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3rd Week of Quantum Information and String Theory 2019 Quantum-Classical Hybrid Algorithm: its advantage and methods for variational optimization KF, arXiv:1803.09954 Mitarai-Negoro-Kitagawa-KF, Phys. Rev. A 98 , 032309 (2019)

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SLIDE 1

Quantum-Classical Hybrid Algorithm: its advantage and methods for variational optimization

3rd Week of Quantum Information and String Theory 2019

KF, arXiv:1803.09954 Mitarai-Negoro-Kitagawa-KF, Phys. Rev. A 98, 032309 (2019) Nakanishi-KF-Todo, arXiv:1903.12166

Keisuke Fujii Graduate School of Science and Engineering, Osaka University JST PRESTO

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SLIDE 2

Overview

Classical computer Quantum computer sampling quantum easy task classical easy task update

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SLIDE 3

Overview

Classical computer Quantum computer sampling quantum easy task classical easy task ・Is there any situation where a quantum-classical hybrid approach provides a complexity theoretic advantage? →adiabatic quantum computation with stoquastic Hamiltonian update

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SLIDE 4

Overview

Classical computer Quantum computer sampling quantum easy task classical easy task ・Is there any situation where a quantum-classical hybrid approach provides a complexity theoretic advantage? →adiabatic quantum computation with stoquastic Hamiltonian ・How should we tune the parameters of (NISQ) quantum computers for quantum-classical variational algorithms. →gradient-based and -free optimizations update

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SLIDE 5

Outline

  • Advantage of quantum-classical hybrid algorithm
  • Adiabatic quantum computation and quantum circuit model
  • Characterization of stoquastic adiabatic quantum computation
  • Quantum speedup in stoqAQC (sampling-based factoring & phase

estimation)

  • Parameter tuning for quantum-classical variational algorithm
  • Gradient-based optimization
  • Gradient-free optimization
  • Numerical comparisons of gradient-based and -free optimizations.
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SLIDE 6

Quantum computational supremacy

Boson Sampling

Experimental demonstrations

  • J. B. Spring et al. Science 339, 798 (2013)
  • M. A. Broome, Science 339, 794 (2013)
  • M. Tillmann et al., Nature Photo. 7, 540 (2013)
  • A. Crespi et al., Nature Photo. 7, 545 (2013)
  • N. Spagnolo et al., Nature Photo. 8, 615 (2014)
  • J. Carolan et al., Science 349, 711 (2015)

Universal linear optics Science (2015)

Linear optical quantum computation Aaronson-Arkhipov ‘13

DQC1

(one-clean qubit model)

I/2n

U

}

|0i

H H

Knill-Laflamme ‘98 Morimae-KF-Fitzsimons ’14 KF et al, ‘18 NMR spin ensemble

IQP

(commuting circuits)

Bremner-Jozsa-Shepherd ‘11 Ising type interaction

|+i |+i |+i |+i

T T

|+i

T

… … Bremner-Montanaro-Shepherd ‘15 Gao-Wang-Duan ‘15 KF-Morimae ‘13 Farhi-Harrow ‘16

non-universals model of quantum computation

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SLIDE 7

Quantum computational supremacy

Boson Sampling

Experimental demonstrations

  • J. B. Spring et al. Science 339, 798 (2013)
  • M. A. Broome, Science 339, 794 (2013)
  • M. Tillmann et al., Nature Photo. 7, 540 (2013)
  • A. Crespi et al., Nature Photo. 7, 545 (2013)
  • N. Spagnolo et al., Nature Photo. 8, 615 (2014)
  • J. Carolan et al., Science 349, 711 (2015)

Universal linear optics Science (2015)

Linear optical quantum computation Aaronson-Arkhipov ‘13

DQC1

(one-clean qubit model)

I/2n

U

}

|0i

H H

Knill-Laflamme ‘98 Morimae-KF-Fitzsimons ’14 KF et al, ‘18 NMR spin ensemble

IQP

(commuting circuits)

Bremner-Jozsa-Shepherd ‘11 Ising type interaction

|+i |+i |+i |+i

T T

|+i

T

… … Bremner-Montanaro-Shepherd ‘15 Gao-Wang-Duan ‘15 KF-Morimae ‘13 Farhi-Harrow ‘16

non-universals model of quantum computation → adiabatic quantum computation with stoquastic Hamiltonian

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SLIDE 8

What is stoquastic Hamiltonian?

