DLAP online Seminar

ニューラルネットワークで探る
量子多体系の表現    

吉岡信行（理研）
Zoom Webinar, 2020.07.09

自己紹介
吉岡

信行 (Nobuyuki Yoshioka)
2015.03 東京大学理学部物理学科

共同研究者のみなさま卒2017.03 東大理物

修士課程修了 (桂研)2020.03 東大理物

博士課程修了 (桂研)博論タイトル「ニューラルネットワークによる物理状態の判定と表現」
現在

Dr. Ryusuke
Hamazaki

Dr. Franco Nori

Dr. Ravindra
Chhajlany

Dr. Clemens
Gneiting

理化学研究所 開拓研究本部 Nori理論量子物理研究室

主な研究内容：量子物理・情報科学の境界から生まれる高速アルゴリズムの探索
・ニューラルネットワークによる量子多体状態の効率的表現

Today

・NIQSアルゴリズム開発
・非平衡量子ダイナミクス
e.g. 物理量のError bounde.g.オンサーガー代数によるPerfect Scarの構築

Prof. Wataru
Mizukami2▸Introduction to Neural Quantum States
Restricted Boltzmann Machine as a quantum state
Variational calculation
Relationship with tensor networks▸Application to open quantum system
Steady state as “ground state” of LindbladianResults by RBM ansatz

3Machine
Learning

Simulation of“Probability Distribution”

Quantum
ComputingMany-body
Problems

Simulation of “Probability distribution”

Classical data
Probability distribution
Single computational basis

Quantum data
Wave function
Unitary/non-unitary mapping

Data-driven learning

Model-driven learning

Obtained via fitting into measurements
No access to “ground truth”

Obtained via optimization ofmodel-based well-defined cost function- Machine learning
e.g. image recognition- Mathematical programminge.g. combinatorial optimization

e.g. data science- Recommendation for Monte Carlo- Quantum tomography- Reproduce quantum state,
Hamiltonian, Liouvilllian etc.
- Error mitigation- Variational simulation- Costless simulation of
eigenvectors, density mat.
- Tensor Nets, Neural Nets,
Quantum circuits, etc.
5Data-driven learning
Optimize the model pλ s.t. “distance” between target P is minimized.
e.g. Kullback-Leibler divergence

DKL (P ||p ) =X

P (x) ln

x

✓

P (x)
p (x)

◆

Target data P(x)Approximant pλ(x)Fit“Measurement” data

POVM Result

Parametrized models

http://yann.lecun.com/exdb/mnist/

No access tounderlying“ground truth”“Actual” target

U
Hand-writing

Quantum state

Further ML task

Physical property

Pretrained model
Generator(e.g. GAN)

Entanglement
Correlator
Fidelity{{

6

Variational simulation
Optimize the ansatz Ψθ w.r.t target depending on what you want to doe.g. Hamiltoniane.g. Product of Liouvillian

Evmc = argmin
✓

Model

H=

X

h

h

✓ |H|
✓|

✓i

✓i

Minimizeenergy†hh⇢✓ |L̂ L̂|⇢✓ ii
0 = argminhh⇢✓ |⇢✓ ii✓
GS approximant ΨθParametrized model{

Steady state ρθ

Non-equilibrium
propertyh q Pqq“Ground truth” accessibleby (super)-exponential cost

Static property

Minimize“energy”

Symmetry breaking

Ordering

Appropriate ansatz, “cost function”, optimization needed7Simulation/Estimation of “Probability distribution”

Classical data
Probability distribution
Single computational basis

Quantum data
Wave function
Unitary/non-unitary mapping

Data-driven

Model-driven

Obtained via fitting into measurements
No access to “ground truth”

Obtained via optimization ofmodel-based well-defined cost function- Machine learning
e.g. image recognition- Mathematical programminge.g. combinatorial optimization

e.g. data science- Recommendation for Monte Carlo- Quantum tomography- Reproduce quantum state,
Hamiltonian, Liouvilllian etc.
- Error mitigation- Variational simulation- Costless simulation of
eigenvectors, density mat.
- Tensor Nets, Neural Nets,
Quantum circuits, etc.
8

