Neural Canonical Transformations
Lei Wang (王磊)https://wangleiphy.github.io

Institute of Physics, CAS

Hamiltonian dynamics
Hamiltonian equations·p = − ∂H∂q·q = + ∂H∂p

Hamiltonian dynamics
Hamiltonian equations·p = − ∂H∂q·q = + ∂H∂pPhase space variables

x = (p, q)
Symplectic metric

I
J=(
−I )

Hamiltonian dynamics
Hamiltonian equations·p = − ∂H∂q·q = + ∂H∂pPhase space variables

Symplectic gradient flow

x = (p, q)·x = ∇ H(x)Jx

Symplectic metric

I
J=(
−I )

Hamiltonian dynamics
Hamiltonian equations·p = − ∂H∂qPhase space variables

Symplectic gradient flow

x = (p, q)·x = ∇ H(x)Jx

Symplectic metric·q = + ∂H
∂p

I
J=(
−I )
1815 × 2646

Neptune

m4 = 0.0000517759138449

11.4707666−25.7294829−10.8169456

explicit Euler, h = 10
0.00288930
0.00114527
0.00039677

Symplectic Integrators
8

Pluto

m5 = 1/ (1.3 · 10 )−15.5387357−25.2225594
P−3.1902382

symplectic
Euler,= 100
implicit
Euler,
h =h10

explicit Euler, h = 10

P

J

S
U

P
N

symplectic Euler, h = 100

P

J

S

0.00276725
S
J−0.00170702
U−0.00136504
N

PJ

J S
U

N

S
U

N

Störmer–Verlet, h = 200
Fig. 2.4. Solutions of the outer solar system

To this system we apply the explicit and implic
S
J
from
Haireretal,
Geometrich = P10, the symplectic
EulerNumericaland the IntegrationStörmer–V

Canonical Transformations
x = (p, q)
which satisfies

Change of variables( ∇x z) J ( ∇x z) = JTz = (P, Q)
symplectic condition

Canonical Transformations
x = (p, q)

Change of variables

which satisfies( ∇x z) J ( ∇x z) = J

one has·z = ∇ K(z)Jz

Tz = (P, Q)
symplectic condition

where

K(z) = H ∘ x(z)Preserves Hamiltonian dynamics in the “latent phase space”

Canonical transformation for Gutzwiller, RMP, ’98
Moon-Earth-Sun 3-body problem

Charles Delaunay↑
More than 1800 pages of this, ~20 years of efforts (1846-1867)

A statistical physics perspective•

Canonical transformation deforms phase space density ρ(x) = e•

Symplectic condition => Jacobian determinant = 1•

Liouville theorem: incompressible flow in phase space−βH(x)Example: Cartesian ↔ Polar(px, py, x, y)

1 2
2
H = (px + py )2
+V(x, y)(pr, pφ, r, φ)

1
1
2
2
K=
pr + 2 pφ)2(r+V(r, φ)How to design “useful” canonical transformations ?

Neural Canonical Transformations

If the mapping f is 1-to-1, then the total area (or volume) should
not change after the transformation from x to z.

Neural transformation in 1d
latent space“neural net”

𝒩(z)

ρ(x)

physicalspace

12

Neural transformations in higher dims

WaveNet 1609.03499 1711.10433
https://deepmind.com/blog/wavenet-generative-model-raw-audio/
https://deepmind.com/blog/high-fidelity-speech-synthesis-wavenet/

Glow 1807.03039
https://blog.openai.com/glow/

Neural transformations in higher dims

WaveNet 1609.03499 1711.10433
https://deepmind.com/blog/wavenet-generative-model-raw-audio/
https://deepmind.com/blog/high-fidelity-speech-synthesis-wavenet/