・Off-diagonal terms are non positive in a standard basis. ・The ground state has positive coefficients in the standard basis. ・No negative sign problem → Quantum Monte-Carlo method.

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SLIDE 9

What is stoquastic Hamiltonian?

・Off-diagonal terms are non positive in a standard basis. ・The ground state has positive coefficients in the standard basis. ・No negative sign problem → Quantum Monte-Carlo method.

・Transverse Ising model:

H = X

ij

JijZiZJ − h X

i

Xi

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SLIDE 10

What is stoquastic Hamiltonian?

・Off-diagonal terms are non positive in a standard basis. ・The ground state has positive coefficients in the standard basis. ・No negative sign problem → Quantum Monte-Carlo method.

・Transverse Ising model:

H = X

ij

JijZiZJ − h X

i

Xi

・Bose-Hubbard model with negative hopping: H = −ω X

ij

(a†

iaj + a† jai) − µ

X

i

ni + U X

i

ni(ni − 1)

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SLIDE 11

What is stoquastic Hamiltonian?

・Off-diagonal terms are non positive in a standard basis. ・The ground state has positive coefficients in the standard basis. ・No negative sign problem → Quantum Monte-Carlo method.

・Transverse Ising model:

H = X

ij

JijZiZJ − h X

i

Xi

・Bose-Hubbard model with negative hopping: H = −ω X

ij

(a†

iaj + a† jai) − µ

X

i

ni + U X

i

ni(ni − 1) ・Heisernberg anti-ferro magnetic on a bipartite graph:

H = J X

ij

(XiXj + YiYj + ZiZj) −

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SLIDE 12

What is stoquastic Hamiltonian?

・Off-diagonal terms are non positive in a standard basis. ・The ground state has positive coefficients in the standard basis. ・No negative sign problem → Quantum Monte-Carlo method.

・Transverse Ising model:

H = X

ij

JijZiZJ − h X

i

Xi

・Bose-Hubbard model with negative hopping: H = −ω X

ij

(a†

iaj + a† jai) − µ

X

i

ni + U X

i

ni(ni − 1) ・Heisernberg anti-ferro magnetic on a bipartite graph:

H = J X

ij

(XiXj + YiYj + ZiZj) −

How powerful is adiabatic quantum computation with these restricted types of Hamiltonians?

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SLIDE 13

Take home messages

  • Non standard basis measurements change the situation

drastically, while they would be relatively easy on an actual quantum machine if it has true quantum coherence.

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SLIDE 14

Take home messages

  • Non standard basis measurements change the situation

drastically, while they would be relatively easy on an actual quantum machine if it has true quantum coherence.

  • StoqAQC with simultaneous measurements can solve

meaningful and important problems like factoring with a quantum-classical hybrid algorithm.

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SLIDE 15

Circuit model and adiabatic model

Circuit model:

H T

universal set of gate

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SLIDE 16

Circuit model and adiabatic model

Circuit model: Adiabatic quantum computation:

a(t) b(t)

→→→→→ trivial state ↓ ↑ ↑↓ ↑ ↓ solution adiabatic theorem

H(t) = a(t)Hinitial + b(t)Hfinal

H T

universal set of gate

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SLIDE 17

Feynman’s seminal idea ‘84

Mapping each step of quantum computation to each site!

Hwork ⊗ Hclock

clock to track the step working system for quantum computation

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SLIDE 18

Feynman’s seminal idea ‘84

Mapping each step of quantum computation to each site!

Hwork ⊗ Hclock

clock to track the step working system for quantum computation

|0i⊗n|0ic U1|0i⊗n|1ic

・ ・ ・

Ut · · · U1|0i⊗n|tic

Clock makes all theses intermediate states orthogonal!

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SLIDE 19

Feynman’s seminal idea ‘84

Mapping each step of quantum computation to each site!

Hwork ⊗ Hclock

clock to track the step working system for quantum computation

|0i⊗n|0ic U1|0i⊗n|1ic

・ ・ ・

Ut · · · U1|0i⊗n|tic

Clock makes all theses intermediate states orthogonal!

