Quantum dynamics simulation
towards Fugaku
The 2nd R-CCS International Symposium

Shigetoshi Sota
RIKEN Center for Computational Science,
Computational Materials Science Research Team1

Quantum many-body systems
 Applications of quantum many-body systems

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•
•
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High-Tc superconductors
Solar battery
Artificial photosynthesis
Spintronic
Optical devices•

Multiferroic materials• Quantum computer

Superconducting transition
temperature
https://www.sankei.com/west/news/171124/wst1711240006-n1.html

Solar battery by strongly
correlated materials
https://www.cems.riken.jp/jp/laboratory/scirg

Next-generation fundamental technologies

 Study of quantum many-body systems

Experiment：

Theory：collaboration

K computer
兵庫県神戸市2Numerical method
 Quantum many-body system(time independent) Schrödinger equation： Hˆ  n En  n(Eigenvalue equation)

N spin-½ system: degree of freedom is not N but 2N !!
up-spinor
down-spin

E.g. N=100  2N ≈ 1031

 Density matrix renormalization group (DMRG) method S. R. White, Phys. Rev. Lett. 69, 2863 (1992).
system:environment:“basis set:“basis set:target state:

Optimized to describe target state.
Singular value decompositionm singular vectors withlargest singular values:mNumerical error of ground state calculation
by DMRG
S. R. White, PRB 48, 10345(1993). 3

Practical calculations by DMRG
(computing resource）

ＰＣ(※Size of optimized Hilbert space: O(m2)）• Ground state calculations on one-dimensional (1-D) systems(DMRG truncation number：m < 500)

（100~1000GFLOPS）• Quantum dynamics on 1-D systems (DDMRG, tDMRG)
Clustermachine（few TFLOPS）

HPC(Univ. etc.)（~ 1PFLOPS）

K computer（10PFLOPS）

Fugaku（~ 1EFLOPS）• Finite temperature calculations on 1-D systems (FT-DMRG)(m < 1,000)• Quantum dynamics on 1-D systems including electron-phonon interaction• Ground state calculations on two-dimensional (2-D) systems• Quantum dynamics on 2-D systems(m < 20,000)• Finite temperature calculations on 2-D systems• Quantum dynamics on 2-D systems including electron-phonon interaction• 3-D systems (ground state, quantum dynamics)

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Developed Massively parallel DMRG
• Dynamical DMRG (SS, H. Matsueda, K. Morita, S. Yunoki, T. Tohyama)Target:
Quantum dynamics
https://www.r-ccs.riken.jp/labs/cms/DMRG/Dynamical_DMRG.html• 2D-DMRG (SS, T. Tohyama, S. Yunoki)Target:Two- (higher-) dimensional systems
https://www.r-ccs.riken.jp/labs/cms/DMRG/2D_DMRG.html• paraDMRG (SS, Y. Imamura, T. Tohyama, T. Nakajima, S. Yunoki)Target:Quantum chemistry (Ab initio DMRG)
https://www.r-ccs.riken.jp/labs/cms/DMRG/paraDMRG.html• QUARTZ (SS, S. Yunoki)Target:Quantum computing simulator
https://www.r-ccs.riken.jp/labs/cms/DMRG/QUARTZ.html(quantum annealing, quantum logic gate)5Performance of developed DMRGFLOPS[%]
100%

Time[s]
32,00016,000

80%

8,000
60%

4,00040%2,000
Strong Elapse
Weak Elapse
Strong FLOPS%
Weak FLOPS%

1,000

500
48192

768

3072
Process

12288

20%

0%

49152

7.8 PFLOPS on K computer
SS, S. Yunoki, T. Tohyama, A. Kuroda,Y. Kitazawa, K. Minami, and F. Shoji

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Quantum dynamics simulation towards Fugaku
 Optimization of our massively parallel DMRG programs for Fugaku

 Quantum dynamics simulation using DDMRG“Dynamical density matrix renormalization group study of spin and charge
T. Tohyama, M. Mori, S. Sota, PRB 97, 235137 (2018).
excitations in the four-leg t-t’-J ladder”
Hamiltonian:•Charge excitation N(q,w)•Spin excitation S(q,w)

doping rate

Fujita group(Tohoku Univ.)

doping ratehole
dopiong

electrondopinghole
dopeelectron
dope7

Real time simulation of quantum many-body systems• Newly developed time-dependent DMRG for higher dimensional systems

Time-dependent DMRG:
Dual optimized Hilbert spaces+(original)Time evolution:Kernel polynomial method• Real time simulation of quantum skyrmion
H  J  S i  S j (Heisenberg spin exchange)
i , j  DiJ  1, D  2  3, B  Bz  1

(S  S )  e  (S  S )  e ii  exxi

i eyy(Dzyaloshinskii–Moriya interaction) B   Sii(Magnetic field to z-direction)
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Quantum computing simulation using Fugaku
 Quantum computing simulation• Developed software: QUARTZ“quantum annealing” and “quantum logic gate”• Quantum annealing by D-Wave

D-Wave (quantum annealing)• Energy-280

http://dwavejapan.com/system/

G=0.10

256
DMRG

-300

-320-340-340-320-300-280

D-WaveHamiltonian is given by random number.
-1150

G=0.10

DMRG

-1200

Increasing thenumber of Qubit

2048

-1250

How to stabilize quantum
annealing?

-1300

-1350

-1400
-1400

-1350

-1300-1250

D-Wave-1200-1150

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Machine learning for quantum many-body systems• Entanglement spectrum• Neural network• Phase diagram

Future work:• Quantum dynamics (arXiv:1901.07900)• Appling to different systems• Data assimilation (arXiv:1908.09335)• Effective model
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Summary
 Past and current works•

Development of massively parallel DMRG programs Future works for Fugaku• Massively parallel DMRG for Fugaku“Quantum dynamics of quantum many-body system”• New method to study quantum many-body systems
 Machine learning Quantum computer“Quantum many-body systems”
 New numerical method
Next-genelationsynchrotron radiation facility(‘23~)

Spring-8
SACLA
Hyogo

Sendai

http://rsc.riken.jp/synchrotron/third.html

https://www.tohoku-epco.co.jp/seven/slit-j/

D-Wave 2000Q
http://dwavejapan.com/system/

IBM-Q
https://www.sbbit.jp/article/cont1/36729