Glow 1807.03039
https://blog.openai.com/glow/

Representation Learning

Page 6
Figure 1.2

Goodfellow, Bengio, Courville, http://www.deeplearningbook.org/

Automatic chemical design
Gomez-Bombarelli et al,1610.02415

Hypothesis generation

Learning representation for science

Galaxy evolution
Schawinski et al, 1812.01114

divergence as

Representation learning in physics
Renormalization group

Monte Carlo update

Effective theory emerges upon
Physics happens on a manifold
http://videolectures.netransformation of the variables
Learn neural nets to unfold that manifold
See Hong-Ye’s and Gurtej’s talk tomorrow morning!

ts unbiased rejection sampling of the physical variables.
nsity distillation, in which the training loss is a variational
h could be useful for automatically identifying collective

Neural Renormalization Group
Normalizing flow with multi-scale neural network Li and LW, PRL ‘18

Generative flow

RG flowhigh-level independent collective variables

physical variables

https://github.com/li012589/NeuralRG

Figure1. (a) The
NeuralRG variables;
network stacksand
bijectorsa hierar- w/ learned representation
Identify
collective
fastinto
sampling
chical structure. The solid dots at the bottom are the physical vari-H(p, q)pqphysical

Symplectic
Neural Net

Neural Canonical Transformations
P

K(P, Q) =∑2
2 2
Pk +ωk Qk

k2

QlatentLearn the network parameter and the latent harmonic frequency

Modular design of the symplectic networkz = 𝒯(x)( ∇x z) J ( ∇x z) = J

𝒯 = 𝒯1 ∘ 𝒯2 ∘ 𝒯3 ∘ ⋯symplectic group

T

Compose symplectic primitives to form a deep neural network

Neural symplectic primitives
Neural coordinate transformationpqSymplectic
Neural Net•

P = p ( ∇Q q)

Q = f(q)invertible
neural net•

Linear transformation: Symplectic Lie algebra•Continuous-time flow: Symplectic generating functions
See also Bondesan, Lamacraft, 1906.04645
Neural ODE, Chen et al, 1806.07366, Monge-Ampère flow, Zhang et al 1809.10188

Training approaches“learn from Hamiltonian”“learn from data”ℒ = dx ρ(x)[ln ρ(x) + βH(x)]∫ℒ = − 𝔼x∼dataset [ln ρ(x)]xz ∼ 𝒩(0, Σ)Sample in the latent spacex ∼ dataset

Symplectic
Neural Net

Density estimation

Symplectic
Neural Net

Variational calculationz

Sample from dataset in the physical space

ADVANCES

IN MATHEMATICS

The

9, 399-407

Superperiod
StringJ. L.

University

TUCK

of California,(1972)of theNonlinear(FPU)

Problem*

Weighted

How is this going to be useful ?

AND

M. T.

MENZELLos Alamos Scientific
Los Alamos, New Mexico 87544

Laboratory,

This paper gives some history of the problem,
and includes
superperiod
datafor quadratic
and cubic nonlinear
terms, together
with a computation
for a prime
numberof particles
in the string. Extensionof the problem
to a circulararray
is discussed,
and there is a bibliography.

Let’s play with examples!

The Maniac I computer(N. Metropolis)
started working
at LosAlamos early in 1952. E. Fermi, who was visiting the Laboratory,J. R. Pasta, and S. M. Ulam entertained themselves by considering what
new problems it opened up for study. One such, in classical fluid theory
in its simplest approximation[ 11, considers a linear array of atoms linked
by nonlinear forces. The results of the first calculations,which werecoded by one of us (M.T.M.),
were so surprisingthat the investigators
were enticed into a study of nonlinear systems generally. The firstsystem to be examined (Fig. I) consisted of a one-dimensionalarrayof mass points linked by light Hookean springs-tension
CC extensionmade nonlinear
by additionof a small term 01(extension) [2] orfl (extension) [3].

