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記
日時：２０１９年０６月０５日（水）１３時３０分〜 Wed, Jun 5, 1:30pm
場所：東京大学 生産技術研究所 千葉実験所 研究実験棟Ｉn210号室
道程：http://hatano-lab.iis.u-tokyo.ac.jp/access.html
講師：桑原知剛さん（理研）Dr. Tomotaka Kuwahara (RIKEN)
演題：Approximate quantum Markov network at finite temperatures
要旨：
In recent years, the Gibbs sampling on quantum computer attracts more and more attentions due to the application to exponential quantum speed up of the semidefinite programming problem [1] and the appearance of machine learning using a quantum Boltzmann machine [2]. Here, the quantum Gibbs states are described by e^{-βH}/Z (β: inverse temperature) for the system Hamiltonian H. As methods of quantum Gibbs sampling, Quantum metropolis sampling algorithm [3] and Davies Gibbs sampling algorithm [4] have been well-known. These algorithms heuristically works well, but the precision analyses are generally extremely difficult and the convergence is often exponentially slower with respect to the system size (eg, spin glass system). Our motivation in this research is to clarify under what conditions the quantum Gibbs sampling is implemented efficiently.
For the purpose, we will first introduce a method to utilize the quantum Markov property. When the system is decomposed into A, B, and C subsystems, we call that a quantum state is approximately Markov if the conditional mutual information I(A,C|B) between A and C via B exponentially decays with respect to the distance between A and C. If the Gibbs state is given by the approximate Markov network, we know that the quantum sampling can be efficiently implemented by a small depth local quantum circuits [5].
In this talk, I will show that such a quantum Markov property always hold for quantum Gibbs states above a certain threshold temperature. In addition to the efficient quantum Gibbs sampling, I will also explain several implications of the quantum Markov property: the strong versions of the area law, the clustering theorem, and existence of the topological entanglement entropy. This is a joint work with Kohtaro Kato and Fernando Brand?o in Caltech IQIM.
参考文献
[1] F. G. S. L. Brand?o and K. M. Svore, IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS), pp. 415 (2017).
[2] M. H. Amin, et al., Phys. Rev. X 8, 021050 (2018).
[3] K. Temme, T. J. Osborne, K. G. Vollbrecht, D. Poulin, and F. Verstraete, Nature 471, 87 (2011).
[4] M. J. Kastoryano and F. G. S. L. Brand?o, Commun. Math. Phys., 344, 915 (2016).
[5] F. G. S. L. Brand?o and M. J. Kastoryano, Commun. Math. Phys., 365, 1 (2019).
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李 宰河
〒277-8574 千葉県柏市柏の葉5-1-5
東京大学生産技術研究所
e-mail: lee@iis.u-tokyo.ac.jp
Tel: 04-7136-6977
Fax: 04-7136-6978
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