H = 1 2 X

i

[(|iihi| + |i 1ihi 1|) (|iihi 1| + h.c)]

|Ψi = 1 p N X

i

|ii

tight-binding Hamiltonian: →ground state:

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SLIDE 20

Universality of non-stoqAQC Aharonov et al ‘04

X Hin =

n

X

i=1

|1ih1|i ⌦ |0ih0|c +

  • energy penalty for the initial state of the working system:

imposing initial state should be all 0s

  • energy penalty for the initial clock state:
  • Hinitial = Hin + (Ic |0ih0|c),
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SLIDE 21

Universality of non-stoqAQC Aharonov et al ‘04

X Hin =

n

X

i=1

|1ih1|i ⌦ |0ih0|c +

  • energy penalty for the initial state of the working system:

imposing initial state should be all 0s

  • energy penalty for the initial clock state:
  • Hinitial = Hin + (Ic |0ih0|c),

site energy

| ih | Hfinal = Hin +

T

X

t=1

1 2[|tiht|c + |t 1iht 1|c (Ut|tiht 1|c + U †

t |t 1iht|c)], n m+n

  • tight-binding Hamiltonian (Kitaev-Shen-Vyalyi ’02) :

Feynman’s Hamiltonian hopping term

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SLIDE 22

Universality of non-stoqAQC Aharonov et al ‘04

X Hin =

n

X

i=1

|1ih1|i ⌦ |0ih0|c +

  • energy penalty for the initial state of the working system:

imposing initial state should be all 0s

  • energy penalty for the initial clock state:
  • Hinitial = Hin + (Ic |0ih0|c),

H(s) = (1 s)Hinitial + sHfinal,

  • adiabatic quantum computation:

The lowest energy gap is always lower bounded by an inverse of polynomial. site energy

| ih | Hfinal = Hin +

T

X

t=1

1 2[|tiht|c + |t 1iht 1|c (Ut|tiht 1|c + U †

t |t 1iht|c)], n m+n

  • tight-binding Hamiltonian (Kitaev-Shen-Vyalyi ’02) :

Feynman’s Hamiltonian hopping term

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SLIDE 23

Universality of non-stoqAQC Aharonov et al ‘04

X Hin =

n

X

i=1

|1ih1|i ⌦ |0ih0|c +

  • energy penalty for the initial state of the working system:

imposing initial state should be all 0s

  • energy penalty for the initial clock state:
  • Hinitial = Hin + (Ic |0ih0|c),

| ih | Hfinal = Hin +

T

X

t=1

1 2[|tiht|c + |t 1iht 1|c (Ut|tiht 1|c + U †

t |t 1iht|c)], n m+n

  • tight-binding Hamiltonian (Kitaev-Shen-Vyalyi ’02) :

Feynman’s Hamiltonian

H(s) = (1 s)Hinitial + sHfinal,

  • adiabatic quantum computation:

The lowest energy gap is always lower bounded by an inverse of polynomial.

|Ψi = 1 p T + 1

T

X

t=0

Ut · · · U1|0i⊗n|Tic

The ground state of the final Hamiltonian (history state):

t

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SLIDE 24

Universality of non-stoqAQC Aharonov et al ‘04

X Hin =

n

X

i=1

|1ih1|i ⌦ |0ih0|c +

  • energy penalty for the initial state of the working system:

imposing initial state should be all 0s

  • energy penalty for the initial clock state:
  • Hinitial = Hin + (Ic |0ih0|c),

| ih | Hfinal = Hin +

T

X

t=1

1 2[|tiht|c + |t 1iht 1|c (Ut|tiht 1|c + U †

t |t 1iht|c)], n m+n

  • tight-binding Hamiltonian (Kitaev-Shen-Vyalyi ’02) :

Feynman’s Hamiltonian

H(s) = (1 s)Hinitial + sHfinal,

  • adiabatic quantum computation:

The lowest energy gap is always lower bounded by an inverse of polynomial.

|Ψi = 1 p T + 1

T

X

t=0

Ut · · · U1|0i⊗n|Tic

The ground state of the final Hamiltonian (history state):

If the propagator Hamiltonian is restricted into stoquastic terms, how quantum computation would be changed?

t

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SLIDE 25

Restriction to stoquastic Hamiltonians

|c (Ut|tiht 1|c + U †

t |t 1iht|c)]

  • ff-diagonal terms:

Each element of U should be non negative!

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SLIDE 26

Restriction to stoquastic Hamiltonians

|c (Ut|tiht 1|c + U †

t |t 1iht|c)]

  • ff-diagonal terms:

Each element of U should be non negative!