Φ
‘i-l

‘itI

‘i
FIGURE

1

ΨAlamos early in 1952. E. Fermi, who was visiting the Laboratory,J. R. Pasta, and S. M. Ulam entertained themselves by considering what
new problems it opened up for study. One such, in classical fluid theory
in its simplest approximation[ 11, considers a linear array of atoms linked
by nonlinear forces. The results of the first calculations,which werecoded by one of us (M.T.M.),
were so surprisingthat the investigators
were enticed into a study of nonlinear systems generally. The firstsystem to be examined (Fig. I) consisted of a one-dimensionalarrayn Hookean springs-tensionof mass points linked by light
CC extension1
2 a small term 2 01(extension) [2] ormade nonlinear H byadditionof=pi + (qi − qi−1) ][fl (extension) [3].
2∑

Example 1: Harmonic Chain

i=1

‘i-l

‘itI

‘i
FIGURE* Workperformed

under

the auspices

1of the U. S. Atomic

EnergyCommission.

Fermi–Pasta–Ulam–Tsingou problem w/o nonlinearity

Learning the normal modes
2.0

(a)

0.2
0.1

r qi Q k

!k

1.5

1.0
πk
ωk = 2 sin( 2n + 2 )0.5

analytical
0.0

(b)

0

8

16

k

24

32

0.0
0.1
k=1
k=2

0.2
0

8

16

24

32

i

Consistency check: neural nets can learn linear coordinate transformations

Example 2: Alanine Dipeptide

250 ns molecular dynamics simulation data at 300 K
https://markovmodel.github.io/mdshare/ALA2/#alanine-dipeptide

More than 3 hours of video …

More than 3 hours of video …

What do biologists see ?
ΦΨslow motion of thetwo torsion angles

stableconformations“Dimensional reduction” to manually designed collective variables

What does the neural net see ?
1.4

(a)

10

mutual information

!k

10

2

1

100

I(Qk : )(b)

I(Qk :

1.2

)

1.0
0.8
0.6
0.4
0.2
0.0

0

6

12

18

k

24

30

1

2

3

4

5

6

k

Unsupervised learning of slow & nonlinear collective variables from data

Latent space interpolationz2
z1

Latent space interpolation

Q2
Q1

Latent space interpolation

Q2 ≈ Φ
Q1 ≈ ΨExample 3: MNIST
Data scientists:“50,000 grayscale
images with 28x28 pixels”

Physical Chemists:“Stable conformationsof a molecule with 784
degrees of freedom”

pq

Symplectic
Neural Nets

Learning slow variables of MNIST
P
Q

K(P, Q) =

∑
k

2
2 2
Pk +ωk Qk

2

q

P

K(P, Q) =∑2
2 2
Pk +ωk Qk

k

2

1.0
0.8

!k

p

Symplectic
Neural Nets

Learning slow variables of MNIST

Q

0.6“Dispersion of
the MNIST”

0.4
0.2
0.0
0

200

400

k

600

800

Conceptual Compression
ω1 < ω2 < ω3 < ω4 < ⋯Encoderfphysical q

ω1
ω2

slow modes, keep

ω3
ω4
⋯

fast modes, throw away
latent Q

Compress by keeping slow collective variables
Kingma et al, 1312.6114 Gregor et al, 1604.08772 Dinh et al, 1605.08803
autoencoders/hierarchical network architecture/hyperbolic latent space…

Neural Alice-Bob game
Aliceslow modes

Encoderf

Bob
Decoderf⋯⋯

throw away

random noises−1?

Conceptual Compression of MNIST
Original

1/784 kept

Conceptual Compression of MNIST
Original

2/784 kept

Conceptual Compression of MNIST
Original

3/784 kept

Conceptual Compression of MNIST
Original

4/784 kept

Conceptual Compression of MNIST
Original

5/784 kept

Conceptual Compression of MNIST
Original

10/784 kept

Conceptual Compression of MNIST
Original

15/784 kept

Conceptual Compression of MNIST
Original

20/784 kept

“A Hamiltonian Extravaganza”—Danilo J. Rezende@DeepMind
Sep 25 ICLR 2020 paper submission deadline
Sep 26 Symplectic ODE-Net, 1909.12077
Sep 27 Hamiltonian Graph Networks with ODE Integrators, 1909.12790
Sep 29 Symplectic RNN, 1909.13334
Sep 30 Equivariant Hamiltonian Flows, 1909.13739
Hamiltonian Generative Network, 1909.13789

http://tiny.cc/hgn

Neural Canonical Transformation with Symplectic Flows, 1910.00024

Hold on tight,
Conceptual leaps!

