesis outlined in this work that the approximation error of each step in the belief-propagation algorithm decreases exponentially with the sliding-window size, under the assumption that the underlying state on which belief-propagation is being performed possesses a so-called thermal boundedness property: a relaxation of the Markov property required for exact convergence.
We have extended the range of application of the decimation renormalization-group (RG) method, using a combination of analytic and numerical techniques in calculating thermal and magnetic properties of a wide variety of one-dimensional quantum spin-(1/2) systems. Efforts to improve the accuracy of the approach include increasing the spatial rescaling factor and investigating the effect of free versus periodic boundary conditions for each renormalization cluster. We have used the RG method to calculate the specific heat and susceptibility for chains with uniaxial exchange anisotropy (ranging from the Ising to the XY limits) in both longitudinal and transverse magnetic fields, as well as for the Heisenberg antiferromagnet with alternating couplings, and have compared our results, when possible, with either exact solutions or with numerical extrapolations from finite-chain calculations. Decimation is a high-temperature approximation, but its range of reliability is found in some cases (e.g., transverse susceptibility for the Ising chain, or susceptibility for the Heisenberg ferromagnet in a field) to extend well into the strong-coupling regime. We note that the decimation results for magnetic properties are, in general, more accurate than those for the thermal response.