X = ✓ 1 1 ◆

CNOT =     1 1 1 1    

Toffoli

Elements of U are non negative, iff U is a unitary version of reversible classical computation.

(I⊗2 − |11⟩⟨11|) ⊗ I + |11⟩⟨11| ⊗ X

Hamiltonian complexity of stoquastic Hamiltonian [Bravyi, DiVencenzo, Oliveira, and Terhal (2008) ]

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SLIDE 27

Stoquastic AQC as a sampling problem

Arbitrary single-qubit measurements! Initial Hadamard transformations are allowed. Unitary gates for reversible classical computation.

|0i |0i |0i … |0i … … …

H H

UT · · · U1 X, CNOT, Toffoli … …

Quantum circuit that can be simulated by stoqAQC.

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SLIDE 28

arbitrary single-qubit measurements measurement-based quantum computation（Graph state, hyper graph state）

Universal QC with stoqAQC

|0i |0i |0i … |0i … … …

H H

UT · · · U1 X, CNOT, Toffoli … …

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SLIDE 29

Universal QC with stoqAQC

|0i |+i

cluster state on a square lattice = universal resource for MBQC

|0i |0i |0i … |0i … … …

H H

UT · · · U1 X, CNOT, Toffoli … …

StoqAQC with adaptive measurements is can simulate universal QC.

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SLIDE 30

With non-adaptive single-qubit measurements?

Universal QC with stoqAQC

|0i |0i |0i … |0i … … …

H H

UT · · · U1 X, CNOT, Toffoli … …

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SLIDE 31

… |+i |+i |+i … … … … … … … …

Z Z Z

…

Ux

e(2πi)0.j1j2...jn e(2πi)0.j2...jn e(2πi)0.jn

e2(πi)0.j1j2...jn

Kitaev’s Phase estimation and Shor’s algorithm

U†

QFT

Inverse QFT

j1 j2

⋮

jn

Ux = ∑

y

|xy mod N⟩⟨y|

U2

x

U2n−1

x

modular exponentiation = unitary reversible classical computation controlled-phase

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SLIDE 32

|0i H

X

U2k

x

|0i H U2k

x

⋱

X

…

⋮

classical processing

}⟨X⟩

R samples

Quantum-classical hybrid phase estimation

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SLIDE 33

|0i H

X

hXi hY i U2k−1

x

|0i H U2k−1

x

⋱

X

…

⋮

⟨X⟩ |0i H

Y

|0i H

⋱

Y

⋮

U2k−1

x

U2k−1

x

…

⋮ ⋮

classical processing

⟨Y⟩

} }

e2πi0.jk...jn

R samples

jk = 0

jk = 1

|¯ 1i

R samples

Quantum-classical hybrid phase estimation

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SLIDE 34

×R

modular exponentiation → reversible classical computation

|0i H

X

(

|¯ 1i

… …

|0i H

…

|0i H

…

|0i H

… …

X Y Y

(

Ux U2n−1

x

Ux U2n−1

x

sampling & classical processing

hXi

hY i

e2πi0.jn

jn = 0

jn = 1

jn

hXi

hY i

e2πi0.j1...jn

j1 = 1

j1 = 0

j1

⋮ ⋮

jk

hXi

hY i

e2πi0.jk...jn

jk = 0

jk = 1

Factoring (phase estimation) can be solved by stoqAQC and classical processing

Quantum-classical hybrid phase estimation

slide-35
SLIDE 35

×R

modular exponentiation → reversible classical computation

|0i H

X

(

|¯ 1i

… …

|0i H

…

|0i H

…

|0i H

… …

X Y Y

(

Ux U2n−1

x

Ux U2n−1

x

sampling & classical processing

hXi

hY i

e2πi0.jn

jn = 0

jn = 1

jn

hXi

hY i

e2πi0.j1...jn

j1 = 1

j1 = 0

j1

⋮ ⋮

jk

hXi

hY i

e2πi0.jk...jn

jk = 0

jk = 1

Factoring (phase estimation) can be solved by stoqAQC and classical processing

Quantum-classical hybrid phase estimation

StoqAQC with simultaneous single-qubit measurements → non-universal model quantum computation Quantum-classical hybrid algorithm allows us to solve nontrivial problems like factoring etc. (Without classical processing, stoqAQC with simultaneous non-standard basis measurements could not decide the problem.)