A reflection on flows

𝒯 = 𝒯1 ∘ 𝒯2 ∘ 𝒯3 ∘ ⋯Change of variables x ↔ z∂zp(x) = p(z) det( ∂x )

Rezende et al, 1505.05770
Dinh et al, 1605.08803
Kingma et al, 1807.03039……

Design principle: efficiency & correct inductive bias
More tractable determinantsMore tractable determinants( ∇x z) J ( ∇x z) = J
T

Autoregressive

Neural RG

Symplectic flow

Continuous-time flows∂ρ(x, t)+ ∇ ⋅ [ρ(x, t)v] = 0∂tSimple density

Complex density

Neural ODE, 1806.07366, 1810.01367 Monge-Ampère flow, Zhang, E, LW, 1809.10188 Hamiltonian flow…

42

Continuous-time flows

fathersfathersof optimaltransport
423 The 3founding
The foundingof optimaltransport

minimize
the total
assumedthat thecost ofcost
oneof oneminimizethe cost.
total Monge
cost. Monge
assumedthattransport
the transport∂ρ(x,t)
unit ofunit
mass
along
a
certaindistance
was givenby thebyproductof theof theof mass along a certain distance
was
given
the
product
+
∇
⋅
ρ(x,
t)v
=
0
[
]
mass bythebydistance.
mass
the distance. ∂t

q(x)
T
x

p(z)
déblais déblais

z

x

T
y

x

y

remblaisremblais

Fig. 3.1.
problemof déblais
and remblais
Fig.Monge’s
3.1. Monge’s
problemof déblais
and remblais

Monge problem (1781): How to transport earth with optimal cost ?
Nowadays
there1810.01367
is a Monge
street street
in Paris,andtherein
one
can
flow,
Zhang,
E, therein
LW,
1809.10188
flow…
Neural ODE,
1806.07366,
Nowadays
thereis a Monge-Ampère
Monge
in
Paris,
andonefindcanHamiltonianfindQuantum Flows ?

🤔

Quantum Flows

SimpleinputsEntangled
qubitsUnitary evolution in a exponentially LARGE Hilbert space(…means a different thing in Cranmer et al 1904.05903 )

ven
ine
mize
to
ckthe
the
ural
alits.
has
tive
red
ese
rate
ion

2

Quantum circuits for generative modelingWith Liu, Zeng, Wu, Hu
PRA ’18, PRA ’19“Born Machine”
Experiments
1801.07686
1812.08862
1811.09905
1901.08047
1904.02214
4

BAS patterns

Non-BAS pattern

Q1

Q2

1

2

Q3

Q4

4

3

1

2

4

3

3

2
1

4

Train quantum circuits as implicit probabilistic models
Strongexpressibility
due toquantum
Figure 1.
Illustrationof the di↵erentiable
QCBM
trainingsampling
scheme. complexity

Variational Quantum Eigensolver
4

θ1
θ1
θ1

θ2

θ6

θ4
θ5

θ3

θ6

θ

⟨H⟩

θ

Peruzzo et al,circuitas aquantum-classicalvariationalalgorithms.
ansatzThese steps
FIG. 1. Illustration of Quantum
the three common
steps of hybrid
have
Nat.
Comm. ’14

