AI-seminar, SLAC, online, 15 Oct 2020

Deep Learning and Quantum Gravity
Koji Hashimoto (Osaka U.)“Neural ODE and Holographic QCD”
ArXiv:2006.00712 w/ H.Y.Hu, Y.Z.You (UCSD)“Deep Learning and AdS/QCD”
ArXiv:2005.02636 w/ T. Akutagawa, T. Sumimoto (Osaka u)“Deep Boltzmann Machine and AdS/CFT”
ArXiv:1903.04951“Deep Learning and AdS/CFT” “Deep Learning and Holographic QCD”
ArXiv:1802.08313, 1809.10536
w/ S. Sugishita (Kentucky), A. Tanaka, A. Tomiya (RIKEN)

Similarity!?

Wormholes in Penrose diagramof maximally extended eternal
AdS Schwarzschild black hole
[Iizuka, Sugishita, KH `17]

Deep Autoencoder

3Roadmap

Quantum
gravityin (d+1)-dim.
‘tHooft `93
Susskind `94
Maldacena `97

Generalspacetime

Anti de Sitter
spacetime?

Quantummechanics
in d-dim.

Swingle `10

Carleo,Troyer `17

Neural network

Tensor network

Roadmap

1.
Quantum
gravityin (d+1)-dim.
‘tHooft `93
Susskind `94
Maldacena `97

Generalspacetime

Anti de Sitter
spacetime?

Quantummechanics
in d-dim.

Swingle `10

Carleo,Troyer `17

Neural network

Tensor network

Roadmap

Quantumgravityin (d+1)-dim.
‘tHooft `93
Susskind `94
Maldacena `97

Generalspacetime?

Quantummechanics
in d-dim.

Anti de Sitter
spacetime2.

Swingle `10

Carleo,Troyer `17

Neural network

Tensor network

Roadmap

3.
Quantum
gravityin (d+1)-dim.
‘tHooft `93
Susskind `94
Maldacena `97

Generalspacetime

Anti de Sitter
spacetime?

Quantummechanics
in d-dim.

Swingle `10

Carleo,Troyer `17

Neural network

Tensor network

Roadmap

4.
Quantum
gravityin (d+1)-dim.
‘tHooft `93
Susskind `94
Maldacena `97

Generalspacetime

Anti de Sitter
spacetime?

Quantummechanics
in d-dim.

Swingle `10Carleo,
Troyer `17

Neural network

Tensor network

Deep Learning and Quantum Gravity
1. Quantum gravity

3 pages

2. Neural network quantum states 3 pages
3. When is NN a spacetime?

5 pages

4. Spacetime emergent from data

4 pages

Discussion: Quantum gravity ⊂ ML ?

1. Quantum Gravity

1/3

Brief History of quantum gravity
1974 ‘tHooft, Veltman:
Perturbation fails in Einstein gravity.
1970 Nambu, Susskind, Nielsen:
String theory of hadrons.

1971 Bekenstein:
Black hole entropy.

1974 Yoneya, Scherk, Schwarz:
String is quantum gravity.

1993 ‘tHooft, Susskind:
Holographic principle.

1997 Maldacena:
AdS/CFT correspondence.
11

1. Quantum Gravity

2/3

AdS/CFT correspondence, no proof[Maldacena, Adv.Theor.Math.Phys. 2 (1998) 231]

“AdS”

“CFT”
“Large N”
Quantum mechanics
in d-dim. spacetime＝ClassicalQuantum gravityin (d+1)-dim. spacetime

・ No proof!
・ Given Left, how can one get Right?
・ Vast amount of evidence known
・ Maldacena’s paper is top-of-top-cite
12

1. Quantum Gravity

3/3

Dictionary : equating partition functions[Gubser, Klebanov, Polyakov, Phys.Lett.B428(1998)105][Witten, Adv.Theor.Math.Phys. 2 (1998) 253]

Partition function ofQuantum mechanics

Z[ 0 ]

Z[ 0 ][Dq(t)]eRdt(L[q,q̇]+ 0 (t)O[q])ettRdtdz

p＝＝

Z

＝

Partition function ofClassical gravity?

g(R[g]+L[ ]+··· )(t, z) Classical sol.

?

gµ⌫ (t, z)

source0 (t)00 (t) =(t, z = 0)z

2. Neural Network Quantum States 1/3
Find ground state wave function ψ(s1, s2, ⋯, sN )Q : Minimize its energy E for a given Hamiltonian H,

A : Use ansatz and optimize parameters!
- Matrix product states
ψ(s1, s2, ⋯) = tr[A (s1) A (s2)⋯]- Tensor network states
ψ(s1, s2, ⋯) =∑m,nBmn Ams1s2 Ans3s417

2. Neural Network Quantum States 2/3
Neural network can be wave functions- Boltzmann machine states[Carleo, Troyer `17],[Nomura, Darmawan, Yamaji, Imada `17], ..

- Deep Boltzmann machine states[Carleo, Nomura, Imada `18], ..

- Feedfoward network states
ψ(s1, ⋯, sN ) =[Saito `18], ..

f σ
W s + bi∑ i∑ ij jij18

2. Neural Network Quantum States 3/3
Better? and Why?
Neural states may beat conventional ones.
Ex) 2-dimensional
antiferromagnetic
Heisenberg model

Energy with
RBM states[Carleo, Troyer `17]
# of hidden units

Discovered intimate relations are there.
1) Boltzmann machine states are tensor network states[Chen, Cheng, Xie, Wang, Xiang `18]2) Tensor states are deep Boltzmann [Gao, Duan `17] [Huang, Moore `17]3) Tensor states are feedforward with “product pooling”[Cohen, Shashua `18]

Ex) Unified approach: MPO-Net [Gao, Cheng, He, Xie, Zhao, Lu, Xiang `19]
19

3. When is NN a spacetime?

1/5

General NN is not a space
No notion of which unit is close to which

Perceptron model[Rosenblatt 1958][Rumelhart, McClelland 1986]

Boltzmann machine[Ackley, Hinton, Sejnowski 1985]

23

3. When is NN a spacetime?

2/5

Sparsity + weight sharing, for NN to be a space
No locality

Locality imposed＝
Convolutional
layer

Fully
connected

Parallellytranslated[K. Fukushima `80]

Input: (n x)
Output:<latexit sha1_base64="nwzfFCLRwUCM0jUUcWfpi10n7t8=">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</latexit>a (n x) + b@x (n x) + c@x2 (n x) + · · ·<latexit sha1_base64="OAM1XRpoTDK6qJPFgGfv9fB7zD0=">AAACv3ichVFNSxtBGH7cWhtj1VgvgpfFYLEUwsQqlp7UeujRr6hoNMxOxjhksrvsToIx+Af8Ax48VZAi/gnBi3/Ag39AEA8WLPTSQ99sFmzrR99lZ5553vd555kZx9cqNIxdtlkv2l92vEp0Jrted/f0pvreLIVeNRAyJzztBSsOD6VWrswZZbRc8QPJK46Wy075czO/XJNBqDx30dR9uV7hJVdtKsENUYXUKrfz/pYacfMzUhtub7+z39sOcTwwiuvC9iNp8Ud6Y/SRgrwoeiYspNIsw6KwH4JsDNKIY9ZLfUMeRXgQqKICCReGsAZHSN8asmDwiVtHg7iAkIryErtIkrZKVZIqOLFlGku0WotZl9bNnmGkFrSLpj8gpY1hdsGO2R07Zyfsmv16slcj6tH0UqfZaWmlX+jdG1j4+V9VhWaDrXvVs54NNvEx8qrIux8xzVOIlr62s3+38Gl+uPGWHbIb8v+VXbIzOoFb+yGO5uT8AZL0ANl/r/shWBrNZD9kxufG0pPT8VMkMIghjNB9T2ASXzCLHO17iivc4rs1ZZUs1/JbpVZbrOnHX2HVfwPAU6ox</latexit>

24

3. When is NN a spacetime?

3/5

NN depth as time
Dynamical system
x·i = fi(x(t))

xi(tn+1) = xi(tn) + Δt ⋅ fi(x(tn))

tn+1 = tn + Δt

Discretized timeResNET (Residual network) : easily trained deep model[K.He et al.,1512.03385]

xi(n)

W

f

xi(n+1)xi(n+1) = f(Wij xj(n)) + xi(n)Skip connection25

3. When is NN a spacetime?

4/5

Hamilton dynamics is a NN
·q = ∂H ,∂p1802.08313·p = − ∂H∂qTime-dependent Hamiltonian = weights/activation00v
0 1 + Δtw11Δtw12
W=
0Δtw21
1 + Δtw120
u0qpWijφiW̃ij

Δtf(x)
1
φi =1Δtg(x)0λ1
W̃ =0001000010000000
λ20λ1
λ21122H = w11 pq + w12 p − w21q + F(vp) − G(uq)22vu(F′ = f, G′ = g)

3. When is NN a spacetime?

5/5

Q. Find a HamiltonianConsider a particle motion x(t) in a given potential
V (x) in 1 dimension, with unknown time-dependent
friction force h(t)ẋ .
One tried many initial conditions (x(t = 0), ẋ(t = 0))
and collected those which stops at t = 10 .
Q. From given data of the initial conditions, find h(t) .
V (x)@V (x)
mẍ = h(t)ẋ +@xx27

4. Spacetime emergent from data 1/4
Gravity side
Classical scalar field theory in unknown 5-dim. spacetime
S=<latexit sha1_base64="TFyvaO4nUBjzbL0k2HMZOKHqRSU=">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</latexit>

Z4d⌘d xds2 =p⇥det g (@⌘ )2V( )1802.08313
1809.10536⇤f ( )dt2 + d 2 + g( )(dx21 + · · · + dx2d 1 )3 2
4+V[ ]=L2
4

Data: ( 0 , Z[ 0 ])t0BlackHole(⌘)( |⌘=1 , @⌘ |⌘=1 , @⌘ |⌘=0 )

1

⌘

031

4. Spacetime emergent from data 2/4
Equation of motion as a feedforward NN
Eq. of motion

2+ h( )

metricDiscretization
Hamilton formV[ ]( +( +=0h( ))= ( )+)= ( )+log

f ( )g( )d 1h( ) ( )V ( ( ))( )( )

Feedfoward neural network for classification=0==0=032

4. Spacetime emergent from data 3/4
Training with data of quark condensate
Data of quantum chromodynamics
simulated by supercomputer<latexit sha1_base64="l+XJxifsHha+KUEDRdBoVx0fszc=">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</latexit><latexit sha1_base64="Y1lb4mzByJPqNHPD1TVSTPNFncc=">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</latexit>@Z[ 0 ] : Quark condensate@ 00 : Quark mass33

4. Spacetime emergent from data 3/4
Training with data of quark condensateh(⌘)<latexit sha1_base64="vTlG9iOIrhww+0CnFBDonAQYKbI=">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</latexit>

Trained values of potential :1/L = 237(3)[MeV] , λ/L = 0.0127(6)344. Spacetime emergent from data 4/4
Emergent metric can predict physics
Interquark potentialh(⌘)<latexit sha1_base64="vTlG9iOIrhww+0CnFBDonAQYKbI=">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</latexit>

Procedures
based on
[Maldacena][Rey,Theisen,Yee][Petreczky, `10]35

Discussion: Quantum gravity ⊂ ML ?
3 steps for quantum gravity
Quantum Mechanics
side

Gravity side
metric gµ⌫ field

ArchitectureLarge DoF limit
Large DoF expansion
Finite DoF

Classical
Classical

Classical
Quantum

Feedforward NN
Deep Boltzmann

QuantumQuantum?

- Neural ODE : free from discretization- Quantum AdS/CFT ⊂ Deep Boltzmann machine- Conventional quantum gravity looks networks
38

Discussion: Quantum gravity ⊂ ML ?
Neural ODE : free from discretization
2006.00712

d (⌘)= f ( (⌘), ⌘, h(⌘))
d⌘Emergent metric

Q Qbar potential
Neural ODE [R.T.Q.Chen, Y.Rubanova,
J.Bettencourt, D.Duvenaud 1806.07366]

39

Discussion: Quantum gravity ⊂ ML ?
Quantum AdS/CFT ⊂ Deep Boltzmann
AdS/CFT[Maldacena 1997]

Deep Boltzman machine[Salakhutdinov, Hinton 2009][KH `19] [You,Yang,Qi `18] (See also [Gan,Shu `17][Howard `18])
40

Discussion: Quantum gravity ⊂ ML ?
Physical picture of Deep Boltzmann
Ground state wave function for given Hamiltonian
is identified as a deep Boltzmann machine[Carleo, Nomura, Imada `18], ..

| ψ⟩ = lim e −τH | any⟩ = e −ΔτH e −ΔτH⋯ | any⟩
τ→∞H<latexit sha1_base64="XagScydK2LYSdBnKEe0o0+PseEo=">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</latexit>

Euclidean time⌧<latexit sha1_base64="/eEqvXEnTni57AUUEIhrBngQ2w0=">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</latexit>(xi ) =X(n)<latexit sha1_base64="AvgKTR8IQdExyctru1rwnMXbrXc=">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</latexit>hj

2{0,1}

2exp 4Xij(0)wij xi hj

XXnij

3(n) (n) (n+1) 5

wij hi hj

41

Discussion: Quantum gravity ⊂ ML ?
AdS/CFT discretized the bulk, but fixed
AdS/MERA

Quantum codes for holography[Swingle `09]

[Pastawski, Yoshida, Harlow, Preskill `15]42