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SLIDE 36

Outline

  • Advantage of quantum-classical hybrid algorithm
  • Adiabatic quantum computation and quantum circuit model
  • Characterization of stoquastic adiabatic quantum computation
  • Quantum speedup in stoqAQC (sampling-based factoring & phase

estimation)

  • Parameter tuning for quantum-classical variational algorithm
  • gradient-based optimization
  • gradient-free optimization designed for hardware efficient ansatz
  • Numerical comparisons of gradient-based and -free optimizations.
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SLIDE 37

Quantum-classical hybrid algorithm

classical computer Quantum computer sampling parameter update classical easy task

Approximated optimization：QAOA (quantum approximate optimization algo.) Ground state：VQE (variational quantum eigensolver) Supervised machine learning：QCL (quantum circuit learning)

.

  • E. Farhi, J. Goldstone, and S. Gutmann, arXiv preprint arXiv:1411.4028 (2014). 


.

  • A. Peruzzo, J. McClean, P. Shadbolt, M.-H. Yung, X.-Q. Zhou, P. J. Love, A. Aspuru-Guzik,

and J. L. O’brien, Nature Communications 5, 4213 (2014). 


  • K. Mitarai, M. Negoro, M. Kitagawa, and K. Fujii Phys. Rev. A 98, 032309 (2018)

trial function of variational methods

slide-38
SLIDE 38

“A variational eigenvalue solver on a photonic quantum processor” Peruzzo, McClean et al, Nature Communication 5:4213 (2014) H-He+ “Hardware-efficient Quantum Optimizer for Small Molecules and Quantum Magnets” Kandala, Mezzacapo et al, Nature 549 242 (2017)

Quantum-classical hybrid algorithm: variational quantum eigensolver

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SLIDE 39

model・ trial function tuning・

  • ptimization

task Neural netwrok W, tanh() backpropagation (gradient) machine learning Rayleigh-Ritz (Hartree-Fock)

  • rthogonal functions

(Slater determinant) diagonalization of Hermitian matrix (HF equation) ground state Tensor network (MPS, PEPS, MERA) tensor network singular value decomposition ground state (dynamics) Variational quantum algorithms parameterized quantum circuit gradient?

[Mitarai-Negoro-Kitagawa-KF ‘18]

  • ther?

[Nakanishi-KF-Todo ’19]

machine learning ground state dynamics

Variational algorithms

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SLIDE 40

Quantum circuit learning: supervised learning

  • n near-term quantum devices

low depth quantum circuit ・・・ ・・・

U({θi})

parameterized quantum circuit

input data loss function： teacher data

  • utput from parameterized

quantum circuit

  • ptimization

using gradient

xj

L = X

j

[f(xj) hA(xj, {θi})i]2

p f(x)|0i + p 1 f(x)|1i

  • K. Mitarai, M. Negoro, M. Kitagawa, and KF Phys. Rev. A 98, 032309 (2018).

and many others

slide-41
SLIDE 41

Generalization of nonlinear functions

teacher prediction

nonlinear classification

Quantum circuit learning: supervised learning

  • n near-term quantum devices
  • K. Mitarai, M. Negoro, M. Kitagawa, and KF Phys. Rev. A 98, 032309 (2018).

and many others

slide-42
SLIDE 42

Generalization of nonlinear functions

teacher prediction

nonlinear classification

Quantum circuit learning: supervised learning

  • n near-term quantum devices
  • K. Mitarai, M. Negoro, M. Kitagawa, and KF Phys. Rev. A 98, 032309 (2018).

and many others

slide-43
SLIDE 43

Variational quantum algorithms and parameterized quantum circuit

U(φ1) U(φ2) U(φ3) U(φ4) U(φ5) U(φ6) U(φ7) U(φ8)

|0i |0i |0i |0i |0i

|ψ({φk})i

for example

Rx(φ(1)

k )

Rx(φ(2)

k )

Rz(φ(3)

k )

Rz(φ(4)

k )

Rx(φ(5)

k )

Rx(φ(6)

k )

Rz(φ(7)

k )

Rz(φ(8)

k )

Rz(φ) = e−i(φ/2)Z Rx(φ) = e−i(φ/2)X

min

{φk}hψ({φk})|H|ψ({φk})i

trial function

  • r

ansatz variational optimization: How can we obtain the gradient ?

