http://www-het.phys.sci.osaka-u.ac.jp/Onogi-workshop/

Neural net for lattice QCD,
and quantum methods fornite-temperature and density
Akio Tomiya (Faculty in IPUT Osaka)
akio_at_yukawa.kyoto-u.ac.jp

PI: Grant-in-Aid for Transformative Research Areas (A)fi,Grant-in-Aid for Early-Career Scientists
CI: Grant-in-Aid for Scienti c Research (C), etc

 fiOsakaU, Sep 23, 2022
10:50-11:30 (40 min)Based on arXiv:
arXiv: 2103.11965
2205.08860 etc

Happy “Kann-reki”(full circle) birthday

Akio Tomiya

Onogi-san and me

還暦おめでとうございます!
When I entered to master course in Osaka university in 2010 (12 years ago!),Tetsuya was the second year as a professor in Osaka university(My supervisor Fukaya-san also joined HEP group as an assistant prof. in 2010)
We had a lot of study groups: String theory, lattice eld theory, RG

http://kabuto.phys.sci.osaka-u.ac.jp/HETphoto/HETCamp2010.html
fi2Happy “Kann-reki”(full circle) birthday

Akio Tomiya

Onogi-san and me2015

おおししょうOnogi-san = Supervisor’s Supervisor’s of me = Great master (大師匠)!
And a mentor of me (he always encouraged me).
He taught me quantum eld theory (perturbation theory/SM/RG/GWW trs),
lattice eld theory, algorithms, how to read a code, general relativity, etc
In particular, my master thesis about many avor QCD.
Thank you and congratulations Onogi-san!
fl

fi

fi

http://www-het.phys.sci.osaka-u.ac.jp/photo2015/Tea%20party/tea%20party.html

3Outline
Two exotic topics
1.What and why QCD/lattice QCD?
2. Lattice QCD + Machine learning
1.“Neural net = Smearing”(t)dUμ (n)dt=(t)(Uμ (n))θ̄

3. Lattice QCD + Quantum algorithm

𝒢

1.Finite temp/dens for QFT

4

Introduction

Akio Tomiya

QCD: a fundamental theory of particles inside of nuclei
QCD (Quantum Chromo-dynamics) in 3 + 1 dimension

1
S = d x[ − tr FμνF μν + ψ̄(i∂/ + gA/ − m)ψ]∫2
4

Fμν = ∂μ Aν − ∂ν Aμ − ig[Aμ, Aν]

Quantization: | ψ(t)⟩ = e−iHt | ψ(0)⟩[Aμ, Eν] = iℏδμν• QCD enables us to calculate (in principle):• Equation of state of neutron star, Tcof quarks and gluons, Parton• Scattering
distributions• Hadron spectrum!

• Strongly coupled quantum system• Use lattice QCD + Monte-Carlo5

Introduction

Akio Tomiya

Lattice path integral > 1000 dim, Trapezoidal int is impossible
K. Wilson 1974

1
S = d x[ + tr FμνFμν + ψ̄(∂/ − igA/ + m)ψ]∫2
4

Imaginary timet = − iτ
Lattice regularization
Uμ = eaigAμ1
S[U, ψ, ψ̄] = a− Re tr Uμν + ψ̄(D/ + m)ψ]∑ [ g2n4cuto = a −1

−1 2 4 2
g a Fμν + O(a 6)Both S give same expectation value for long range Re Uμν ∼2Path integral formalism

1
1
−S
⟨ ⟩=
U ψ̄ ψe
(U) =
Z∫
Z∫
1
=
Ue −Se [U] (U)
Z∫
∏ 4∏

=

𝒪

𝒟

𝒪

𝒟

type integral like Trapezoid, Simpson etc.
We cannot control numerical error

ff

𝒟

𝒟

𝒟

𝒪

n∈{ℤ/L} μ=1dUμ(n) >1000 dim. We cannot use Newton–Cotes

𝒪

4

Ue −Sgauge[U] det(D + m) (U)ff

6Introduction

Akio TomiyaMonte-Carlo integration is available
1
⟨ ⟩=
Z∫

M. Creutz 1980

Ue −Se [U] (U)Se [U] = Sgauge[U] − log det(D/ [U] + m)

1 −S [U]Monte-Carlo: Generate eld con gurations with “P[U] = e e”. It gives expectation value
Z
U2
U3
U1
P[U]

サンプ → サンプ

P[U]

P[U]

→ サンプ

→…

HMC: Hybrid (Hamiltonian) Monte-Carlo
De-facto standard algorithm

𝒪
ff

ff

ff

𝒟

𝒪

1 2
S(x, y) = (x + y 2 + xy)
2

fi

fi

7

Introduction

Akio TomiyaMonte-Carlo integration is available
M. Creutz 1980

1
⟨ ⟩=
Z∫

Ue −Se [U] (U)Se [U] = Sgauge[U] − log det(D/ [U] + m)

1 −S [U]Monte-Carlo: Generate eld con gurations with “P[U] = e e”. It gives expectation value
Z
U2
U3
U1
P[U]

サンプ → サンプ

P[U]

P[U]

→ サンプ

→…

Error of integration is determined by the number of sampling1

Nsample ∑k𝒪Nsample)𝒪[Uk] ± O(

1

ff

ff

ff

𝒟

𝒪

⟨ ⟩=
𝒪

Nsamplefi

fi

8Introduction

Akio Tomiya

Correlation between samples = ine ciency of calculation
1 N⟨O[ϕ]⟩ =O[ϕk] ± O(N∑kNindep =ϕ1

1
NindepΓ̄(t))

e −t/τac

Nsamplet2τac1
Γ̄(t) =(O[ϕk+t] − Ō)(O[ϕk] − Ō) ∼ e −t/τac
N−t∑
k

ϕ2

ϕ3

サンプルA → サンプルB
Correlated→ サンプルC

Correlated→…

Slightly Correlated

Large τac means, such simulation is ine cient
ffi

ffi

9

Introduction

Akio Tomiya

Summary for now: long autocorrelation = ine ciency
1 N⟨O[ϕ]⟩ =O[ϕk] ± O(N∑kNindep =1
NindepΓ̄(t))

e −t/τac

Nsamplet2τac1
Γ̄(t) =(O[ϕk+t] − Ō)(O[ϕk] − Ō) ∼ e −t/τacN−t∑kτac is given by an update algorithm (N. Madras et. al 1988)• Autocorrelation time τac quanti es similarity between samples• τac is algorithm dependent quantity• If τac becomes half, we can get doubly precise results in the same time cost
Can we make this mild using machine learning?
ffi

fi

10

Introduction

Akio Tomiya

Neural net can make human face images

11Introduction

Akio Tomiya

Neural net can make human face images
Neural nets can generate realistic human faces (Style GAN2)

Realistic Images can be generated by machine learning!
Con gurations as well? (con guration ~ images?)

fi

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12

ML for LQCD is needed• Machine learning/ Neural networks• Data processing techniques for 2d image indaily life (pictures = pixels = a set of real #)• Neural network can generate images!
(arpproximately)• Lattice QCD is more complicated than pictures• 4 dimension• Non-abelian gauge d.o.f. and symmetry• Fermions• Exactness of algorithm is necessary• Q. How can we deal with?

http://www.physics.adelaide.edu.au/theory/sta /leinweber/VisualQCD/QCDvacuum/ff

13

Introduction

Akio Tomiya

Con guration generation with machine learning is developing
Year

Group

ML

Dim.

TheoryGauge sym

Exact?

Fermion?

Lattice2021/ref

2017

AT, Akinori
Tanaka

RBM
+ HMC

2d

Scalar-No

No

arXiv: 1712.03893

2018 K. Zhou+

GAN

2d

Scalar-No

No

arXiv: 1810.12879

2018

J. Pawlowski +

GAN
+HMC

2d

Scalar

-

Yes?

No

arXiv: 1811.03533

2019

MIT+

2d

Scalar-Yes

No

arXiv: 1904.12072

2020

MIT+

2d

U(1)

Equivariant

Yes

No

arXiv: 2003.06413

2020

MIT+

Flow
Flow
Flow

2d

SU(N)

Equivariant

Yes

No

arXiv: 2008.05456

2020

AT, Akinori
Tanaka +

SLMC

4d

SU(N)

InvariantYes

PartiallyarXiv: 2010.11900

2021

M. Medvidovic´+

A-NICE

2d

Scalar-No

No

arXiv: 2012.01442

2021 S. Foreman L2HMC

2d

U(1)

Yes

Yes

No

2021

AT+

SLHMC

4d

QCD

Covariant

Yes

YES!

2021

L. Del
Debbio+2dScalar, O(N)

-

Yes

No

2021

MIT+

Flow
Flow

2d

Yukawa

-

Yes

Yes

2021

S. Foreman,AT+

Flowed
HMC

2d

U(1)

Equivariant

Yes

No but compatible

2021

XY Jing

2d

U(1)Equivariant

Yes

No

2022

J. Finkenrath

Neural
net
Flow

2d

U(1)

Equivariant

Yes

Yes (diagonalization)

arxiv: 2201.02216

2022

MIT+

Flow

2d, 4dU(1), QCD

Equivariant

Yes

Yes

arXiv:2202.11712 +

2022

AT+

Flow

2d, 3dScalarThis talkarXiv: 2112.01586

Yrs
＋…

fi

14

LQCD + Machine learning
How to deal gauge sym.