Q1
aThe
c procedure are illustrated in Fig. 2,
H2, remove two qubits associated with the spin–parity
resultsfromanoptimization
lowest-energy
interatomic
distances(bonddistanQ2
Q3
reducing the Hamiltonian to a six-qubit problem that using
the Hamiltonian for BeH2 at theQ1 |0〉interatomic distance of 1.7 Å.
Q7inSupplementary
Information.
Q4
ht spin orbitals. A similar approach is used to map LiH Althoughusing a large number of entanglers
UENT helps to achieve
Q5
Q2 |0〉
Q6
bits. The Hamiltonians for H2, LiH and BeH2 at their betterQ8 energy estimates in the absence of noise, the combined effectQ3 |0〉
y interatomic distances (bond distance) are given explicitly of decoherence and finite
sampling
sets
theoptimaldepth
foroptia 2p 1s 1s′Q4 |0〉
2p
1s′
1s
2s
2s
ntary Information.
mizationson our quantum0.4
hardware to 0–2 entanglers. The resultsb
SCALABLE QUANTUM SIMULATION OF MOLECULAR ENERGIES
PHYS.
REV. X 6, 031007 (2016)
Q5 |0〉
100
H2 molecule with 2 qubits
BeH2 moleculewith
6qubitsQ1
Q6
Q6 |0〉
b
c 0.2
(a)
d
(b)
–6.6
d
–12.0
U1,0 (Tk)

U1,1(Tk)

I X–π/2–
Yπ/2

U2,0 (Tk)

U2,1(Tk)

I X–π/2
Yπ/2

U3,0 (Tk)

U3,1(Tk)

I X–π/2
Yπ/2

U4,1(Tk)

I X–π/2
Yπ/2

U5,0 (Tk)

U5,1(Tk)

I X–π/2
Yπ/2

U6,0 (Tk)

U6,1(Tk)

I X–π/2
Yπ/2

–13.5

–0.4
–0.6
–0.8

of parametrized quantum gates, U1 ðθ1 ÞU2 ðθ2 Þ…U n ðθn Þ.
In this work, we parametrize our circuit according tounitary coupled cluster theory [20,22,23]. As describedin Appendix D, unitary coupled cluster theory predicts thatthe ground state of Eq. (1) can be expressed asE ðθ~1 Þ is then measured and returned to the classical outerloop. This continues for m iterations until the energy
converges to a minimum value E ðθ~m Þ, which represents
the VQE approximation to E 0 .
Because our experiment requires only a single variational parameter, as in Eq. (3), we elect to scan 1000
different values of θ ∈ ½−π; πÞ inNorder to obtain expectation values that define the entire potential energy curve.
,d
We do this to simplify the classical feedbackqroutine
but atthe cost of needing slightly more experimental trials. Theseq =1
expectation values are shown in Fig. 2(a) and the corresponding energy surfaces at different
bond lengths areNshown in Fig. 2(b). The energy surface in Fig. 2(b) is
q ,0 an on-thelocally optimized at each bond length to emulate
fly implementation.
q =1
Figure 3(a) shows the exact and experimentally determined energies of molecular hydrogen at different bondlengths. The minimum energy bond length (R ¼ 0.72 Å)corresponds to the equilibrium bond length, whereas theasymptote on the right-hand part of the curve corresponds–8.0

Q5 |0〉

–1.0
–15.5

20

6–
5

R

U

ENT
CR4–5

Q7
Q7 H

Q4

CR2–1

Q5

Q6 |0〉

Figure 1 | Quantum chemistry on a superconducting quantum
processor. Solving electronic-structure problems on a quantum computer
FIG. 1. Hardware and software schematic of the variational quantum eigensolver. (Hardware) micrograph shows two Xmon transmon–14.0
reliesonformappings
betweenfermionic
and qubit operators. a, Parity
qubits and microwave pulse sequences to perform single-qubit
rotations (thick lines),
dc pulses
two-qubit entangling
gates (dashed
–7.4
Li
H
lines), and microwave spectroscopy tones for qubit measurements (thin lines). (Software) quantum circuit diagram shows preparation of
mappingeight
spin
orbitalsthe Hartree-Fock state, followed by application of the unitary coupled cluster ansatz
in Eq. (3) andof
efficientpartialtomography(R ) to (drawn in blue and red, not to scale) onto
measure the expectation values in Eq. (1). Finally, the total energy is computed according
to Eq.
(4) and provided
to a classicaloptimizer
eight
qubits,
which
are
then
reduced to six qubits owing to fermionicwhich suggests new parameters.
–14.5
spin
and
paritysymmetries.
The
lengthofthebars
indicatetheparityof–7.6
where the thegγ are spin
scalars orbitals
and the Hγ are
local areHamiltonians~ ≡ UðθÞjϕi.
~that
encoded in each qubit. b, False-coloured opticaljφðθÞi
Even if jϕi is a simple product state and~ is a very shallow circuit, jφðθÞi~ can contain complex
and thecorresponding
as in Eq. (1).
The initial guessofθ~0theUðθÞmicrographsuperconducting
quantum processor with seven–15.0
many-body correlations and span an exponential number ofobjective value E ðθ~0 Þ are then fed to a classical greedytransmon
Thesewhich
qubitsstandard basis states.
minimization
routine (e.g.,qubits.
gradient descent),
then are coupled via two coplanar waveguide–7.8~ as a concatenation
We can express the mapping UðθÞsuggests aresonators
new setting of theparametersand
θ~1 . The
energy(violet)have
individual coplanar waveguide resonatorst

Q3 |0〉
Q4 |0〉

–4

–13.0
–0.2

Q2 |0〉

Q3

–3

Q3

Q5

Q1 |0〉

40

Q4Q1

–4

Q2

–12.50

Q6

R2

–7.2

Q7

1 mm

0

Q2

C

H

Q2

–4

R2

H

–4

R2

Q7

Q3

Q1

R2

–7.0

Q4
C

C

0

Q5

50

R1
C

Q3

–3

Q4

20

Q5

R1

Q5

–6.8

C

50

Q1

Q6

CR2–1

Q1

Q6

Q6

Q4 40

C

100

PHYS. REV. X 6, 031007 (2016)

Energy (hartree)

SCALABLE QUANTUM SIMULATION OF MOLECULAR ENERGIES

x

C

x

U4,0 (Tk)UENT

VQE on actual quantum devices
Q3

Q2

0

0

d
Q2
H

for control and read-out.Be
c, Hardware-efficient quantum circuit for trialstate preparationand energy estimation, H
shown here for six qubits. For
Heach iteration k, the circuit is composed of a sequence of interleaved
single-qubit rotations Uq,d(θk) and entangling unitary operations UENT
that entangle all of the qubits in the circuit. A final set of post-rotations(I, X− π/2 or Yπ/2) before the qubits are read out is used to measure theexpectation values of the individual Pauli terms in the Hamiltonian and to
estimate the energy of the trial state. d, An example of the pulse sequence
for the preparation of a six-qubit trial state, in which UENT is implemented
as a sequence of two-qubit cross-resonance gates.