∂ ∂φk hψ({φk})|H|ψ({φk})i

slide-44
SLIDE 44

Analytical differentiation of quantum circuits

U({φk}) = Y

k

Wke−i(φk/2)Pk

Parameterized quantum circuit:

hermitian & unitary such as Pauli operators fixed unitary

slide-45
SLIDE 45

Analytical differentiation of quantum circuits

U({φk}) = Y

k

Wke−i(φk/2)Pk

Parameterized quantum circuit:

hermitian & unitary such as Pauli operators fixed unitary

Expectation value:

hA({φk})i = hψ({φk})|A|ψ({φk})i

where |ψ({φk})i = U({φk})|0i⊗n

slide-46
SLIDE 46

Analytical differentiation of quantum circuits

U({φk}) = Y

k

Wke−i(φk/2)Pk

Parameterized quantum circuit:

hermitian & unitary such as Pauli operators fixed unitary

Expectation value:

hA({φk})i = hψ({φk})|A|ψ({φk})i

where |ψ({φk})i = U({φk})|0i⊗n

Analytical differentiation:

@ @l hA({k})i = hA({1, ..., l + ✏, l+1, ...}i hA({1, ..., l ✏, l+1, ...}i 2 sin ✏ ✏ = ⇡/2

slide-47
SLIDE 47

Analytical differentiation of quantum circuits

U({φk}) = Y

k

Wke−i(φk/2)Pk

Parameterized quantum circuit:

hermitian & unitary such as Pauli operators fixed unitary

Expectation value:

hA({φk})i = hψ({φk})|A|ψ({φk})i

where |ψ({φk})i = U({φk})|0i⊗n

Analytical differentiation:

@ @l hA({k})i = hA({1, ..., l + ✏, l+1, ...}i hA({1, ..., l ✏, l+1, ...}i 2 sin ✏ ✏ = ⇡/2

hA(θ)i = hψ|e−i(θ/2)P Aei(θ/2)P |ψi ∂ ∂θhA(θ)i = hψ|(iP/2) ˜ A|ψi + hψ| ˜ A(iP/2)|ψi = i(1/2)(hψ|P ˜ Aψi hψ| ˜ AP|ψi)

˜ A = e−i(θ/2)P Aei(θ/2)P

(

(

hA(✓ + ✏)i = cos2(✏/2)h ˜ Ai + sin2(✏/2)hP ˜ APi i sin(✏/2) cos(✏/2)hP ˜ Ai + i cos(✏/2) sin(✏/2)h ˜ APi

slide-48
SLIDE 48

Analytical differentiation of quantum circuits

・・・ RZ(φ1) RX(φ2) RZ(φ3) RZ(φ4) RX(φ5) RZ(φ6) ・・・

slide-49
SLIDE 49

Analytical differentiation of quantum circuits

・・・ RZ(φ1) RX(φ2) RZ(φ3) RZ(φ4) RX(φ5) RZ(φ6) ・・・

→ The gradient can be obtain differently from measurements of two observables.

e−i(φl/2)Pl ∂ ∂φl

=

e−i

l+✏ 2

Pl

ー

1 2 sin ✏(

(

e−i

l−✏ 2

Pl

✏ = ⇡/2

slide-50
SLIDE 50

Analytical differentiation of quantum circuits

See also

  • M. Schuld, et al. (Xanadu) "Evaluating analytic gradients on quantum hardware." Physical

Review A 99, 032331 (2019) → PennyLane

  • Z. Y. Chen, et al. "VQNet: Library for a Quantum-Classical Hybrid Neural Network." arXiv

preprint arXiv:1901.09133 (2019). → 本源量子

  • K. Mitarai, and K. Fujii. "Methodology for replacing indirect measurements

with direct measurements." arXiv preprint arXiv:1901.00015 (2018). ・・・ RZ(φ1) RX(φ2) RZ(φ3) RZ(φ4) RX(φ5) RZ(φ6) ・・・

→ The gradient can be obtain differently from measurements of two observables.

e−i(φl/2)Pl ∂ ∂φl

=

e−i

l+✏ 2

Pl

ー

1 2 sin ✏(

(

e−i

l−✏ 2

Pl

✏ = ⇡/2

slide-51
SLIDE 51

Outline

  • Advantage of quantum-classical hybrid algorithm
  • Adiabatic quantum computation and quantum circuit model
  • Characterization of stoquastic adiabatic quantum computation
  • Quantum speedup in stoqAQC (sampling-based factoring & phase

estimation)