15

Introduction

Akio Tomiya

Neural network is a universal approximator of functions
Image classi cation, cats and dogs
100x100
Flatten

⟹

0.000
0.000
0.8434 Image is a vector
0.756 (this is 10,000 dimension)
0.3456⋮

10,000 dimension

Input

f: Neural net

0 Label is 2 dim vectordog =(1) (cat = (1, 0)t)
dog =

0
(1)

f(Image of dog) =

0.1
(0.9)

2 dimension
Images of “dog”

cat =

1
(0)

Images of “cat”

fi16Introduction

Akio Tomiya

A ne transformation + element-wise transformation
Fully connected neural networksfθ( x ⃗) = σ

(l=2)

(W

(l=2) (l=1)

σ

(W

(l=1)

x ⃗+ b

(l=1)
⃗

)+ b

(l=2)
⃗

)

θ represents a set of parameters: eg wij(l), bi(l), ⋯ (throughout this talk!)⃗ = x⃗
Component of neural net: l = 2,3,⋯ and u (1){(l)(l) (l−1)(l)zi =wij uj+ bi∑j(l)(l) (l)ui = σ (zi )Matrix productvector addition(w, b determined in
the training)element-wise (local)
Non-linear transf.
Typically σ ~ tanh shape

Neural network = (Variational) map between vector to vector
ffi

17

Introduction

Akio Tomiya

Neural network is a universal approximator of functions
Image classi cation, cats and dogs
100x100
Flatten

⟹

0.000
0.000
0.8434 Image is a vector
0.756 (this is 10,000 dimension)
0.3456⋮

10,000 dimension

Input

f: Neural net

0 Label is 2 dim vectordog =(1) (cat = (1, 0)t)
dog =

0
(1)

f(Image of dog) =

0.1
(0.9)

2 dimension
Images of “dog”

Fit ansatz

cat =

1
(0)

Images of “cat”

Fact: neural network can mimic any function! (universal app. thm)
In this example, neural net mimics a map between
image (10,000-dim vector) and label (2-dim vector)

Mathematical Physics Studies

Akinori Tanaka
Akio Tomiya
Koji Hashimoto

Deep Learning
and Physicsfi

18

What is the neural networks?

Akio Tomiya

Convolution layer = trainable lter
Filter on image

Laplacian lter

*

0

1

0

1

-2

1

0

1

0

=

Edge detection(Discretization of ∂ 2)

IMPORTANT: If inputs are shifted to right, outputs are shifted to right= translationally equivaliant (similar to covariance, operation just commute)
Convolution layer

Fukushima, Kunihiko (1980
Zhang, Wei (1988) + a lot!

Trainable lter*w11 w12 w13
w21 w22 w23
w31 w32 w33→

Edge detection
Smoothing(Gaussian lter)

Gaussian lter

1
16

…

1

2

1

2

4

2

1

2

1

This can be any lter which helps feature extraction
but still transitionally equivariant!
fi

)

fi

fi

fi

fi

fi

19

Training can be done with back propagation

w11 w12 w13
dog =w21 w22 w23
w31 w32 w33

Translationequivariant map
with trainable
parameters

G.A.
Pooling/
atten

Dense
net

loss function
quanti es
error of output

0
(1)

cat =

1
(0)

feed

L

Feedback = training(Steepest descent)

20
fi

fl

Convolution neural network

Akio Tomiya

Smearing

Akio Tomiya

Smoothing improves global properties
Coarse image

Eg.

Smoothened image

Gaussian lter
1
16

1

2

1

2

4

1

1

2

1

Numerical derivative is unstable

Numerical derivative is stable

We want to smoothen gauge con gurations
with keeping gauge symmetry
Two types:

APE-type smearing
Stout-type smearing

M. Albanese+ 1987
R. Ho mann+ 2007
C. Morningster+ 2003

fi

fi

ff

21

Smearing

Akio Tomiya

Smoothing with gauge symmetry, APE type
M. Albanese+ 1987
R. Ho mann+ 2007

APE-type smearing
Covariant sum

Uμ(n) → Uμfat(n) =α †(1 − α)Uμ(n) + Vμ [U](n)[]6Vμ†[U](n) =̂ ν†(n + μ)̂ + ⋯Uν(n)Uμ(n + ν)U

Schematically,

∑
μ≠ν

= [

+

(1 − α)Normalization

[M] =

M
M †M

Or projection

Vμ†[U ](n)& Uμ(n) shows same transformation→Uμfat[U ](n) is as well

+

α
6 ∑
ν

In the calculation graph,

]

1−α

V

Mult
Sum

α /6

+

[⋯]

U (1)

𝒩

𝒩

𝒩

U

ff

𝒩

22

Smearing

Akio Tomiya

Smoothing with gauge symmetry, stout type
C. Morningster+ 2003

Stout-type smearing

Uμ(n) → Uμfat(n) = eQUμ(n)Covariant sum= Uμ(n) + (eQ − 1)Uμ(n)Q: anti-hermitian traceless plaquette

This is less obvious but this actually obeys same transformation=e=

Schematically,(In the calculation graph,UQ(e − 1)U

)

+ e
(

+

U

)

−1

fat
23

Smearing

Akio Tomiya

Smearing decomposes into two parts
General form of smearing (covariant transformation){ Uμfat(n) =(zμ(n))

Gauge covariant sum
A local function𝒢𝒩zμ(n) = w1Uμ(n) + w2 [U]24

Smearing

Akio Tomiya

Smearing 〜 neural network with xed parameter!
AT Y. Nagai arXiv: 2103.11965

General form of smearing (covariant transformation){ Uμfat(n) =zμ(n) = w1Uμ(n) + w2 [U](zμ(n))

Gauge covariant sum
A local function

It has similar structure with neural networks,{(l)(l) (l−1)(l)zi =wij uj+ bi∑j(l)(l) (l)ui = σ (zi )Matrix productvector addition
element-wise (local)
Non-linear transf.
Typically σ ~ tanh shape𝒢

𝒩

Actually, we can nd a dictionary between them

fi

fi

25

Gauge covariant neural network= trainable smearing

Akio Tomiya

AT Y. Nagai arXiv: 2103.11965(convolutional)
Neural network

Smearing in LQCD

Input

Image
(2d data, structured)gauge con g(4d data, structured)

OutputImage(2d data, structured)gauge con g(4d data, structured)

Dictionary

Symmetry

Translation

with Fixed paramImage lter

Translation, rotation(90°),
Gauge sym.
Gauge sym
(APE/stout …) Smearing

Local operation

Summing up nearest
neighbor with weights

Summing up staples
with weights

Activation function

projection/normalization
Tanh, ReLU, sigmoid, …
in Stout/HYP/HISQ

Formula for chain rule

Backprop“Smeared force
calculations” (Stout)

Training?

Backprop + Delta rule

AT Nagai 2103.11965

Well-known(Index i in the neural net corresponds to n & μ in smearing. Information processing with NN is evolution of scalar eld)

fi

fi

fi

fi

26

Takeaway message
Gauge Covariant Neural networks= trainable smearing, training for SU(N) fields

27Gauge covariant neural network= trainable smearing

Akio Tomiya

AT Y. Nagai arXiv: 2103.11965

Gauge covariant neural network = general smearing with trainable parameters w{ (zμ(l+1)(n))

zμ(l+1)(n) = w1(l)Uμ(l)(n) + w2(l)Uμ(l+1)(n)[U (l)] :

(l)
[U]
θ̄

(Weight “w” can be depend on n and μ = fully connected like. Less symmetric, more parameters)

UμNN(n)[U] = Uμ(3)(n) Uμ(2)(n) Uμ(1)(n)[Uμ(n)]
[
]]
[

e.g.

Good properties: Obvious gauge symmetry. Translation, rotational symmetries.
(Analogous to convolutional layer, this fully uses information of the symmetries)

NN
NN
Uμ(n) ↦ Uμ (n) = Uμ (n)[U]1. Gauge covariant composite function:
Input = gauge eld, Output = gauge eld

𝒢𝒩

2. Parameters in the network can be trainable using ML techniques.
fi

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28

fl

Gauge covariant neural network

Akio Tomiya

Training can be done with (extended) back propagation
AT Y. Nagai arXiv: 2103.11965

Gauge inv. loss function can be constructed by gauge invariant actions
Usual neural network
w11 w12 w13
dog =w21 w22 w23
w31 w32 w33
Translationequivariant map
with trainable
parameters

G.A.
Pooling/
atten

Gauge covariant map withtrainable parameters
e.g.

