Clustering of Conditional Mutual Information for Quantum Gibbs States above a Threshold Temperature

Kuwahara, Tomotaka; Kato, Kohtaro; Brandão, Fernando G. S. L.

doi:10.1103/physrevlett.124.220601

Cited by 56 publications

(50 citation statements)

References 90 publications

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“…Using the absence of zeros to prove the decay of qCMI is another interesting question. This would extend the result of [29] to lower temperatures down to the phase transition point. Since the approach of [29] resembles some of the techniques we use, this looks like a promising direction.…”

Section: Discussion and Open Questionsmentioning

confidence: 83%

“…This has been achieved for the classical Ising model [34,37] by relating the derivatives of the partition function to combinatorial objects that can be efficiently counted. Another approach for an improved running time is recently introduced in [29], where the authors apply a multivariate version of the cluster expansion that we use to compute the derivatives of log Z β (H ) efficiently.…”

Section: Discussion and Open Questionsmentioning

confidence: 99%

“…This result has been used to find algorithmic schemes for preparing the Gibbs state on a quantum computer [9] or estimating the free energy in 1D [27,30]. A recent result of [29] uses cluster expansions along with a technique very similar to the one we use in Theorem 10 (i.e. showing the low-order derivatives of the correlation function are zero) to establish the decay of conditional mutual information above some constant temperature.…”

Section: Previous Workmentioning

confidence: 99%

See 2 more Smart Citations

Classical algorithms, correlation decay, and complex zeros of partition functions of Quantum many-body systems

Harrow

Mehraban

Soleimanifar

2020

Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing

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We present a quasi-polynomial time classical algorithm that estimates the partition function of quantum many-body systems at temperatures above the thermal phase transition point. It is known that in the worst case, the same problem is NP-hard below this point. Together with our work, this shows that the transition in the phase of a quantum system is also accompanied by a transition in the hardness of approximation. We also show that in a system of n particles above the phase transition point, the correlation between two observables whose distance is at least Ω(log n) decays exponentially. We can improve the factor of log n to a constant when the Hamiltonian has commuting terms or is on a 1D chain. The key to our results is a characterization of the phase transition and the critical behavior of the system in terms of the complex zeros of the partition function. Our work extends a seminal work of Dobrushin and Shlosman on the equivalence between the decay of correlations and the analyticity of the free energy in classical spin models. On the algorithmic side, our result extends the scope of a recent approach due to Barvinok for solving classical counting problems to quantum many-body systems.

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Section: Discussion and Open Questionsmentioning

confidence: 83%

Section: Discussion and Open Questionsmentioning

confidence: 99%

Section: Previous Workmentioning

confidence: 99%

See 1 more Smart Citation

Classical algorithms, correlation decay, and complex zeros of partition functions of Quantum many-body systems

Harrow

Mehraban

Soleimanifar

2020

Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing

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“…Conditional Mutual Information (CMI) [34] represents the conditional dependence of two genes X and Y under the condition of the third gene Z, which can be expressed as…”

Section: Conditional Mutual Informationmentioning

confidence: 99%

Dynamic Bayesian Network Modeling Based on Structure Prediction for Gene Regulatory Network

Wang

et al. 2021

IEEE Access

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Gene regulatory network can intuitively reflect the interaction between genes, and an indepth study of these relationships plays a significant role in the treatment and prevention of clinical diseases. Therefore, correct reconstruction of gene regulatory network has become the first critical step in the study of disease treatment and prevention at the genetic level. Among the methods for gene regulatory network reconstruction, the Bayesian network model has been widely concerned because of its advantages of expressing both the regulatory relationship and the degree of strength between genes. Nevertheless, the complexity of the Bayesian network model in structure learning is extremely high, making the efficiency of the reconstruction network is low and the scale is limited. Therefore, this paper proposed a dynamic Bayesian network modeling based on structure prediction (DBN-SP). The method combines the correlation model with the dynamic Bayesian network model. On the premise of making full use of the search strategy of dynamic Bayesian network model structure learning, the candidate parent node set is selected based on the structure prediction method firstly. Based on this, in the process of structure learning, some redundant information can be removed and the search space can be reduced. After the network reconstructed, structure optimization by using the conditional mutual information method can further remove redundant edges and make the network more accurate. The experimental results show that DBN-SP greatly improves the efficiency and scale of the gene regulatory network reconstruction, and the accuracy and other indexes are also improved. DBN-SP is freely accessible at https://github.com/quluxuan/DBN-SP.git.

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“…scitation.org/journal/jmp methods have also been used to design classical algorithms for quantum models. 1,2,9,[15][16][17] The goal of this paper is to convey the simplicity and flexibility of the third method that results from using the cluster expansion formalism. We emphasize that ideas of this type have previously been used to establish similar algorithms: for |β| ≤ (10e 2 Δ) −1 with quasi-polynomial runtime 1 and for |β| ≤ (16e 3 Δ) −1 with polynomial runtime.…”

Section: Articlementioning

confidence: 99%

Efficient algorithms for approximating quantum partition functions

Mann

Helmuth

2021

Journal of Mathematical Physics

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We establish a polynomial-time approximation algorithm for partition functions of quantum spin models at high temperature. Our algorithm is based on the quantum cluster expansion of Netočný and Redig and the cluster expansion approach to designing algorithms due to Helmuth, Perkins, and Regts. Similar results have previously been obtained by related methods, and our main contribution is a simple and slightly sharper analysis for the case of pairwise interactions on bounded-degree graphs.

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Clustering of Conditional Mutual Information for Quantum Gibbs States above a Threshold Temperature

Cited by 56 publications

References 90 publications

Classical algorithms, correlation decay, and complex zeros of partition functions of Quantum many-body systems

Classical algorithms, correlation decay, and complex zeros of partition functions of Quantum many-body systems

Dynamic Bayesian Network Modeling Based on Structure Prediction for Gene Regulatory Network

Efficient algorithms for approximating quantum partition functions

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