Neural Networks as variational ansatz

Carleo&Troyer (’17),

e.g. Restricted Boltzmann Machine
Dimension-free variational ansatz inspired by machine learning

| i=

X

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X
h

e

Wij

Optimize a,b,W that minimize the GS energy,h |H| i
EGS = argmin= argminh | iai ! aii hj +ai

h1

i +bj hj

Wij : coupling

Tracing out aux. space

1

2

Physical spins
N 2 {±1}

Restricted Boltzmann Machine

GS energy optimization of 1d AF Heisenberg (L=22)

Optimization by variational Monte Carloe.g. stochastic gradient descent,Mag. fields

Interaction

Hidden spinshM 2 {±1}

P

,

0

⇤

P

( )H , 0 ( )| ( )|2⌘@aihHihHi
Taken from NetKet website

9

RBM as ground state ansatz
e.g. Restricted Boltzmann Machine
Dimension-free variational ansatz inspired by machine learning

| i=

X

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X
hInteractione

WijMag. fields

i hj +ai

h1

i +bj hj

Tracing out aux. space

Hidden spinshM 2 {±1}
Wij : coupling

1

2

Physical spins
N 2 {±1}

Restricted Boltzmann MachineComparison with other ansatz Carleo&Troyer Science 355(’17)
State-of-art GS energy accuracy achieved by variational imaginary-time evolution
1d Traverse-field Ising model
80 spins, periodic boundary,h : field, alpha: (# of hidden neuron)

1d AF Heisenberg model
80 spins, periodic boundary

2d AF Heisenberg model
10x10 spins, periodic boundary

10

About RBM-type ansatz
Entanglement property Deng etal. PRX (’17)

Hidden spinse.g.) Maximized half-chain Renyi entropy for 2n spinswith 𝒪(n) parameters
i⇡ i⇡
4 4

A
S↵ = n log 2, 8↵

i⇡
4

i⇡
4

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A
S↵ ⌘

1

1

↵

↵
log[Tr(⇢A )]Physical spinse.g.) Random RBM states
von Neumann Entanglement scaling- EE(RBM) < Page value (average EE of Haar random)
add
hidden spinsNot all random states efficiently captured- In fact you need exponentially many parameters to
fit Haar random states- Level spacing of entanglement ham obeys Poisson(expected to correspond to those of integrable system)

11

Other types of neural quantum stateFeed-forward neural nets

non-linear( ) / Nn L n · · · N 2 L 2 N 1 L 1 ( )Intermediate output given as

1
2

linear( )yn = Nn (Ln (yn 1 ))e.g. Fully-connected Saito (’17), Cai&Liu (’18), Choo et al. (’17){

Linear:zj = (L())j =X

Wij i + bji

Non-linear:yj = N (zj ) = tanh(zj )Input 1st hidden 2nd

3rd

OutputMomentum-resolved energy
1dAFH model (L=36 sites) Choo et al., PRL (’17)- Better performance in excited states for 1d AFH- Entanglement spreading is slow,
not so powerful in d>1 systems
12Other types of neural quantum stateFeed-forward neural nets

non-linear( ) / Nn L n · · · N 2 L 2 N 1 L 1 ( )lineare.g. Convolutional neural nets Choo et al., (’19), Szabo&Castelnovo (’20)- Local receptive fields acting on finite regions- MERA-like structure with higher eﬃciency forvolume-law entanglement

Levine et al., PRL (‘19)

RBM: O(N) parameters in 2d
CNN: O( N) parameters in 2d

Convolution

Non-linear

Ground-state energy
J1J2 model (J2/J1 = 0.5, 10x10) Ferrari et al., (’19)
CNN w/o S2=0
RBM w/ S2=0

- Performance not the best without implementing symmetry(e.g. 2D J1-J2 Heisenberg model)
Ferrari et al., (’19) Nomura&Imada (’20)

13Other types of neural quantum state
Combination with conventional technique Nomura et al., (2017)

( )=

RBM (

) PP ( )Pair productoriginally introduced to study!
X system Y †fermonic| PP i =PP ()

Based on two-body interaction between visible spins:| PP i =XY †ci |0iPP ( )

0

i

X

!