=e
(

w1 +w2

)

cat =

1
(0)

Parametrized
actionCovariantU

UμNN(n)[U]

0
(1)

feed

L

Translation equivariant =the image is shifted,
output image is also shifted

Covariant neural networks

U

Dense
net

Invariantloss function

NN

to
d
e
e

D

nst

ruc

t lo

D [UμNN(n)[U]]Sferm [U NN]

feed

L

Splaq [U NN]

feed

L

Parametrized Dirac operator

FCo
distorted

irac

op

Invariantloss function

ops

Covariant

W [UμNN(n)[U]]Parametrized loop operators(e.g. plaquette, Polyakov loop)

Q [U NN]

Topological chargeL

cf. Gauge equivariant neural net (M Favoni+)

29

Gauge covariant neural network

Akio Tomiya

Neural ODE of Cov-Net = “gradient ow”⃗
u \(l−1)

ResNetContinuum
Layer
Limit

+

(t)
⃗

du
dt

=

(t)
⃗

(u )

arXiv: 1512.03385

arXiv: 1806.07366(Neural IPS 2018 best paper)

𝒢

𝒢

Neural ODE

⃗
u (l)

fl

30

Gauge covariant neural network

Akio Tomiya

Neural ODE of Cov-Net = “gradient ow”⃗
u \(l−1)

ResNetContinuum
Layer
Limit

+

(t)
⃗

du
dt

Neural ODE

U

Gauge-cov net
Continuum
Layer
Limit

Neural ODE
for Gauge-cov NN(t)dUμ (n)

(t)
⃗

=

(l)

dt

⃗
u (l)

(u )
θ̄

=

arXiv: 1806.07366(Neural IPS 2018 best paper)

U (l+1)+(t)(Uμ (n))θ̄

arXiv: 1512.03385

AT Y. Nagai arXiv: 2103.11965“Gradient” ow(not has to be gradient of S)

𝒢“Continuous stout smearing is the Wilson ow”
𝒢𝒢𝒢2010 M. Luscher

fl

fl

fl

31

Gauge covariant neural network

Akio Tomiya

Short summary

Symmetry

Fixed parameter

Continuum
limit of layers

How to Train

Conventional
neural network

Convolution:
Translation

Convolution:
Filtering(e.g Gaussian/Laplasian)

ResNet:Neural ODE

Delta rule andbackprop
Gradient opt.

Gauge cov. net

Gauge covarianceTranslation equiv,
90° rotation equiv“Gradient ow”

Extended Delta
rule andbackprop
Gradient opt.

AT Y. Nagai arXiv: 2103.11965

Smearing

Next, I show a demonstration

Re-usable stout
force subroutine(Implementation is easy &no need to use ML library)fl32

An application
Self-learning HMC

33

Application for the staggered in 4d

Akio Tomiya

Problems to solve
arXiv: 2103.11965

Our neural network enables us to parametrize gauge symmetric action
covariant way. It can be used in variational ansatz in gauge theory.

S NN[U] = Splaq [UμNN(n)[U]]e.g.

S NN[U] = Sstag [UμNN(n)[U]]

Test of our neural network?

Can we mimic a di erent Dirac operator using neural net?
Arti cial example for HMC:

{Action in MD

Target action

S[U] = Sg[U] + Sf[ϕ, U; m = 0.3],

Sθ[U] = Sg[U] + Sf[ϕ, UθNN[U]; mh = 0.4],

Q. Simulations with approximated action can be exact?
-> Yes! with SLHMC (Self-learning HMC)

ff

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34

SLHMC = Exact algorithm with ML

Akio Tomiya

SLHMC for gauge system with dynamical fermions
HMC

arXiv: 2103.11965 and reference therein

U

U

U

U

U

U

￼

￼

￼

Eom

U

Metropolis

Eom

U

U

U′
π′
ϕ

Metropolis

ϕ

Eom

Gauss

U

￼

Eom

U

Eom

U

π

π′
ϕ

Eom

U

U′

ϕ

Eom

Self
Lerning
HMC

π

Eom

Gauss

U

Eom

Eom

Eom

Metropolis

Both use

1
HHMC =
π 2 + Sg + Sf
2∑
Non-conservation of H cancels since
the molecular dynamics is reversible

Metropolis1H=
π 2 + Sg + Sf[U]
2∑
Eom

1
H=
π 2 + Sg+Sf[U NN[U]]2∑
Neural net approximatedfermion action but exact

SLHMC works as an adaptive reweighting!

35

Application for the staggered in 4d

Akio Tomiya

Lattice setup and question
arXiv: 2103.11965

TargetTwo color QCD (plaquette + staggered)

Algorithms

SLHMC, HMC (comparison)

Parameter

Four dimension, L=4, m = 0.3, beta = 2.7, Nf=4 (non-rooting)Target action

S[U] = Sg[U] + Sf[ϕ, U; m = 0.3],

Action in MD(for SLHMC)

Sθ[U] = Sg[U] + Sf[ϕ, UθNN[U]; mh = 0.4],

Observables

Code

For Metropolis Test

Plaquette, Polyakov loop, Chiral condensate ⟨ψ ψ⟩
Full scratch,
fully written in Julia lang.

AT+ (in prep)(But we added some functions on the public version)36

Lattice QCD code
We made a public code in Julia Language
What isAkio Tomiya

AT & Y. Nagai in prep

? 1.Open source scienti c language (Just in time compiler)
2.Fast as C/Fortran (sometime, faster), Productive as Python
3.Machine learning friendly (Julia ML packages + Python libraries w/ PyCall)
4.Supercomputers support Julia(O cial package) : Laptop/desktop/PC-cluster/Jupyter (Google colab)

SU(Nc)-heatbath/SLHMC/SU(Nc) Stout/(R)HMC/staggered/Wilson-Clover
Domain-wall/Measurements (Now updating to v1.0, MPI ver is ready)
1. Download Julia binary
3 steps in 5 min
2. Add the package through Julia package manager
3. Execute!
https://github.com/akio-tomiya/LatticeQCD.jl

fi

ffi

37

Details (skip)

Akio Tomiya

Results: Loss decreases along with the training
Loss function:

2

arXiv: 2103.11965

1
Lθ[U] =
Sθ[U, ϕ] − S[U, ϕ] , ~ -log(reweighting factor)
2

Training iteration history

Without training, e^(-L)<< 1, this means that candidate with approximated action
never accept.
After training, e^(-L) ~1, and we get practical acceptance rate!
38

Application for the staggered in 4d

Akio Tomiya

Results are consistent with each otherarXiv: 2103.11965

Expectation value

Acceptance = 40%

Future work: Domain-wall/Overlap SLHMC (?)
39

Other architecture:
Flow based sample algorithm

40

Related works

Akio Tomiya

Gradient ow as a trivializing map
Trivializing map for lattice QCD has been demanded…

1
⟨ ⟩=
⋯
dϕx,y,z,te−S(ϕ) [ϕx,y,z,t]∏ ∏ ∏ ∏
Z ∫ ∫ x∈100
y∈100 z∈100 t∈100

ϕ̃ = ℱτ(ϕ)

Flow equation (change variable)

If the solution satis es S(ℱτ(ϕ)) + ln det(Jacobian) =∑n

ϕ̃2n ,

𝒪

arxiv 1904.12072, 2003.06413, 2008.05456

𝒪

M. Luscher arXiv:0907.5491
fi

fl

41

Flow based sampling algorithm

Akio Tomiya

Normalizing ow ~ Change of variables
Simplest example: Box Mullerz=e− 12 (x 2+y 2){tan θ = y/xChangeof variables∞∞∫−∞ ∫−∞dx

dy e− 12 x 2− 12 y 2

Original integral: hard

2π

1

1
=
dθ dz
∫0
2 ∫0
Easy

Point: Make problem easier with change of variables (make the measure at)
RHS is at measure→We can sample like right eq.

ξ1 ∼ (0,2π){ ξ2 ∼ (0,1)

We can reconstructa “ eld con g” x, y
for original theory
like right eq.

x = r cos θ{ y = r sin θθ = ξ1r = −2 log ξ2A change of variable which Dϕe −S[ϕ] makes at = Trivializing map
fl

fl

fl

fi

fl

fi

42

Related works

Akio Tomiya

Gradient ow as a trivializing map
Trivializing map for lattice QCD has been demanded…

1
⟨ ⟩=
⋯
dϕx,y,z,te−S(ϕ) [ϕx,y,z,t]∏ ∏ ∏ ∏
Z ∫ ∫ x∈100
y∈100 z∈100 t∈100

ϕ̃ = ℱτ(ϕ)

Flow equation (change variable)

If the solution satis es S(ℱτ(ϕ)) + ln det(Jacobian) =∑n

ϕ̃2n ,1− ∑ ϕ̃2n⟨ ⟩=⋯dϕ̃ [ℱτ(ϕ)]e∏ ∏ ∏ ∏
Z ∫ ∫ x∈100
y∈100 z∈100 t∈100
It becomes Gaussian integral! Easy to evaluate!!
However, the Jacobian cannot evaluate easily, so it is not practical.
Life is hard.