  • Parameter tuning for quantum-classical variational algorithm
  • gradient-based optimization
  • gradient-free optimization designed for hardware efficient ansatz
  • Numerical comparisons of gradient-based and -free optimizations.
slide-52
SLIDE 52

Sequential minimal optimization for quantum- classical hybrid algorithms

Ken M. Nakanishi, Keisuke Fujii, Synge Todo, arXiv:1903.12166 ・・・ ・・・

U({θi})

parameterized quantum circuit

fix except for

{θi}

θj hA(θ)i = hψ|e−i(θ/2)P Aei(θ/2)P |ψi = cos2(θ/2)hAi + sin2(θ/2)hPAPi + cos(θ/2) sin(θ/2)ih[A, P]i = α sin(θ + β) + γ

Unknown parameters are only three.

slide-53
SLIDE 53

Sequential minimal optimization for quantum- classical hybrid algorithms

Ken M. Nakanishi, Keisuke Fujii, Synge Todo, arXiv:1903.12166

  • 1
  • 0.5

0.5 1 0.5π 1.0π 1.5π 2.0π

Estimation of the cost function at three independent points exact minimum ・・・ ・・・

U({θi})

parameterized quantum circuit

fix except for

{θi}

θj hA(θ)i = hψ|e−i(θ/2)P Aei(θ/2)P |ψi = cos2(θ/2)hAi + sin2(θ/2)hPAPi + cos(θ/2) sin(θ/2)ih[A, P]i = α sin(θ + β) + γ

Unknown parameters are only three.

slide-54
SLIDE 54

Benchmark task：5qubit, 100 parameters

Comparison between gradient-based and -free

  • ptimizations

・・・ ・・・ ・・・

randomly chosen fixed unitary

U †(θ∗)

U(θ)

parameterized quantum circuit

|0i |0i |0i |0i

fidelity

・Gradient based: BFGS, CG ・Gradient like: SPSA ・Gradient free: sequential minimum optimization(SMO) , Nelder-Mead, Powell Optimization methods:

slide-55
SLIDE 55
  • Gradient based methods outperforms NealderMead, Powell, and SPSA .
  • Sequential minimal optimization substantially outperforms others especially

in the presence of statistical error.

  • (steps) counts the total number of call of a quantum computer.

initial random choice of the parameter

Comparison between gradient-based and -free

  • ptimizations

Ken M. Nakanishi, Keisuke Fujii, Synge Todo, arXiv:1903.12166

SMO

# of steps (= # of observables estimated on QC) are changed.

slide-56
SLIDE 56
  • Gradient based methods outperforms NealderMead, Powell, and SPSA .
  • Sequential minimal optimization substantially outperforms others especially

in the presence of statistical error.

  • (steps) counts the total number of call of a quantum computer.

Comparison between gradient-based and -free

  • ptimizations

Ken M. Nakanishi, Keisuke Fujii, Synge Todo, arXiv:1903.12166

SMO

# of samples to estimate an observable is changed.

slide-57
SLIDE 57

Variations of variational approaches:

model・ trial function tuning・

  • ptimization

task Neural netwrok W, tanh() backpropagation (gradient) machine learning Rayleigh-Ritz (Hartree-Fock)

  • rthogonal functions

(Slater determinant) diagonalization of Hermitian matrix (HF equation) ground state Tensor network (MPS, PEPS, MERA) tensor network singular value decomposition ground state (dynamics) Variational quantum algorithms parameterized quantum circuit which kind? gradient?

[Mitarai-Negoro-Kitagawa-KF ‘18]

  • ther?

[Nakanishi-KF-Todo ’19]

machine learning ground state dynamics advantage?

slide-58
SLIDE 58

Summary

  • We have seen an example where a non-universal model of

quantum computation can solve non-trivial problem by a quantum-classical hybrid approach.

  • Gradient can be directly obtained from analytical

differentiation of parameterized quantum circuits.

  • We can design a new optimization scheme, which is robust

against the statistical error, based on the property of the parameterized quantum circuits.

Todo, Nakanishi@UTokyo Mitarai, Negoro, Kitagawa @ Osaka U

Collaborators: