TY - JOUR

T1 - Purifying Deep Boltzmann Machines for Thermal Quantum States

AU - Nomura, Yusuke

AU - Yoshioka, Nobuyuki

AU - Nori, Franco

N1 - Funding Information:
Y. N. is supported by Grant-in-Aids for Scientific Research (JSPS KAKENHI) (Grants No. 16H06345, No. 17K14336, No. 18H01158, and No. 20K14423) and MEXT as “Program for Promoting Researches on the Supercomputer Fugaku” (Basic Science for Emergence and Functionality in Quantum Matter —Innovative Strongly-Correlated Electron Science by Integration of “Fugaku” and Frontier Experiments—) (Projects ID: hp200132 and hp210163). N. Y. is supported by the Japan Science and Technology Agency (JST) (via the Q-LEAP program). F. N. is supported in part by: NTT Research, Army Research Office (ARO) (Grant No. W911NF-18-1-0358), Japan Science and Technology Agency (JST) (via the CREST Grant No. JPMJCR1676), JSPS (via the KAKENHI Grant No. JP20H00134 and the JSPS-RFBR Grant No. JPJSBP120194828), the Asian Office of Aerospace Research and Development (AOARD) (via Grant No. FA2386-20-1-4069), and the Foundational Questions Institute Fund (FQXi) via Grant No. FQXi-IAF19-06. Part of numerical calculations were performed using NetKet . Some calculations were performed using the computational resources of supercomputers in RIKEN (Fugaku and HOKUSAI) and the Institute of Solid State Physics at the University of Tokyo.
Funding Information:
Y. N. is supported by Grant-in-Aids for Scientific Research (JSPS KAKENHI) (Grants No. 16H06345, No. 17K14336, No. 18H01158, and No. 20K14423) and MEXT as "Program for Promoting Researches on the Supercomputer Fugaku" (Basic Science for Emergence and Functionality in Quantum Matter 'Innovative Strongly-Correlated Electron Science by Integration of "Fugaku" and Frontier Experiments') (Projects ID: hp200132 and hp210163). N. Y. is supported by the Japan Science and Technology Agency (JST) (via the Q-LEAP program). F. N. is supported in part by: NTT Research, Army Research Office (ARO) (Grant No. W911NF-18-1-0358), Japan Science and Technology Agency (JST) (via the CREST Grant No. JPMJCR1676), JSPS (via the KAKENHI Grant No. JP20H00134 and the JSPS-RFBR Grant No. JPJSBP120194828), the Asian Office of Aerospace Research and Development (AOARD) (via Grant No. FA2386-20-1-4069), and the Foundational Questions Institute Fund (FQXi) via Grant No. FQXi-IAF19-06. Part of numerical calculations were performed using NetKet [72]. Some calculations were performed using the computational resources of supercomputers in RIKEN (Fugaku and HOKUSAI) and the Institute of Solid State Physics at the University of Tokyo.
Publisher Copyright:
© 2021 American Physical Society.

PY - 2021/8/6

Y1 - 2021/8/6

N2 - We develop two cutting-edge approaches to construct deep neural networks representing the purified finite-temperature states of quantum many-body systems. Both methods commonly aim to represent the Gibbs state by a highly expressive neural-network wave function, exemplifying the idea of purification. The first method is an entirely deterministic approach to generate deep Boltzmann machines representing the purified Gibbs state exactly. This strongly assures the remarkable flexibility of the ansatz which can fully exploit the quantum-to-classical mapping. The second method employs stochastic sampling to optimize the network parameters such that the imaginary time evolution is well approximated within the expressibility of neural networks. Numerical demonstrations for transverse-field Ising models and Heisenberg models show that our methods are powerful enough to investigate the finite-temperature properties of strongly correlated quantum many-body systems, even when the problematic effect of frustration is present.

AB - We develop two cutting-edge approaches to construct deep neural networks representing the purified finite-temperature states of quantum many-body systems. Both methods commonly aim to represent the Gibbs state by a highly expressive neural-network wave function, exemplifying the idea of purification. The first method is an entirely deterministic approach to generate deep Boltzmann machines representing the purified Gibbs state exactly. This strongly assures the remarkable flexibility of the ansatz which can fully exploit the quantum-to-classical mapping. The second method employs stochastic sampling to optimize the network parameters such that the imaginary time evolution is well approximated within the expressibility of neural networks. Numerical demonstrations for transverse-field Ising models and Heisenberg models show that our methods are powerful enough to investigate the finite-temperature properties of strongly correlated quantum many-body systems, even when the problematic effect of frustration is present.

UR - http://www.scopus.com/inward/record.url?scp=85112367053&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85112367053&partnerID=8YFLogxK

U2 - 10.1103/PhysRevLett.127.060601

DO - 10.1103/PhysRevLett.127.060601

M3 - Article

C2 - 34420335

AN - SCOPUS:85112367053

VL - 127

JO - Physical Review Letters

JF - Physical Review Letters

SN - 0031-9007

IS - 6

M1 - 060601

ER -