𝒪

𝒪

𝒪

arxiv 1904.12072, 2003.06413, 2008.05456

𝒪

M. Luscher arXiv:0907.5491
fi

fl

43

Related works

Akio Tomiya

Flow based algorithm = neural net represented ow algorithm
MIT + Google brain 2019~

Train a neural net as a “ ow” ϕ̃ = ℱ(ϕ)
If it is well approximated, we can sample from a Gaussian
It can be done “Normalizing ow” (Real Non-volume preserving map)Moreover, Jacobian is tractable!
arxiv 1904.12072, 2003.06413, 2008.05456
fl

fl

fl

44

Related works

Akio Tomiya

Flow based algorithm = neural net represented ow algorithm
MIT + Google brain 2019~

Their sampling strategy
sample gaussian → inverse trivializing map → QFT con gurations
Tractable Jacobian (by even-odd strategy)
After sampling, Metropolis-Hastings test (Detailed balance)→ exact!
arxiv 1904.12072, 2003.06413, 2008.05456
fl

fi

45

Flow based sampling algorithm
Flow based ML for QFT∫

Dϕe

MIT + Deepmind + …

−S[ϕ]O[ϕ] ∝∏∫idφie

Original integral: hard

Sample∏Sample−V(φi) −1

J [φ]O[F[φ]]

Easy
Vol

Flow-based sampling algorithm

Akio Tomiyae −V(φi)i

SampleTrivial theory(no kinetic term, no topology)

SampleNo auto-correlation
No correlation for points
Flow

Flow

Flow

Flow

Flow

No auto-correlation
Approx.correlation for points“Cooling = change of variable”
via trained neural net“un-trivializing map”
Metropolis-Hastings withe −S /e −V(φi)J −1[φ]Small auto-correlation
Correct correlations
Reject(Use left conf.)
arXiv: 2101.08176 and ref therein

Reject(Use left conf.)46

Normalizing ow in Julia

AkioAdTomiyaWe made a public code in Julia Language

https://arxiv.org/abs/2208.08903

flhttps://github.com/AtelierArith/GomalizingFlow.jl

A public code for
Flow-based samplingalgorithm
not only 2d but also 3d

47

LQCD + Quantum galtorithm

48

Motivation

Akio Tomiya

Sign problem prevents using Monte-Carlo• Monte-Carlo enables us to evaluate expectation values for “statisticalsystem”, like lattice QCD in imaginary time
1

Nconf

Nconf ∑c

⟨O[U]⟩ =O[Uc] +(

Nconf )
1

1 −S[U]Uc ← P(U) = e∈ ℝ+
ZGreat successes!
Sign problem

μarXiv:0906.3599• If we turn on the baryon chemical potential μ, Monte-Carlo methods do notwork because e −S[U] becomes complex. This is no more probability. (signproblem)• Operator formalism does not have such problem! But it requires huge memory
to store quantum states, which cannot be realized even on supercomputer.
𝒪• Quantum states should be stored on quantum device (Feynman)

Motivation

Akio Tomiya

μ = 0 is good for Classical, T=0 is good for Quantum
Classical machine: Lattice eld theory calculations rely on

1 −S[U]
P(U) = edet(D[U] + m)2 ∈ ℝ+
Z

Since 1980 (M. Creutz)~- This P(U) cannot be regarded as probability if μ ≠ 0 (sign problem)
Quantum machines can realize (any) unitary evolutions (Solovay Kitaev theorem),U(t) = e−iHt̂

Phys.Rev.D 105 (2022) 9, 09450
and references therein- No problem for μ≠0 because we can only use unitary gates (operators)- “Short time evolution” (shallow circuit) is preferred for near-term devices*

We need a method to calculate T>0 and μ≠0 for QCD
and for near-term quantum devices
3fi*https://indico.hiskp.uni-bonn.de/event/40/contributions/484/attachments/358/630/Powers%20Talk%20Final%20Draft%20Updated.pdf

50Summary of this talk

Akio Tomiya

Hybrid = Quantum algorithm + machine learning
AT arXiv: 2205.08860

Fukushima , Hatsuda
Rept.Prog.Phys.74:014001,2011

I investigated T-mu phase diagram using a quantum algorithm & neural network(β-VQE, No sign problem) for Schwinger model (toy model of QCD)
51QFT with Hamiltonian

Akio Tomiya

Hamiltonian vs Lagrangian

Operator formalism (This work)

H : Hamiltonian in QFT
Real time
Minkowski in M^{d+1}

U(t) = e−itH

Euclid(t → τ)t = − iτ
Minkowski(τ → t)

Finite temperature/imaginary time
Euclid in S^1 x M^d−τH

U(τ) = e⟨OO(τ)⟩ = Tr[O(0)O(τ)ρ]ρ = U(τ)/Z- Typical use case of quantum algorithm is for real-time. Unitary.
- Time evolution: Correlators (e.g. 2pt on light-cone), etc- Main interest: ⟨Ω | O | Ω⟩, where | Ω⟩ is the exact ground state- Di culty: State preparation for exact ground state of H

ffi

52

State preparation, VQE and Beta-VQE

Akio Tomiya

State preparation is hard
We are interested in expectation value with true ground state for Hamiltonian⟨O⟩ = ⟨Ω | O | Ω⟩

For the actual ground state H | Ω⟩ = E0 | Ω⟩
The exact ground state can be prepared using adiabatic state preparation = longunitary evolution with gradually changing Hamiltonian

−iHt

e

−iHkint/N −iHmasst/N

≈ (e

e

N

⋯)

B Chakraborty, M
Honda, T Izubuchi, Y
Kikuchi, AT
Phys.Rev.D 105 (2022)
9, 094503

BUT, Near term quantum devices are only capable to deal with simple (short) circuit!
Variational approaches help to evaluate the ground state to evaluate the expectation value= Variational Quantum Eigen-solver (VQE), a quantum-classical hybrid algorithm
53

VQE and Beta VQE 1/2

Akio Tomiya

Background: VQE is a variational method• Quantum machine: Exact ground state | Ω⟩ preparation is hard. In particular, it isdi cult on near term devices• Variational method for a pure state with a short circuit (VQE, variation quantum
eigen-solver).

• Quantum/Classical hybrid algorithm, iterative. Uθ is a short circuit.
• Parametrized unitary circuit (~parametrized state | θ⟩, θ: a set of parameters)
VQE: Iterative approx
Quantum

| 0⟩
| 0⟩
| 0⟩

ffi

| 0⟩

Uθ[ ⊗ | 0⟩] = | θ⟩UθhkClassical

Expectation value(re) construct⟨H⟩θ =∑k⟨θ | hk | θ⟩minimize ⟨H⟩θ tuning θTuned parameter• Systematic error since | θ⟩ = Uθ[ ⊗ | 0⟩] ≠ | Ω⟩ but cheap

QFT with Hamiltonian

Akio Tomiya

Hamiltonian vs Lagrangian

Operator formalism (This work)

H : Hamiltonian in QFT
Real time
Minkowski in M^{d+1}−itH

U(t) = e

Euclid(t → τ)t = − iτ
Minkowski(τ → t)

Finite temperature/imaginary time
Euclid in S^1 x M^d−τH

U(τ) = e⟨OO(τ)⟩ = Tr[O(0)O(τ)ρ]ρ = U(τ)/Z- Thermal state in quantum system?
-> Density matrix formalism
55

Density matrix

Akio Tomiya

uni es description of pure states and mixed states
Pure states: System is purely quantum

ρpure = | Ψ⟩⟨Ψ |⟨O⟩ = Tr[Oρpure] = ⟨Ψ | O | Ψ⟩Mixed states: States are classically mixed (≠ superposition)

ρmixed =∑iwi | ψi⟩⟨ψi |⟨O⟩ = Tr[Oρmixed] =∑i

wi⟨ψi | O | ψi⟩wi ∈ ℝ+ represents probability to nd a pure state | ψi⟩
Thermal states (Grand-canonical):1 − 1 (H−μ̂ N)̂ρT,μ = e TZ⟨O⟩T,μ = Tr[OρT,μ](Alternative approach TPQ: AT Yuki Nagai APLAT, 2020)

Thermal-quantum average in general⟨O⟩ = Tr[Oρ]
fi

fi

56

Density matrix

Akio Tomiya

Quantum version of probability distribution
Thermal-quantum average in general⟨O⟩ = Tr[Oρ]
General Properties of density matrix ρ• It uni es discretions of pure states and mixed states• Normalized as Tr[ρ] = 1• ρ can be regarded as quantum version of probability distribution p(x)•e.g.) S = −<—>∫dx p(x)log p(x) (Shannon entropy)
S = − Tr[ρ log ρ] (Von-Neumann entropy)• Distance between two density matrices = quantum relative entropy (next)fi

57

VQE and Beta VQE 2/2

Akio Tomiya

Beta VQE is a variational method for mixed states• KL divergence for ρ = Kullback–Leibler Umegaki divergence (Pseudo-distance for ρ)•Classical ver: D(p | q) =∫dx p(x)log p(x)/q(x) (KL divergence)• Relative entropy. Di erence of two distributions (~distance)• Positive de nite, Used in machine learning• D=0 if and only if p, q are equal• Quantum D(ρ1 | ρ2) = Tr[ρ1 log ρ1 / ρ2] (KL-Umegaki divergence ~ distance)• Positive de nite
• D=0 if and only if ρ1, ρ2 are equal• Kullback–Leibler Umegaki divergence can be used for variational approaches
Ansatz for ρ?
ff

fi

fi

58

VQE and Beta VQE

Akio Tomiya

Beta VQE is a variational method for mixed states
J. -Guo Liu+ 1902.02663*M. Germain+ 1502.03509• Variational ansatz for thermal quantum system:
•

ρΘ =

∑

{ x }⃗⃗ x ⃗ | Uθ† , Θ = θ ∪ ϕ (parameters)
pϕ[ x ⃗] Uθ | x ⟩⟨x ⃗ = (x1, x2, x3, ⋯, xk, ⋯)⊤, and xk ∈ {0,1} : (roughly) fermion occupation•⊤
tox prepare• | x ⟩⃗ = | x1⟩ ⊗ | x2⟩ ⊗ | x3⟩ ⊗ ⋯ : easy
⃗
=
(x
,
x
,
x
,
⋯,
x
,
⋯)
, and
•
1 2 3
k
: (roughly)
fermionthe{0,1}conventional
VQE• Uθ | x ⟩⃗ : parametrized pure states, similar xtok ∈occupationa con guration of x ,⃗• pϕ[ x ⃗]: Classically approximated distribution• | x ⟩⃗ = | xfor1⟩ ⊗ | x2⟩ ⊗ | x3⟩ ⊗ ⋯ :Neural network (MADE*) is used. ϕ = parameters
Product state⃗xThis
can generate
congurationsof
Neural(Easy to prepare)
Variational
quantum circuit(Entanglement)
fi

59
fi

network(Thermal)Beta VQE

Akio Tomiya

Extended VQE for mixed states
Jin-Guo Liu+ 1902.02663

1
̂
̂
|
We
minimize
ℒ(Θ)
=
D(ρ
ρ
)
−
ln
Z
=
Tr[ρ
ln
ρ
]
+
Tr[ρ
(
H
−
μ
N
)]
Θ T,μ
T,μ
Θ
Θ
Θ
•
T

• Variational bound: ℒ(Θ) ≥ − log ZT,μ• Advantage of beta VQE• No sign problem, even with the chemical potential• Bounded variational approximation• Disadvantage• Systematic error• Need numerical resource if we use a classical machine60

Simulation results

Akio Tomiya

Simulation setup (mostly skip)• We apply beta-VQE for Schwinger model (= QED in 1+1d).