0

= PG @

1Nsite /2

X
i,j

i

ci |0i

1Nsite /2"# † † A
fi,j ci" cj#|0iprohibits double occupancy"# † † A
f
c
c
|0i
i,j
i"
j#
Result for J1J2 Heisenberg model Nomura & Imada (2020)
i,j= PG @

14

Relation with other variational ansatz
Tensor networks- Class of variational ansatz based on tensor contraction

| i=

X

Cont

O

T↵

↵

!

| i

Many-body Hilbert space
MPS
const bond-dim

1d area-law
states- Advantageous especially for area-law entangled statese.g. Matrix Product States| i = Tr

Y

[ i]
Ai

i

Fannes et al. (’92), White (’92),!

| i- Killer app in gapped 1d when optimized via DMRGMPS ⇆ RBM is hard Chen et al. (’18),
Non-linear equation
generally not solvable- Not scalable for generic d>2 systems

A1 A2
1

2

· · · AN···N

Exponential bond-dim neededin general

15

Relation with other variational ansatz- Product of MPSs in subset of system

| i=

Y

Tr

S

Y

[ i]
Ai,S

i2S

!

(

| i

cf. MPS| i = Tr

Glasser etal. PRX (’18)

Y
i

[ i]
Ai

!

| i

(

RBM as subclass of ”tensor network”: String Bond States

RBM, defined as follows, can also be written in the above form

0
1
X
X
X
( )/
exp @
Wij i hj +b j hj A=

hj =±1
Y P

(e

ij

i Wij

i +bj

j

+e

j

P

i Wij

i

bj

)

- Very generic, highly-dependenton “string" choices.
✓ P

Y

i Wij

i +bj

◆

e
0
P
Equivalentbond dimension= to single
Tr MPS with exponential W
ij i bj
i
0
e
j
!
Y
Y [ ]
i
=
Tr
A
i,S
- DMRG not available but allows
Monte Carlo sampling Schuch etal. PRL (’08)
S

i2S

16

Variational simulation by neural networks
Algorithm

AnsatzGround state (condensed matter)
Carleo&Troyer Science (’17) Cai&Liu, PRB (’18)
Nomura etal. PRB (’17)
Glasser etal. PRX (‘18)

Ground state (quantum chemistry)

Convolutional neural nets
Choo etal. PRB (’19)

Szabo&Castelnovo (’20)

Autoregressive models

Choo et al. (’20)

Pfau et al. (’19)

Levine etal. PRL (’20) Davis&Wang (’20)

Hermann et al. (’19)

Yoshioka, Mizukami, Nori, in prep.

Hibat-Allah et al. (’20)

Tomorrow 15:30 @FQCS2020

Real Time evolution
Carleo&Troyer (’17)
Schmitt&Heyl (’19)

Czischek etal. PRB (’17)
Lopez-Gutierrez&Mendl (’19)

Volume-law entanglement in NQSDeng et al. PRX (’17) Levine etal. PRL (’19)

Steady State in open system
Yoshioka&Hamazaki PRB (’19) Nagy&Savona PRL (’19)
Hartmann&Carleo PRL (’19)

Vincenti etal. PRL (’19)

Disclaimer: Table not exclusive17▸Introduction to Neural Quantum States
Restricted Boltzmann Machine as a quantum state
Variational calculation
Relationship with tensor networks▸Application to open quantum system
Steady state as “ground state” of LindbladianResults by RBM ansatz

18

Open quantum system

Yoshioka&Hamazaki, Phys. Rev. B 99, 214306 (2019).

System w/ dissipation

Environment

Microscopic & coherent

Macroscopic, heat-bath likee.g.) Superconducting qubitse.g.) Cold atom, Trapped ionsinteractTraceoute.g.) Measurement equipmente.g.) Laser shining

ions

Taken from Garcia-Repoll, J.Phys B (’05)

19Setup: Time evolution in open quantum system
Unitary
evolution

SystemEnvironment

⇢
ˆ
=
⇢
ˆ
⌦
⇢
ˆ
tot
B⌦ ⇢
⇢ˆtot = ⇢ˆ⇢ˆ⌦
⇢
ˆ
=
⇢
ˆ
ˆB
tot B

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Environment<latexit sha1_base64="QvYso0FEGBhCoc8adS9KnCsbV/k=">AAACqXichVHLSuRAFD1mdEZbZ2xnNoKbxkYRhKaiA4owIOrCpa22NhppkkxpFyapkFQ3OCE/MD/gwpWCiPoZbly6cdH+gbhUcOPC2+mA+L4hVafOvefWqSrLd0SoGGu0aV/aO75+6+zKdPd8/9Gb7fu5EspaYPOSLR0ZlC0z5I7weEkJ5fCyH3DTtRy+am3PNvOrdR6EQnrLasfnG6655YlNYZuKqEp2zqiaKjKCqowrkeGaqhq4kZIqjv88ZXKGVMLlYe6t4pk4rmTzrMCSyL0GegrySGNBZo9g4C8kbNTggsODIuzAREjfOnQw+MRtICIuICSSPEeMDGlrVMWpwiR2m8YtWq2nrEfrZs8wUdu0i0N/QMochtglO2a37Jydsmv28G6vKOnR9LJDs9XScr/S+79/6f5TlUuzQvVJ9aFnhU1MJl4FefcTpnkKu6Wv/9u9XZpaHIqG2QG7If/7rMHO6ARe/c4+LPLFPWToAfSX1/0arIwV9PGCXvydn55Jn6ITAxjECN33BKYxjwWUaN8TXKCBK21UK2plba1VqrWlml94Fpr9CBLkp2A=</latexit>⇢ˆtot = ⇢ˆ ⌦ ⇢ˆB<latexit sha1_base64="QvYso0FEGBhCoc8adS9KnCsbV/k=">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</latexit>

Trace out
env.
SystemEnvironment

⇢
ˆ
=
⇢
ˆ
⌦
⇢
ˆ
tot
B⌦ ⇢
⇢ˆtot = ⇢ˆ⇢ˆ⌦
⇢
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=
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ˆB
tot B
<latexit sha1_base64="QvYso0FEGBhCoc8adS9KnCsbV/k=">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</latexit><latexit sha1_base64="QvYso0FEGBhCoc8adS9KnCsbV/k=">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</latexit><latexit sha1_base64="QvYso0FEGBhCoc8adS9KnCsbV/k=">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</latexit>⇢ˆtot = ⇢ˆ ⌦ ⇢ˆB<latexit sha1_base64="QvYso0FEGBhCoc8adS9KnCsbV/k=">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</latexit>

Non-unitaryevolution⇣0†⇢ˆtot := Û ⇢ˆtot Ûunitary<latexit sha1_base64="R/myDq7oKOcQYpjaRxouOsMUvSM=">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</latexit>

Trace out
env.
System00
Environment⇢ˆ := trB [ˆ⇢tot ]<latexit sha1_base64="XT9pTPN8WqUevyEm8eLK5zRNI7Q=">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</latexit>

0=

X

†
0
⇢
ˆ
M̂
s
⇢ˆ := trM̂
[ˆ
⇢
s ]
B tot
<latexit sha1_base64="XT9pTPN8WqUevyEm8eLK5zRNI7Q=">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</latexit><latexit sha1_base64="i+GCIYlFeslLLJxHzZb+80tLawU=">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</latexit>s⇣X<latexit sha1_base64="XqDtschhK1vaxeP/kWoYi5S/ft8=">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</latexit>0Tr[ˆ⇢] = Tr[ˆ⇢]=1
Non-unitary but
Trace-preserving map⌘†
Û Û = Iˆ⌘†M̂s M̂s = Iˆs Kraus

operator

20

GKSL formalism and stationary state

Yoshioka&Hamazaki, Phys. Rev. B 99, 214306 (2019).

Impose quantum master equation to satisfy{

1. Completely positive and trace preserving (CPTP)
2. Time-locality (Markovianity, short correlation time in env.)

Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) equation Lindblad (1976) Gorini, Kossakowski&Sudarshan (1976)dˆ⇢(t)= L(t)ˆ
⇢(t) :=
dt Liouvilliani[Ĥ(t), ⇢ˆ(t)] + D(t)[ˆ⇢(t)]

Unitary dynamics

D(t)[ˆ
⇢(t)] =

Non-unitary,
but trace-preserving

If Ĥ(t) = Ĥ and D(t) = D , i.e., time-homogeneous,at least one non-equilibrium stationary state assuredXˆ s (t)ˆ
⇢(t) ˆ † (t)sso
1 nˆ† ˆˆ(t)s (t) s (t), ⇢2e.g. Two-lev. system with damping|1i<latexit sha1_base64="qx2VKsbQp/EdXkzrSrwtIgI6lJw=">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</latexit>dˆ⇢SS
Solution for=0dt<latexit sha1_base64="7G4p5vgMnCoR+uXmxPOhfVRYIPM=">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</latexit>|0i<latexit sha1_base64="fwcvpo5AdI3JD1JHUpuStWLsvbk=">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</latexit>

stationary state
21

Variational Search for stationary states

Step3

Ansatz selection

Define optimization task

| || iii
N<latexit sha1_base64="xzIqpMaRWCJrMgeC+6hg27H0ykE=">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</latexit>hh⇢|L̂ L̂|⇢ii
arg minhh⇢|⇢ii⇢d|⇢(t)ii= L̂|⇢(t)ii
dt|⇢ii =<latexit sha1_base64="1ZqLIPVSxLl3hLU/JSA4Phdqb40=">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</latexit><latexit sha1_base64="IUKb4jDEiWUOdaTncgrDf8jsf7Y=">AAACeXichVHLSsNAFD2Nr1pf8bEQ3FRLRVyUiRUUV6Ibl7a1KthSkji2oWkSkmmhrf6AP+DCjQoi1c9w4w+48BPEpYILXXibBkRFvSFzz5y5586ZGc0xDU8w9hCSurp7evvC/ZGBwaHhEXl0bNuzq67Os7pt2u6upnrcNCyeFYYw+a7jcrWimXxHK6+313dq3PUM29oSdYfnK2rRMg4MXRVEFeSJw5xbsnOuahVNHg1yQY6xBPMj+hMoAYghiE1bvkIO+7Cho4oKOCwIwiZUePTtQQGDQ1weTeJcQoa/znGECGmrVMWpQiW2TGORZnsBa9G83dPz1TrtYtLvkjKKOLtnLfbM7tgNe2Tvv/Zq+j3aXuqUtY6WO4WR48nM67+qCmWB0qfqT88CB1j2vRrk3fGZ9in0jr7WOHnOrKTjzVl2wZ7I/zl7YLd0Aqv2ol+mePoUEXoA5ft1/wTbCwklmVhILcZW14KnCGMKM5ij+17CKjawiSzt28AZWrgOvUnT0pw03ymVQoFmHF9CSn4AehySXg==</latexit>

1<latexit sha1_base64="kNY/L7q0sXyOblw3t0zEPrwy1TM=">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</latexit>⌧2<latexit sha1_base64="F21V9/wXGYY1GUlzBoWFk67PlUg=">AAACaXichVG7SgNBFD1ZXzE+kmgTtBGXiFWYqKBYBW0s8zBR0BB2NxNd3Re7k0AM/oCVnaiVgoj4GTb+gEU+QSwj2Fh4s0kQkegdZubOmXvunLlXdQzdE4w1A9LA4NDwSHA0NDY+MRmORKcKnl11NZ7XbMN2d1XF44Zu8bzQhcF3HZcrpmrwHfV4s32/U+Oup9vWtqg7vGgqB5Ze0TVFEFTY18q28EoROZlgvs31d2R0LW1H7rGPMmxoqMIEhwVBvgEFHo09JMHgEFZEgzCXPN2/5zhFiLhViuIUoRB6TOsBnfa6qEXndk7PZ2v0ikHTJeYc4uyFPbAWe2aP7JV99s3V8HO0tdRpVztc7pTCZ7Hcx78sk3aBw2/Wn5oFKljzteqk3fGR9i+0Dr92ctHKrWfjjQV2y95I/w1rsif6gVV71+4yPHuNEDWA9cr92+k1oLCUSC4nljIrcmqj24ogZjGPRar3KlLYQhp5evcI57jEVeBNikoxaaYTKgW6nGn8MEn+Ar8yjC8=</latexit><latexit sha1_base64="0MPiP3lQfyZD9yg9ZQ4NWW7xenY=">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</latexit>

1<latexit sha1_base64="F21V9/wXGYY1GUlzBoWFk67PlUg=">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</latexit>

···

h1

<latexit sha1_base64="lhrTmau4vnnhzSukVjnfn+JUvhU=">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</latexit>· ·h·M<latexit sha1_base64="/bqvD60dPUz8Q1zUvx+tEkvfmZo=">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</latexit><latexit sha1_base64="/bqvD60dPUz8Q1zUvx+tEkvfmZo=">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</latexit><latexit sha1_base64="JBGUWSech30LuzI6v1/ode8Zaww=">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</latexit>