maximal entanglement23. We set our two-qubit gate times at 150 ns, to
12
3
4
1 to minimize
2 the effect3of decoherence
4 without 5compromising thetry–1.2 accuracy of the optimization outcome; see Supplementary Information.
jφðθÞi ¼ e
j01i;
ð3Þ
Interatomic distance (Å)
Interatomic distance N(Å)
Interatomic
distance
(Å)
0
1
2
3
4
After
each
trial
state
is
prepared,
weestimatetheassociatedenergyq,d1−where jϕi ¼ j01i is the Hartree-Fock (mean-field) state(θ )] × ! × UENT ×|Φ(θ )〉 = ∏ [U (θ )] × UENT × ∏ [Uof molecular hydrogen in the representation of Eq. (1).
by measuring
expectation
valuesof the individual Pauli terms in
q = 1each interatomic distance.
plication to quantum
chemistry.
a–c,Experimental
results
outcomes at
The topthe
insets
in
each
panel
As discussed
in Appendix D, unitary
coupledclusterInteratomic
distance
FIG. 3. Computedenergy
by
both
VQE
and
PEA.
VQE
theory is widelyH
believed
to be classically curve
intractable andand errors. (a) Energy surface of molecular hydrogen as determined
the Hamiltonian.
These
estimatesareaffectedby (Å)
stochastic fluctua2
known
to be strictly(dottedmore powerful thanthe “gold and density plots
rcles), exact energyisstandard”
surfaceslines)
highlight
thequbitsusedfortheexperimentandthecross-resonance
tions
due
to
finitesampling.
Different post-rotations are applied after−4of classical electronic structure theory, coupled∏[U(θ )]hartree|00 … 0〉 (error bars on VQE data are smaller than markers).
PEA
approachapproachshows
dissociationenergy
error
of
ð8
%
5Þ
×
10
cluster theory [43–46]. The gate model circuit that
;
where
‘c’
denotesthecontrol
qubit
and
‘t’
the
colour scales) of outcomesfrom
numerical
simulations,gates(arrows,
labelledCR
trial-state
preparationforsampling
different
Pauli
operators (Fig. 1c, d).
c–t
performs this unitary mapping is shown in the software−2
section of Fig.
1.
Webottomgroup
thePauli operators
into
tensorproduct basis sets that require
shows
dissociationenergy
error
of
ð1
%
1Þ
×
10
hartree.
(b)
Errorsin
VQEenergysurface.
Red
dots
show
error
in
the
experimentally
(a),
LiH
(b)
and
BeH
(c).
The
.
The
insetsarerepresentations
targetqubit)
that
constituteUeratomic distances for
H
~
2
2
ENT
VQE solves for the parameter vector θ with a classicalpost-rotations. We
that such grouping
the qubits
areenergies
all initialized
in theirgroundstate been|0〉, theobtained
first the same
optimization routine. One first prepares an initial ansatz
determined energies. Green diamonds show Because
the error
in the
that
would
have
experimentally
bynumericallyrunning show
the circuit−iθX0 Y 1

the state |00…0〉, applying d entanglers UENT that alternate with N Euler
1345
rotations,giving2

PRX ‘16Nature‘17
Figure 3 | Application to quantum chemistry. a–c,

(black filled circles), exact energy surfaces (dotted liFlow with classical neural net& quantum circuit
V

W2

W1

x
b1

x1
x2

x
b2

x3

x
b3

Autoencoder
x

| x⟩

Uθ

†
pϕ(x)Uθ | x⟩⟨x | Uθ = ρ
∑
x

= MW

p(x

2

MA

1

= MV

A classical mixture of quantum states parametrizes density matrices
= MW

Masks

x ∼ pϕ(x)Verdon et al, 1910.02071

† −βHUθ e Uθ = pϕ“β”-VQE
variational free energyℒ = βTr(ρH) + Tr(ρ ln ρ) ≥ − ln Z
=1

10

=2

=3

=4

20

3x3 quantum
Ising model @ β=130

with Jin-Guo Liu,
unpublished

0

100

200

300

400

500

epoch

Study quantum thermodynamics with classical & quantum flows

Yao: Extensible, Efficient Framework for
Quantum Algorithm Design
https://github.com/QuantumBFS/Yao.jl++=

Xiu-Zhe Luo (IOP, CAS → Waterloo & PI)
Jin-Guo Liu (IOP, CAS → Harvard)

Features:
Differentiable programming quantum circuits• Batch parallelization with GPU acceleration• Quantum block intermediate representation•

Neural Canonical Transformation with Symplectic Flows
Shuo-Hui Li, Chen-Xiao Dong, Linfeng Zhang, LW, 1910.00024

Yao.jl: Extensible, Efficient Framework for Quantum Algorithm Design
Xiu-Zhe Luo, Jin-Guo Liu, Pan Zhang, LW, up coming

Thank You!