Toy model of QCD, con nement, chiral symmetry breaking

1
S = d x[ − FμνF μν + ψ̄(i∂/ − gA/ − m)ψ]
∫
4
2

1 2
H = dx − iψγ (∂1 + igA1)ψ + mψ ψ + Π
∫ [
2 ]
1

∂x E = g ψ̄ γ 0ψ

• Staggered fermion••

Jordan-Wigner transformation. Open Boundary condition.
g = 1, Nx = (4, 6), 8, 10, 1/T = [0.5-20.0], mu= [0-1.4], 4 lattice spacings 1/2a = [0.5-0.35]
We do not take large volume limit but take continuum limit••(Practically, Nx>10 cannot be calculated on our numerical resources)(My previous work shows data from Nx>12 are essential to take stable large volume limit though)Setup for beta VQE:••

Unitary part = SU(4) ansatz
Classical weight = Masked Auto-Encoder for Distribution Estimation (MADE)

Training epoch is 500. Sampling = 5000 for classical distribution• Observables• Variational free energy (exact and variational one)• (Translationally invariant) Chiral condensate• Check point: Dependence of variational error on temperature and mu
61
fi

•
•
•

Simulation results

Akio Tomiya

Variational free energy is O(1), Nx=10

T

AT. 2205.08860

β = 1/T

μ1.Mild dependence on μ (not fatal
2.Hard for T -> 0 (large deviation) as expected)62

Summary for T-mu

Akio Tomiya

AT. 2205.08860
Fukushima , Hatsuda
Rept.Prog.Phys.74:014001,2011⟨ψ ψ⟩T,μ⟨ψ ψ⟩0.05,0• We investigate T-μ phase diagram for Schwinger model• Continuum extrapolation has been evaluated(except for additive mass renormalization by 2206.05308)• The variational approach does not show di culty for our parameter regime• Towards to go large volume, optimization of code, GPU version, tensor
network. (noise-free) real device!

ffi

63

Summary
1.What and why QCD/lattice QCD?
1.Problem: Long auto-correlation, Sign problem
2. Lattice QCD + Machine learning
1.Trainable smearing + SLHMC = adaptive reweighting
3. Lattice QCD + Quantum algorithm
1.Sign problem + non-unitary -> classical/quantum hybrid!

(t)dUμ (n)
𝒢

dt

=

(t)
(Uμ (n))θ̄Congratulations again, Onogi-san!

64

65

Self-introduction

Akio Tomiya

Lattice QCD & Machine learning
What/who am I?
I am a particle physicist, working on lattice QCD.
I want to apply machine learning + quantum alg. on it.
My papers

https://scholar.google.co.jp/citations?user=LKVqy_wAAAAJ

Phase transition detection with NN

Axial anomaly at T>0 with Mobius Domain-wall fermionsPhase diagram via Quantum/Classical algorithm

Biography
2010 - 2015 : Osaka university (Master& PhD)
2015 - 2018 : Postdoc in CCNU (Wuhan, China)
2018 - 2021 : SPDR in RIKEN/BNL (Brookhaven, US)
2021 : Faculty in IPUT Osaka
KAKENHI (Grants-in-Aid for Scienti c Research)
PI: Grant-in-Aid for Transformative Research Areas (A)Grant-in-Aid for Early-Career Scientists
CI: Grant-in-Aid for Scienti c Research (C), etc
fifi

https://cometscome.github.io/DLAP2020/66

Details (skip)

Akio Tomiya

Network: trainable stout (plaq+poly)
arXiv: 2103.11965

All ρ is weight
O meas an loop operator

Structure of NN(Polyakov loop+plaq
in the stout-type)

TA: Traceless, anti-hermitian operation

Uμ(l+1)(n) = exp(Qμ(l)(n))Uμ(l)(n)UμNN(n)[U] = Uμ(2)(n) Uμ(1)(n)[Uμ(n)][]Neural network
Parametrized action:

Sθ[U] = Sg[U] + Sf[ϕ, UθNN[U]; mh = 0.4],
2
1Lθ[U] =
Sθ[U, ϕ] − S[U, ϕ] ,2

Loss function:

2- layered stout
with 6 trainable parameters

Action for MD is built bygauge covariant NN

Invariant under,rot, transl, gauge trf.

Training strategy: 1.Train the network in prior HMC (online training+stochastic gr descent)2.Perform SLHMC with xed parameter

fi

Gauge covariant neural network and full QCD simulation

67

Details (skip)

Akio Tomiya

Results: Loss decreases along with the training
2

1
Lθ[U] =
Sθ[U, ϕ] − S[U, ϕ] ,2

Loss function:

arXiv: 2103.11965

Intuitively, e^(-L) is understood as
Boltzmann weight or reweighting factor.

Prior HMC run (training)

Training history

Ω: sum of un-traced loops
C: one U removed ΩΛ: A polynomial of U. (Same object in stout)

Without training, e^(-L)<< 1,
this means that candidate with approximated action
never accept.
After training, e^(-L) ~1, and we getpractical acceptance rate!
We perform SLHMC with these values!
Gauge covariant neural network and full QCD simulation

68

Gauge covariant neural network

Akio Tomiya

Training can be done with (extended) back propagation
AT Y. Nagai arXiv: 2103.11965

Gauge inv. loss function can be constructed by gauge invariant actions

S NN[U] = S [UμNN(n)[U]]Loss function

Lθ[U] = f (SS: gauge action or fermion action[U])f : mean-square for example,NNmini-batch(c.f. Behler-Parrinello type neural net)

Training: We can use “gradient descent” (also “Adam” (adaptive-momentum) is applicable)
Repeat update(until converge)

θ(l)∂Lθ[U]←θ −η∂θ (l)(l)θ (l) is parameters in l-th layer

Example ofGradient descent

Gauge covariant neural network and full QCD simulation

69

Gauge covariant neural network

Akio Tomiya

Training can be done with (extended) back propagation
AT Y. Nagai arXiv: 2103.11965

Gauge inv. loss function can be constructed by gauge invariant actions

S NN[U] = S [UμNN(n)[U]]Loss function

Lθ[U] = f (SS: gauge action or fermion action[U])f : mean-square for example,NNmini-batch(c.f. Behler-Parrinello type neural net)

Training: We can use “gradient descent” (also “Adam” (adaptive-momentum) is applicable)
Repeat update(until converge)

θ(l)∂Lθ[U]←θ −η∂θ (l)(l)θ (l) is parameters in l-th layer

The second term requires the chain rule for matrix elds, we developed extended delta rule:∂Lθ[U] ∂L ∂f ∂S NN ∂U (l+1) ∂z (l+1)=∂θ (l)∂f ∂S NN ∂U (l+1) ∂z (l+1) ∂θ (l)This matrix derivative is common to the stout force(namely well known)fiGauge covariant neural network and full QCD simulation70

Simulation results

Akio Tomiya

Continuum extrapolation for Nx = 8, 10
AT. 2205.08860

Continuum limit with a polynomial ansat
it looks good So far*
Nx = 8Nx = 10*(I did not include additive mass shift (Ross Dempsey+ arXiv: 2206.05308)
I thank to Takis Angelides (DESY) and Etsuko Itou (RIKEN) for letting me know this important reference!)

We use Nx = 10 results for the phase diagram
.

z

71

VQE and Beta VQE 2/2

Akio Tomiya

Beta VQE is a variational method for mixed states
J. -Guo Liu+ 1902.02663*M. Germain+ 1502.03509• Variational method for mixed states: Variational method on ρ
•

ρΘ =

∑

{ x }⃗⃗ x ⃗ | Uθ† , Θ = θ ∪ ϕ (parameters)
pϕ[ x ⃗] Uθ | x ⟩⟨x ⃗ = (x1, x2, x3, ⋯, xk, ⋯)⊤, and xk ∈ {0,1} : (roughly) fermion occupation•• | x ⟩⃗ = | x1⟩ ⊗ | x2⟩ ⊗ | x3⟩ ⊗ ⋯ : easy to prepare• Uθ | x ⟩⃗ : parametrized pure states, similar to the conventional VQE• pϕ[ x ⃗]: Classically approximated distribution for a con guration of x ,⃗Neural network (MADE*) is used. ϕ = parameters
This can generate con gurations of x ⃗

fi

fi

72

MADE?