2<latexit sha1_base64="/bqvD60dPUz8Q1zUvx+tEkvfmZo=">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</latexit><latexit sha1_base64="BUF/hQn+wkxPFDeQxhQFse6/CB4=">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</latexit>···<latexit sha1_base64="5LFOlW0T1FqKMKsse/dgWM15/rw=">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</latexit>⌧NhM<latexit sha1_base64="F21V9/wXGYY1GUlzBoWFk67PlUg=">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</latexit><latexit sha1_base64="teKw6pI/NSt1VTO5jDKA3zf6vp0=">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</latexit>

N

Step4

Run optimization<latexit sha1_base64="0hJH5Fi+NeasErEslF6AWJwryFI=">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</latexit>|iihj| 7! |ii ⌦ |ji<latexit sha1_base64="xdgZ4tf3sOf7HvN3bEY7zd4KKiY=">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</latexit>ˆhhL† L̂ii<latexit sha1_base64="yCu/un2sORBxG+3lO7J/4eNCOww=">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</latexit>

†

⌧h1

2

Lindblad eqn. invector representation

Step2

1

Step1

Yoshioka&Hamazaki, Phys. Rev. B 99, 214306 (2019).

Precisely zero if exact
Iteration

22

Step1: Vector representation

Yoshioka&Hamazaki, Phys. Rev. B 99, 214306 (2019).

Under the representation of density matrix as pure state vector
by doubling the number of qubits.

( )

⇢ˆ =
<latexit sha1_base64="g5kzesKzhZhFLd/Fk9wFvXWVi00=">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</latexit><latexit sha1_base64="PAteIQ+iIH66HU6YuElYCXKbCX8=">AAACZHichVHLSsNAFD2N7/potQiCIGJRXJUbFRRXRTcuW7VVUJEkjjo0TUKSFrT4A7pVXLhSEBE/w40/4KI/IIjLCm5ceJMGREW9w8ycOXPPnTMzumNKzyeqx5SW1rb2js6ueHdPb18i2T9Q9OyKa4iCYZu2u65rnjClJQq+9E2x7rhCK+umWNNLi8H+WlW4nrStVf/AEVtlbc+Su9LQfKbyk9vJNGUojNGfQI1AGlHk7OQNNrEDGwYqKEPAgs/YhAaP2wZUEBzmtlBjzmUkw32BI8RZW+EswRkasyUe93i1EbEWr4OaXqg2+BSTu8vKUYzTI91Sgx7ojp7p/ddatbBG4OWAZ72pFc524nho5e1fVZlnH/ufqj89+9jFXOhVsncnZIJbGE199fC8sTK/PF6boCt6Yf+XVKd7voFVfTWu82L5AnH+APX7c/8ExamMOp1R8zPp7EL0FZ0Yxhgm+b1nkcUScijwuQInOMVZ7EnpUVLKYDNViUWaFL6EMvIBV0iJqw==</latexit><latexit sha1_base64="nyPloKqtmtOdehDkHlVs+eISDrg=">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</latexit>

T<latexit sha1_base64="g5kzesKzhZhFLd/Fk9wFvXWVi00=">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</latexit><latexit sha1_base64="MYnfLBkC0klSOJRY2D2p9cYAEVI=">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</latexit>

)

(

⇢ˆ =
7! |⇢ii<latexit sha1_base64="leyKFJh8UEpVDpFmqDf1Xd/Z2pY=">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</latexit>

Cui etal. PRL (’15)<latexit sha1_base64="PAteIQ+iIH66HU6YuElYCXKbCX8=">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</latexit><latexit sha1_base64="CVaQuyTPajY1xdDDNAHHR4Ny+8A=">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</latexit>⇢ˆ =<latexit sha1_base64="HXLlDEWB1F5tf8c/FgNoq3MngiU=">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</latexit>X⌧⇢ ⌧ | ih⌧ |<latexit sha1_base64="CVaQuyTPajY1xdDDNAHHR4Ny+8A=">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</latexit>

1 X
7 |⇢ii =!
⇢ ⌧ | , ⌧ ii
C ⌧“physical” “fictitious”

d|⇢(t)ii=
L̂|⇢(t)ii
Vector representationof Lindblad equation Goal: Find L̂|ˆ
⇢SS ii = 0
dt
!
⇣
⌘
X
d|⇢(t)iiTˆ
i Ĥ ⌦ 1̂ 1̂ ⌦ Ĥ+D̂[
= L̂|⇢(t)ii =ii ] |⇢(t)ii
dt!
i
non anti-hermitian
Unitary ⌘dynamics⇣
Dissipation,XTˆNon-unitary=i Ĥ ⌦ 1̂ 1̂ ⌦ Ĥ+D̂[]|⇢(t)ii
ii<latexit sha1_base64="zkDoiJFt7OHbhMawIo1EDiHGjQE=">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</latexit><latexit sha1_base64="iGnWcjxaiUjIri8b9EfbdMK5FKc=">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</latexit><latexit sha1_base64="iGnWcjxaiUjIri8b9EfbdMK5FKc=">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</latexit>

where D̂[ ˆ i ] = ˆ i ⌦ ˆ ⇤i<latexit sha1_base64="YOaPKL04VqLlwQAIxyXdafMJNVA=">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</latexit>

1 ˆi † ˆi i ⌦ 1̂2

1 ˆT ˆ⇤1̂ ⌦i i
2(e.g. ˆ i = i )<latexit sha1_base64="Mr9O8p+hYUWwzg4h5kHoHulBG70=">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</latexit>

23

Step2: Steady state as “ground state”

Idea: Ground-state search problem Cui etal. PRL (’15)

Stationary state as zero eigenvector
Right eigenvectors of Lindblad operator L̂

One can easily show that<latexit sha1_base64="AqBesRvs05Dw1ioMyFLmFPpMqIo=">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</latexit>L̂|⇢n ii =<latexit sha1_base64="vcp1QGRNOypNOE/Ach1rH94EOnY=">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</latexit>L̂tn |⇢n ii

Yoshioka&Hamazaki, Phys. Rev. B 99, 214306 (2019).

e |⇢n ii = ent<latexit sha1_base64="zAKUGV9awXuOSrF86QPpYYj9RLQ=">AAACxHichVHNShxBEP6c/KgbEzfxIngZXJSclh4VFCEgBoIHD+5uVgXXDD1j7+5gzw8zvQs6GR8gL+DBk4KI+BgSyAvksI8QPKrkkkNqZieIkZgapuvrr+qrqu62AulEirHegPbk6bPng0PDhRcjL1+NFl+/WY/8TmiLuu1LP9y0eCSk44m6cpQUm0EouGtJsWHtvk/jG10RRo7vfVR7gdh2ectzmo7NFVFm0RSf4kabq7jhctW2uYxXk0RXif5Zb4Rt34y9hAD3WlL88e9SiaQWOzwLP5psFkuszDLTHwIjByXktuYXz9DADnzY6MCFgAdFWIIjom8LBhgC4rYRExcScrK4QIICaTuUJSiDE7tLa4t2Wznr0T6tGWVqm7pI+kNS6phi39k5u2bf2AX7wX79s1ac1Uhn2SNv9bUiMEe/jNd+/lflkldo36kenVmhiYVsVodmDzImPYXd13f3D69ri9WpeJqdsCua/5j12CWdwOve2KcVUT1CgR7A+Pu6H4L1mbIxWzYqc6Wl5fwphjCBSbyl+57HElawhjr1/Yor3OBW+6BJLdI6/VRtINeM4Z5pB78BJVuw6Q==</latexit>|⇢n ii

Eigenvalues satisfy
1. Re[ n ]  0<latexit sha1_base64="k8JX4OKWzgwubShC0QunfyiVa04=">AAACf3ichVG7SgNBFD1Z3/EVtRFtgsFHFe6q4AME0cZSo1EhCWF3nejivtidBDQI1v6AhZWCiljoP9j4AxZ+glgq2Fh4s1kQFfUOM3PmzD13zszonmUGkugxpjQ0NjW3tLbF2zs6u7oTPb3rgVv2DZE1XMv1N3UtEJbpiKw0pSU2PV9otm6JDX13sba/URF+YLrOmtzzRMHWth2zZBqaZKqYGMjbmtzx7WpGHOTyFgu3tKJTSM4lqZhIUZrCSP4EagRSiGLZTVwijy24MFCGDQEHkrEFDQG3HFQQPOYKqDLnMzLDfYEDxFlb5izBGRqzuzxu8yoXsQ6vazWDUG3wKRZ3n5VJDNMDXdEL3dM1PdH7r7WqYY2alz2e9bpWeMXuo/7Vt39VNs8SO5+qPz1LlDAdejXZuxcytVsYdX1l//hldTYzXB2hM3pm/6f0SHd8A6fyapyviMwJ4vwB6vfn/gnWx9PqRFpdmUzNL0Rf0YpBDGGM33sK81jCMrJ87iEucINbJaaMKmmF6qlKLNL04UsoMx+a3pMy</latexit>

Steady state realized at t ! 1<latexit sha1_base64="ATZNUECbx+0JFzLUx02nZNzYp6k=">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</latexit>

Steady state is obtained variationally by considering<latexit sha1_base64="LN/NiqZby1haE9RmczfVPUJljWg=">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</latexit>

Our goal<latexit sha1_base64="1LlfOyuouNaDSdTGugO7mbWMxTA=">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</latexit>'##
Spectrum
Spectrum ofof ℒℒ 'ℒℒ##

Im
Im !!

“Spiral”“Spiral”

Real,
Real, non-negative
non-negative
spectrum
spectrumRe
Re !!

Albert&Jiang, PRA (’14)

L̂ L̂|⇢ii = 0<latexit sha1_base64="AwShLZxDAivpbUsbnHGzMpskVx0=">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</latexit>hh⇢|L̂ L̂|⇢ii
0 = argminhh⇢|⇢ii⇢

2. At least one Re[ n ] = 0 (usually unique)

Decaying
Decaying
modes
modes

L̂|⇢ii = 0†<latexit sha1_base64="lZBU5wOSXEjUCxaNWxCga00kLTk=">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</latexit>

Spectrum
Spectrum ofof ℒℒ##

†

Re(Λ)
Re(Λ)
Stationary
Stationary
state
state

24

Step4: Run optimization

Yoshioka&Hamazaki, Phys. Rev. B 99, 214306 (2019).

Optimization for 1d transverse-field Ising model (TFIM) under V=1, g=1, γ=1, Nspin = 8.
3ˆ†Convergence at hhL L̂ii < 10
, fidelity = 0.996 for α=4<latexit sha1_base64="hX3CpkjsL34qWM56BDNDwvApYWA=">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</latexit>

Optimization with Nit = 1500, Nsamp = 2000

Variational Monte Carlo Algorithm
0. Choose initial parameter p0
1. Estimate parameter update δpˆhhL† L̂ii- Stochastic reconfiguration, i.e.,δp for imaginary time evolution by MC sampling
per update step

<latexit sha1_base64="0hJH5Fi+NeasErEslF6AWJwryFI=">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</latexit>- O(Nsampα 2 N 2 + Np3)

2. Update as p ← p − δp
3. Repeat 1-2 until convergence.

Iteration
Model in above calculation : 1d TFIM + damping

Barreior etal. Nature (’11) Carr etal. PRL (’13)

Realized in trapped ions, Rydberg atoms
with uniform damping as

25

Results

Yoshioka&Hamazaki, Phys. Rev. B 99, 214306 (2019).

e.g. Transverse-field Ising model w/ damping

Barreior etal. Nature (’11) Carr etal. PRL (’13)

Realized in trapped ions, Rydberg atoms
with

1d TFIM with damping

Real, Exact

Real, RBM

Imag, Exact

Imag, RBM

- Fidelity > 0.995 achieved up to L=16 (32 spins) at g/V = 0.3- 40-fold #parameter reduction at strong fieldcompared to MPS (L=16, reported by Hartmann&Carleo)

2d TFIM with damping

Jin etal. PRB (’18)- Fidelity > 0.999 achieved for 2x2, 3x3 at g/V = 1- Cost function optimized (~10-3) up to 5x5
Real/Imaginary part of density matrices
2d TFIM, V=1, g=1, γ=1, fidelity>0.999

26

Summary
Introduction to neural quantum states- Diverse application: GS, excited states, real-time evolution…
- Properties quite different from tensor nets
e.g. quantum entanglement

hM 2 {±1}h1
Wij : coupling1

N 2 {±1}2- Mainly direct problem. What about inverse problem?

Solving dissipative many-body systems

Yoshioka&Hamazaki, Phys. Rev. B 99, 214306 (2019).

- Variational method for steady state in open quantum systemdemonstrated in{- 1d/2d TFIM with damping- 1d XYZ with damping    - Both real-time and imaginary-time evolution possible
Hartmann&Carleo PRL (’19) Nagy&Savona PRL (’19)

27