Akio Tomiya(masked) Auto-encoder for binary variable distribution• MADE (neural network) mimics joint probability distribution e.g.
p(x1, x2, x3) , whose input is binary array (x1, x2, x3), xi = 0, 1

Reconstructed MNIST (Binarized)

Auto-encoder with a mask -> Generative model for binary array(Please ask me later in detail)*M. Germain+ 1502.03509

73

Beta VQE

Akio Tomiya

Extended VQE for mixed states
Jin-Guo Liu+ 1902.02663

NN

VQE

1 − 1 (H−μ̂ N)̂T
⃗ x ⃗ | Uθ†
pϕ[ x ⃗] Uθ | x ⟩⟨We approximate ρ = eby ρΘ =∑•
Z{ x }⃗•⟨O⟩T,μ ≈ Tr[ρΘO] =∑{ x }⃗

pϕ[ x ⃗] ⟨ x ⃗ | Uθ†OUθ | x ⟩⃗• Quantum machine can store a state Uθ | x ⟩⃗ (test wave function)• Classical machine can sample thermal distribution from pϕ[ x ⃗] (neural net)
• All parameters are tuned such that minimizing D(ρΘ | ρ)
• Optimization of parameters is done with a optimizer (as in machine learning)74Motivation

Akio Tomiya

Sign problem prevents using Monte-Carlo• Monte-Carlo enables us to evaluate expectation values for “statisticalsystem”, like lattice QCD in imaginary time
1

Nconf

Nconf ∑c

⟨O[U]⟩ =O[Uc] +(

Nconf )
1

1 −S[U]Uc ← P(U) = e∈ ℝ+
ZGreat successes!
Sign problem

μarXiv:0906.3599• If we turn on the baryon chemical potential μ, Monte-Carlo methods do notwork because e −S[U] becomes complex. This is no more probability. (signproblem)• Operator formalism does not have such problem! But it requires huge memory
to store quantum states, which cannot be realized even on supercomputer.
𝒪• Quantum states should be stored on quantum device (Feynman)

Motivation

Akio Tomiya

μ = 0 is good for Classical, T=0 is good for Quantum
Classical machine: Lattice eld theory calculations rely on

1 −S[U]
P(U) = edet(D[U] + m)2 ∈ ℝ+
Z

Since 1980 (M. Creutz)~- This P(U) cannot be regarded as probability if μ ≠ 0 (sign problem)
Quantum machines can realize (any) unitary evolutions (Solovay Kitaev theorem),U(t) = e−iHt̂

Phys.Rev.D 105 (2022) 9, 09450
and references therein- No problem for μ≠0 because we can only use unitary gates (operators)- “Short time evolution” (shallow circuit) is preferred for near-term devices*

We need a method to calculate T>0 and μ≠0 for QCD
and for near-term quantum devices
3fi*https://indico.hiskp.uni-bonn.de/event/40/contributions/484/attachments/358/630/Powers%20Talk%20Final%20Draft%20Updated.pdf

76Summary of this talk

Akio Tomiya

Hybrid = Quantum algorithm + machine learning
AT arXiv: 2205.08860

Fukushima , Hatsuda
Rept.Prog.Phys.74:014001,2011

I investigated T-mu phase diagram using a quantum algorithm & neural network(β-VQE, No sign problem) for Schwinger model (toy model of QCD)
77QFT with Hamiltonian

Akio Tomiya

Hamiltonian vs Lagrangian

Operator formalism (This work)

H : Hamiltonian in QFT
Real time
Minkowski in M^{d+1}

U(t) = e−itH

Euclid(t → τ)t = − iτ
Minkowski(τ → t)

Finite temperature/imaginary time
Euclid in S^1 x M^d−τH

U(τ) = e⟨OO(τ)⟩ = Tr[O(0)O(τ)ρ]ρ = U(τ)/Z- Typical use case of quantum algorithm is for real-time. Unitary.
- Time evolution: Correlators (e.g. 2pt on light-cone), etc- Main interest: ⟨Ω | O | Ω⟩, where | Ω⟩ is the exact ground state- Di culty: State preparation for exact ground state of H

ffi

78

State preparation, VQE and Beta-VQE

Akio Tomiya

State preparation is hard
We are interested in expectation value with true ground state for Hamiltonian⟨O⟩ = ⟨Ω | O | Ω⟩

For the actual ground state H | Ω⟩ = E0 | Ω⟩
The exact ground state can be prepared using adiabatic state preparation = longunitary evolution with gradually changing Hamiltonian

−iHt

e

−iHkint/N −iHmasst/N

≈ (e

e

N

⋯)

B Chakraborty, M
Honda, T Izubuchi, Y
Kikuchi, AT
Phys.Rev.D 105 (2022)
9, 094503

BUT, Near term quantum devices are only capable to deal with simple (short) circuit!
Variational approaches help to evaluate the ground state to evaluate the expectation value= Variational Quantum Eigen-solver (VQE), a quantum-classical hybrid algorithm
79

VQE and Beta VQE 1/2

Akio Tomiya

Background: VQE is a variational method• Quantum machine: Exact ground state | Ω⟩ preparation is hard. In particular, it isdi cult on near term devices• Variational method for a pure state with a short circuit (VQE, variation quantum
eigen-solver).

• Quantum/Classical hybrid algorithm, iterative. Uθ is a short circuit.
• Parametrized unitary circuit (~parametrized state | θ⟩, θ: a set of parameters)
VQE: Iterative approx
Quantum

| 0⟩
| 0⟩
| 0⟩

ffi

| 0⟩

Uθ[ ⊗ | 0⟩] = | θ⟩UθhkClassical

Expectation value(re) construct⟨H⟩θ =∑k⟨θ | hk | θ⟩minimize ⟨H⟩θ tuning θTuned parameter• Systematic error since | θ⟩ = Uθ[ ⊗ | 0⟩] ≠ | Ω⟩ but cheap

QFT with Hamiltonian

Akio Tomiya

Hamiltonian vs Lagrangian

Operator formalism (This work)

H : Hamiltonian in QFT
Real time
Minkowski in M^{d+1}−itH

U(t) = e

Euclid(t → τ)t = − iτ
Minkowski(τ → t)

Finite temperature/imaginary time
Euclid in S^1 x M^d−τH

U(τ) = e⟨OO(τ)⟩ = Tr[O(0)O(τ)ρ]ρ = U(τ)/Z- Thermal state in quantum system?
-> Density matrix formalism
81

Density matrix

Akio Tomiya

uni es description of pure states and mixed states
Pure states: System is purely quantum

ρpure = | Ψ⟩⟨Ψ |⟨O⟩ = Tr[Oρpure] = ⟨Ψ | O | Ψ⟩Mixed states: States are classically mixed (≠ superposition)

ρmixed =∑wi | ψi⟩⟨ψi |⟨O⟩ = Tr[Oρmixed] =iwi represents probability to nd a pure state | ψi⟩∑iwi⟨ψi | O | ψi⟩

Thermal states (Grand-canonical):1 − 1 (H−μ̂ N)̂ρT,μ = e TZ⟨O⟩T,μ = Tr[OρT,μ](Alternative approach TPQ: AT Yuki Nagai APLAT, 2020)

Thermal-quantum average in general⟨O⟩ = Tr[Oρ]
fi

fi

82

Density matrix

Akio Tomiya

Quantum version of probability distribution
Thermal-quantum average in general⟨O⟩ = Tr[Oρ]
General Properties of density matrix ρ• It uni es discretions of pure states and mixed states• Normalized as Tr[ρ] = 1• ρ can be regarded as quantum version of probability distribution p(x)•e.g.) S = −<—>∫dx p(x)log p(x) (Shannon entropy)
S = − Tr[ρ log ρ] (Von-Neumann entropy)• Distance between two density matrices = quantum relative entropy (next)fi

83

VQE and Beta VQE 2/2

Akio Tomiya

Beta VQE is a variational method for mixed states• KL divergence for ρ = Kullback–Leibler Umegaki divergence (Pseudo-distance for ρ)•Classical ver: D(p | q) =∫dx p(x)log p(x)/q(x) (KL divergence)• Relative entropy. Di erence of two distributions (~distance)• Positive de nite, Used in machine learning• D=0 if and only if p, q are equal• Quantum D(ρ1 | ρ2) = Tr[ρ1 log ρ1 / ρ2] (KL-Umegaki divergence ~ distance)• Positive de nite
• D=0 if and only if ρ1, ρ2 are equal• Kullback–Leibler Umegaki divergence can be used for variational approaches
Ansatz for ρ?
ff

fi

fi

84

VQE and Beta VQE 2/2

Akio Tomiya

Beta VQE is a variational method for mixed states
J. -Guo Liu+ 1902.02663*M. Germain+ 1502.03509• Variational method for mixed states: Variational method on ρ
•

ρΘ =

∑

{ x }⃗⃗ x ⃗ | Uθ† , Θ = θ ∪ ϕ (parameters)
pϕ[ x ⃗] Uθ | x ⟩⟨x ⃗ = (x1, x2, x3, ⋯, xk, ⋯)⊤, and xk ∈ {0,1} : (roughly) fermion occupation•• | x ⟩⃗ = | x1⟩ ⊗ | x2⟩ ⊗ | x3⟩ ⊗ ⋯ : easy to prepare• Uθ | x ⟩⃗ : parametrized pure states, similar to the conventional VQE• pϕ[ x ⃗]: Classically approximated distribution for a con guration of x ,⃗Neural network (MADE*) is used. ϕ = parameters
This can generate con gurations of x ⃗

fi

fi

85

MADE?

Akio Tomiya(masked) Auto-encoder for binary variable distribution• MADE (neural network) mimics joint probability distribution e.g.
p(x1, x2, x3) , whose input is binary array (x1, x2, x3), xi = 0, 1

Reconstructed MNIST (Binarized)

Auto-encoder with a mask -> Generative model for binary array(Please ask me later in detail)*M. Germain+ 1502.03509

86

Beta VQE

Akio Tomiya

Extended VQE for mixed states
Jin-Guo Liu+ 1902.02663

1 − 1 (H−μ̂ N)̂T
⃗ x ⃗ | Uθ†
pϕ[ x ⃗] Uθ | x ⟩⟨We approximate ρ = eby ρΘ =∑•
Z{ x }⃗•⟨O⟩T,μ ≈ Tr[ρΘO] =∑{ x }⃗

pϕ[ x ⃗] ⟨ x ⃗ | Uθ†OUθ | x ⟩⃗• Quantum machine can store a state Uθ | x ⟩⃗ (test wave function)• Classical machine can sample thermal distribution from pϕ[ x ⃗] (neural net)
• All parameters are tuned such that minimizing D(ρΘ | ρ)
• Optimization of parameters is done with a optimizer (as in machine learning)87

Beta VQE 4/4

Akio Tomiya

Extended VQE for mixed states
Jin-Guo Liu+ 1902.02663

1
̂ − μN̂ )]We
minimizetheloss
function
ℒ(Θ)
=
D
−
ln
Z
=
Tr[ρ
ln
ρ
]
+
Tr[ρ
(
H
Θ
Θ
Θ
•
T

• Variational bound: ℒ(Θ) − log ZT,μ ≥ 0• We use SU(4) ansatz for each 2 qubits for Uθ
• Advantage of beta VQE• No sign problem, even with the chemical potential• Bounded variational approximation• Disadvantage• Systematic error• Need numerical resource if we use a classical machine
88Simulation results

Akio Tomiya

Simulation setup (mostly skip)• We apply beta-VQE for Schwinger model (= QED in 1+1d).

Toy model of QCD, con nement, chiral symmetry breaking

1
S = d x[ − FμνF μν + ψ̄(i∂/ − gA/ − m)ψ]
∫
4
2

1 2
H = dx − iψγ (∂1 + igA1)ψ + mψ ψ + Π
∫ [
2 ]
1

∂x E = g ψ̄ γ 0ψ

• Staggered fermion••

Jordan-Wigner transformation. Open Boundary condition.
g = 1, Nx = (4, 6), 8, 10, 1/T = [0.5-20.0], mu= [0-1.4], 4 lattice spacings 1/2a = [0.5-0.35]
We do not take large volume limit but take continuum limit••(Practically, Nx>10 cannot be calculated on our numerical resources)(My previous work shows data from Nx>12 are essential to take stable large volume limit though)Setup for beta VQE:••

Unitary part = SU(4) ansatz
Classical weight = Masked Auto-Encoder for Distribution Estimation (MADE)

Training epoch is 500. Sampling = 5000 for classical distribution• Observables• Variational free energy (exact and variational one)• (Translationally invariant) Chiral condensate• Check point: Dependence of variational error on temperature and mu
89
fi

•
•
•

Simulation results

Akio Tomiya

Variational free energy is O(1), Nx=10

T

AT. 2205.08860

β = 1/T

μ1.Mild dependence on μ (not breaking
2.Hard for T -> 0 (large deviation) as expected)

90

Simulation results

Akio Tomiya

Continuum extrapolation for Nx = 8, 10
AT. 2205.08860

Continuum limit with a polynomial ansat
it looks good So far*
Nx = 8Nx = 10*(I did not include additive mass shift (Ross Dempsey+ arXiv: 2206.05308)
I thank to Takis Angelides (DESY) and Etsuko Itou (RIKEN) for letting me know this important reference!)

We use Nx = 10 results for the phase diagram
.

z

91

Summary for T-mu

Akio Tomiya

AT. 2205.08860
Fukushima , Hatsuda
Rept.Prog.Phys.74:014001,2011⟨ψ ψ⟩T,μ⟨ψ ψ⟩0.05,0• We investigate T-μ phase diagram for Schwinger model• Continuum extrapolation has been evaluated (except for additive mass
renormalization by 2206.05308)• The variational approach does not show di culty for our parameter regime• Towards to go large volume, optimization of code, GPU version, tensor
network. (noise-free) real device!

ffi

92

93

QFT with Hamiltonian

Akio Tomiya

Same hamiltonian
H : Hamiltonian for QFT)
Finte temperature

Real time
To Euclid (t → τ)U(t) = e−itHt = − iτOperator formalism

perturbation

To Minkowski (τ → t)

⟨OO(t)⟩ = ⟨Ω | ̂ O(0)O(t) | Ω⟩U(τ) = e−τH

⟨OO(τ)⟩ = Tr[O(0)O(τ)ρ]ρ = U(τ)/Z| Ω⟩ ∼ lim U(t) | 0⟩path intZ=
perturbation

M

S =

∫
∞

Path int

iS M

ψ̄ ψe

dt d x ℒ (x, t)
∫−∞ ∫
d

M

This work

Z=
SE =

∫

ψ̄ ψe

∫0

dτ d d x ℒE(x, τ)
∫

1/T

−S E

Conventional QFT

𝖳𝒟

ψ(τ + 1/T, x ⃗) = − ψ(τ, x ⃗)

𝒟

𝒟

𝒟

Fermion has Anti-PBC for imaginary time direction.
This is necessary to connect get trace formulain the operator formalism

94Simulation results

Akio Tomiya

Variational free energy is O(1), Nx=10
ApproxExact~1/aApproxAT. 2205.08860

Exact

1.Mild dependence on
2.Hard for T -> 0 (large deviation) as expected
95
μ

~1/a

SU(4)

Akio Tomiya

Variational ansatz
J. -Guo Liu+ 1902.02663

The general gate consists of 15 single qubit gates and 3 CNOT gates.
Each two qubit unitary is parametrized by 15 parameters in the rotational gates, whichparametrizes the SU(4) group.

96

VQE and Beta VQE 1/2

Akio Tomiya

Background: VQE is a variational method• Quantum machine: Exact ground state preparation is hard. In particular, it isdi cult on near term devices• Variational method for a pure state with a short circuit (VQE, variation quantum
eigen-solver).

• Quantum/Classical hybrid algorithm, iterative• Parametrized unitary circuit (~parametrized state | θ⟩, θ: a set of parameters)
̂
̂
|
|
|
|
θ⟩
=
U(θ)
0⟩
0⟩
0⟩
⋯
U(θ)
,
and
is a short circuit (entanglement +( 1 2 3 )•rotations)• If ⟨θ | H | θ⟩ = 0, | θ⟩ ≈ | Ω⟩, where | Ω⟩ is the exact ground state= Variational approach for quantum systemffi•

VQE and Beta VQE (skip)

Akio Tomiya

Beta VQE is a variational method for mixed states• Variational method for mixed states: Variational method on ρ
•

ρΘ =

∑

{ x }⃗

J. -Guo Liu+ 1902.02663*M. Germain+ 1502.03509⃗ x ⃗ | Uθ† , Θ = θ ∪ ϕ (parameters)
pϕ[ x ⃗] Uθ | x ⟩⟨x ⃗ = (x1, x2, x3, ⋯, xk, ⋯)⊤, and xk ∈ {0,1} : (roughly) fermion excitation•• Uθ | x ⟩⃗ : parametrized pure states, similar to the conventional VQE• pϕ[ x ⃗]: Classically approximated distribution for a con guration of x ,⃗Neural network (MADE*) is used. ϕ = parametersexact), we get approximated a set of states (= thermal)
Minimizing D(ρΘ | ρT,μ•• Shifted one (by a constant) is used in practice:

98
fi

•

1
exact
̂
ℒ(Θ) ≡ D(ρΘ | ρT,μ ) − ln Z = Tr[ρΘ ln ρΘ] + Tr[ρΘ(Ĥ − μN)]
T
⏟
const

The two language problem and solution?

99

Akio Tomiya

The two language problem and solution?
• Programs for machine learning are usually implemented in Python• LatticeQCD is in C++ (+CUDA)• Two di erent languages used = “2 (programming) language problem”• Use of one language is better for productivity• Python + LQCD: GPT for Grid, PyBridge++ for Bridge++, PyQCD• Julia language* could be a solution of the problem• High performance as C++, Write like Python• NASA uses Julia 😲. Works on supercomputers• Machine learning, GPU and MPI friendly (Flux.jl, CUDA.jl, MPI.jl etc)
• LatticeQCD.jl, AT & Y. Nagai (updating to 1.0):

MPI-Parallel, stout smearing, domain-wall, staggered, (R)HMC, improved gauge
actions, SU(Nc), gauge-covariant-neural net, ILDG support, etc…

* introduction: https://www.youtube.com/c/TheJuliaLanguage/playlists, https://akio-tomiya.github.io/julia_in_physics/ (japanese)
Akio Tomiya
ff100The two language problem and solution?
• Programs for machine learning are usually implemented in Python• LatticeQCD is in C++ (+CUDA)• Two di erent languages used = “2 (programming) language problem”• Use of one language is better for productivity• Python + LQCD: GPT for Grid, PyBridge++ for Bridge++, PyQCD• Julia language* could be a solution of the problem• High performance as C++, Write like Python• NASA uses Julia 😲. Works on supercomputers• Machine learning, GPU and MPI friendly (Flux.jl, CUDA.jl, MPI.jl etc)
• LatticeQCD.jl, AT & Y. Nagai (updating to 1.0):

MPI-Parallel, stout smearing, domain-wall, staggered, (R)HMC, improved gauge
actions, SU(Nc), gauge-covariant-neural net, ILDG support, etc…

* introduction: https://www.youtube.com/c/TheJuliaLanguage/playlists, https://akio-tomiya.github.io/julia_in_physics/ (japanese)

Akio Tomiya

ff

101

Beta VQE 2/4

Akio Tomiya

Extended VQE for mixed states

1 − 1 (H−μ̂ N)̂T
• How can we realize ρΘ ≈ ρ for ρ = Z e

Jin-Guo Liu+ 1902.02663• Minimize Kullback–Leibler–Umegaki divergence (pseudo-distance)
ρΘ
|
D(ρ
ρ)
=
Tr[ρ
ln
]
=
Tr[ρ
ln
ρ
]
−
Tr[ρ
ln
ρ]
Θ
Θ
Θ
Θ
Θ
•
ρ
• Relative entropy for density matrices (Classical ver. is called KL div.)• This is bounded D(ρΘ | ρ) ≥ 0 and saturated i ρΘ = ρ• In practice, we minimize shifted one,1̂ℒ(Θ) = D(ρΘ | ρ) − ln Z = Tr[ρΘ ln ρΘ] + Tr[ρΘ(Ĥ − μN)]
T
⏟
const

ff

102

Note (skip)

Akio Tomiya

ρT,μ =

1 − 1 (H−μ
̂ N̂ )
e T
ZT,μ

T,μ

(const in Θ)

103Beta VQE 3/4

Akio Tomiya

Extended VQE for mixed states
Jin-Guo Liu+ 1902.02663

1
̂
̂
ℒ(Θ)
=
Tr[ρ
ln
ρ
]
+
Tr[ρ
(
H
−
μ
N)]
Θ
Θ
Θ
•
T

•

Tr[ρΘ log ρΘ] =∑{ x }⃗pϕ( x ⃗)log pϕ( x ⃗)• We need two derivatives∂∂ℒ(Θ) =pϕ( x ⃗)[log pϕ( x ⃗)] : Classical
• ∂ϕ
∂ϕ {∑
p: a neural network
x }⃗
∂
1 ∂
†
⃗
⃗
|
|
: Quantum
ℒ(Θ)
=
⟨
x
U
ℋU
x
⟩]
θ
• ∂θ
θ
T ∂θ

-> gradient descent

REINFORCE algorithm

104

MADE: Masked Auto-encoder for Distribution Estimation
1502.03509

I (mostly) skip this section in the seminar

105

arXiv: 1502.03509

Summary of MADE

(simple) Neural network for probability estimation• MADE = Masked Auto-encoder for Distribution Estimation• Auto-encoder is a neural network• It can mimic a joint distribution of binary variables (0, 1)• (x1, x2, x3, x4) is distributed as p(x1, x2, x3, x4) ≡ p[ x ⃗]• It is categorized as a generative model (as the normalizing ow)

fl• It is correctly normalizedarXiv: 1502.03509

Basics (skip)
Product rule in the probability theory• A con guration of variables (x1, x2, x3, x4) is distributed asp(x1, x2, x3, x4) ≡ p[ x ⃗]• Probability distribution is normalized.
• For binary variables,
1=

1

1

1

1

∑∑∑∑

fi

x1=0 x2=0 x3=0 x4=0p(x1, x2, x3, x4) =∑{ x }⃗

pϕ[ x ⃗]

arXiv: 1502.03509

Basics (skip)
Product rule in the probability theoryp(x1, x2)
• de nition of the conditional probability is p(x2 | x1) ≡ p(x1)• Equivalently, p(x1, x2) = p(x1)p(x2 | x1) : Product rule
• We can generalize to more than 2 variablesp(x1, x2, x3)
|
|
|
p(x
x
,
x
)
=
⇔
p(x
,
x
,
x
)
=
p(x
x
,
x
)p(x
x
)p(x
)
3
1
2
1
2
3
3
1
2
2
1
1
•
p(x1, x2)

• p(x1, x2, x3, x4) = p(x1)p(x2 | x1)p(x3 | x1, x2)p(x4 | x1, x2, x3)fi•

We abbreviate this as p(x1, x2, x3, x4) =

4

∏
k=1

p(xk | x<k)Bernoulli process (skip)

arXiv: 1502.03509

un-fair coin
• A (un-)fair coin, which takes face for a probability p, Tail for 1-p• This process is called “Bernoulli trial” in Math
• Let us denote it as Bernoulli(p)Basics (skip)

arXiv: 1502.03509

Product rule in the probability theory• Neural network (NN) mimics
p(x1, x2, x3) = p(x1)p(x2 | x1)p(x3 | x1, x2), whose input is binary
array (x1, x2, x3): 3 correlated coins• We can draw a sample using x1̂ ∼ y1 ≈ p(x1)• How can we construct x2̂ ∼ y2 ≈ p(x2 | x1)• input only depends on x1• How can we construct x3̂ ∼ y3 ≈ p(x3 | x1, x2)• input only depends on x1, x2

arXiv: 1502.03509

Auto-encoder (skip)
Auto-encoder ~ (un normalized) ow

x2
x3

y1

−E[x] =y2y3

x log yi + (1 − xi)log(1 − yi)∑ iie −E[x] =

∏
i

∑
{x}

fl

x1

e

yi−xi(1 − yi)−(1−xi)

−E[x]

≠1

Not-normalizedarXiv: 1502.03509

Auto-regressive property (skip)
Product ruley1 = p(x1 = 1), y2 = p(x2 = 1 | x1), y3 = p(x3 = 1 | x1, x2)

p(x1 = 0) = 1 − y1, p(x2 = 0 | x1) = 1 − y2, p(x3 = 0 | x1, x2) = 1 − y3yd = p(xd = 1 | x<d)p(x1, x2, x3, x4) =

4

∏
k=1

p(xd = 0 | x<d) = 1 − ydp(xk | x<k)−log p(x1, x2, x3, x4) = −

4

∑
k=1

log p(xk | x<k)

arXiv: 1502.03509

MADE (skip)

Masked auto-encoder for density estimation
x1
x2
x3

1
3
1
1
2
2

2

Assign numbers on node:
Input& output node = assign

3 y1

x1̂ ∼ y1 ≈ p(x1 | x2, x3)

1 y2

x2̂ ∼ y2 ≈ p(x2)2 y3x3̂ ∼ y3 ≈ p(x3 | x2)

arXiv: 1502.03509

MADE (skip)

Masked auto-encoder for density estimation
x1
x2
x3

1
3
1
1
2
2

2

3 y1

x1̂ ∼ y1 ≈ p(x1 | x2, x3)

1 y2

x2̂ ∼ y2 ≈ p(x2)2 y3x3̂ ∼ y3 ≈ p(x3 | x2)

arXiv: 1502.03509

MADE (skip)

Masked auto-encoder for density estimation
x1
x2
x3

1
3
1
1
2
2

2

3 y1

x1̂ ∼ y1 ≈ p(x1 | x2, x3)

1 y2

x2̂ ∼ y2 ≈ p(x2)2 y3x3̂ ∼ y3 ≈ p(x3 | x2)

arXiv: 1502.03509

MADE (skip)

Masked auto-encoder for density estimation
x1
x2
x3

1
3
1
1
2
2

2

3 y1

x1̂ ∼ y1 ≈ p(x1 | x2, x3)

1 y2

x2̂ ∼ y2 ≈ p(x2)2 y3x3̂ ∼ y3 ≈ p(x3 | x2)

arXiv: 1502.03509

MADE (skip)

Masked auto-encoder for density estimation
x1
x2
x3

1
3
1
1
2
2

2

3 y1

x1̂ ∼ y1 ≈ p(x1 | x2, x3)

1 y2

x2̂ ∼ y2 ≈ p(x2)2 y3x3̂ ∼ y3 ≈ p(x3 | x2)

arXiv: 1502.03509

MADE (skip)

Masked auto-encoder for density estimation
x1
x2
x3

1
3
1
1
2
2

2

3 y1

x1̂ ∼ y1 ≈ p(x1 | x1, x3)

1 y2

x2̂ ∼ y2 ≈ p(x2)2 y3x3̂ ∼ y3 ≈ p(x3 | x2)

arXiv: 1502.03509

MADE (skip)

Masked auto-encoder for density estimation
x1
x2
x3

1
3
1
1
2
2

2

3 y1 ≈ p(x1 | x2, x3)

x1̂ ∼ y1(x2, x3)

1 y2 ≈ p(x2)x2̂ ∼ y2

2 y3 ≈ p(x3 | x2)

x3̂ ∼ y3(x2)

We can draw a set of sample (x1̂ , x2̂ , x3̂ ) from pϕ(x1, x2, x3) where ϕ is network param.