quantum-classical hybrid algorithm: its advantage and ...qist2019/slides/3rd/fujii.pdf · •...

Quantum-Classical Hybrid Algorithm: its advantage and methods for

variational optimization

3rd Week of Quantum Information and String Theory 2019

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

Keisuke FujiiGraduate School of Science and Engineering,

Osaka UniversityJST PRESTO

OverviewClassical computerQuantum computer

sampling

quantum easy task classical easy task

update

OverviewClassical computerQuantum 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

OverviewClassical computerQuantum 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

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.

Quantum computational supremacy

Boson Sampling

Experimental demonstrationsJ. 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 opticsScience (2015)

Linear optical quantum computation

Aaronson-Arkhipov ‘13

DQC1(one-clean qubit model)

I/2n U}|0i HH

Knill-Laflamme ‘98Morimae-KF-Fitzsimons ’14KF 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 ‘15Gao-Wang-Duan ‘15

KF-Morimae ‘13

Farhi-Harrow ‘16

non-universals model of quantum computation

Quantum computational supremacy

Boson Sampling

Experimental demonstrationsJ. 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 opticsScience (2015)

Linear optical quantum computation

Aaronson-Arkhipov ‘13

DQC1(one-clean qubit model)

I/2n U}|0i HH

Knill-Laflamme ‘98Morimae-KF-Fitzsimons ’14KF 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 ‘15Gao-Wang-Duan ‘15

KF-Morimae ‘13

Farhi-Harrow ‘16

non-universals model of quantum computation

→ adiabatic quantum computation with stoquastic Hamiltonian

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.

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 � hX

i

Xi

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 � hX

i

Xi

・Bose-Hubbard model with negative hopping:

H = �!X

ij

(a†iaj + a†jai)� µX

i

ni + UX

i

ni(ni � 1)

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 � hX

i

Xi

・Bose-Hubbard model with negative hopping:

H = �!X

ij

(a†iaj + a†jai)� µX

i

ni + UX

i

ni(ni � 1)

・Heisernberg anti-ferro magnetic on a bipartite graph:

H = JX

ij

(XiXj + YiYj + ZiZj)� hX

Zi

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 � hX

i

Xi

・Bose-Hubbard model with negative hopping:

H = �!X

ij

(a†iaj + a†jai)� µX

i

ni + UX

i

ni(ni � 1)

・Heisernberg anti-ferro magnetic on a bipartite graph:

H = JX

ij

(XiXj + YiYj + ZiZj)� hX

Zi

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

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.

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.

Circuit model and adiabatic modelCircuit model:

HT

universal set of gate

Circuit model and adiabatic modelCircuit model:

Adiabatic quantum computation: a(t) b(t)

→→→→→

trivial state↓↑↑↓↑↓

solution

adiabatic theorem

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

HT

universal set of gate

Feynman’s seminal idea ‘84Mapping each step of quantum computation to each site!

Hwork

⌦Hclock

clock to track the stepworking system for quantum computation

Feynman’s seminal idea ‘84Mapping each step of quantum computation to each site!

Hwork

⌦Hclock

clock to track the stepworking system for quantum computation

|0i⌦n|0ic U1|0i⌦n|1ic ・・・ Ut · · ·U1|0i⌦n|tic

Clock makes all theses intermediate states orthogonal!

Feynman’s seminal idea ‘84Mapping each step of quantum computation to each site!

Hwork

⌦Hclock

clock to track the stepworking 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)]

| i = 1pN

X

i

|ii

tight-binding Hamiltonian:

→ground state:

Universality of non-stoqAQC Aharonov et al ‘04

H(s) = (1� s)Hinitial + sHfinal,

Hinitial = Hin + (Ic � |0ih0|c),

Hfinal = Hin +TX

t=1

1

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

t |t� 1iht|c)],

Hin =nX

i=1

|1ih1|i ⌦ |0ih0|c +m+nX

j=n+1

|�ih�|j ⌦ |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:H(s) = (1� s)Hinitial + sHfinal,

Hinitial = Hin + (Ic � |0ih0|c),

Hfinal = Hin +TX

t=1

1

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

t |t� 1iht|c)],

Hin =nX

i=1

|1ih1|i ⌦ |0ih0|c +m+nX

j=n+1

|�ih�|j ⌦ |0ih0|c,

Universality of non-stoqAQC Aharonov et al ‘04

H(s) = (1� s)Hinitial + sHfinal,

Hinitial = Hin + (Ic � |0ih0|c),

Hfinal = Hin +TX

t=1

1

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

t |t� 1iht|c)],

Hin =nX

i=1

|1ih1|i ⌦ |0ih0|c +m+nX

j=n+1

|�ih�|j ⌦ |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:H(s) = (1� s)Hinitial + sHfinal,

Hinitial = Hin + (Ic � |0ih0|c),

Hfinal = Hin +TX

t=1

1

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

t |t� 1iht|c)],

Hin =nX

i=1

|1ih1|i ⌦ |0ih0|c +m+nX

j=n+1

|�ih�|j ⌦ |0ih0|c,

site energy

H(s) = (1� s)Hinitial + sHfinal,

Hinitial = Hin + (Ic � |0ih0|c),

Hfinal = Hin +TX

t=1

1

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

t |t� 1iht|c)],

Hin =nX

i=1

|1ih1|i ⌦ |0ih0|c +m+nX

j=n+1

|�ih�|j ⌦ |0ih0|c,

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

Feynman’s Hamiltonian

hopping term

Universality of non-stoqAQC Aharonov et al ‘04

H(s) = (1� s)Hinitial + sHfinal,

Hinitial = Hin + (Ic � |0ih0|c),

Hfinal = Hin +TX

t=1

1

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

t |t� 1iht|c)],

Hin =nX

i=1

|1ih1|i ⌦ |0ih0|c +m+nX

j=n+1

|�ih�|j ⌦ |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:H(s) = (1� s)Hinitial + sHfinal,

Hinitial = Hin + (Ic � |0ih0|c),

Hfinal = Hin +TX

t=1

1

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

t |t� 1iht|c)],

Hin =nX

i=1

|1ih1|i ⌦ |0ih0|c +m+nX

j=n+1

|�ih�|j ⌦ |0ih0|c,

H(s) = (1� s)Hinitial + sHfinal,

Hinitial = Hin + (Ic � |0ih0|c),

Hfinal = Hin +TX

t=1

1

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

t |t� 1iht|c)],

Hin =nX

i=1

|1ih1|i ⌦ |0ih0|c +m+nX

j=n+1

|�ih�|j ⌦ |0ih0|c,

• adiabatic quantum computation:

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

site energy

H(s) = (1� s)Hinitial + sHfinal,

Hinitial = Hin + (Ic � |0ih0|c),

Hfinal = Hin +TX

t=1

1

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

t |t� 1iht|c)],

Hin =nX

i=1

|1ih1|i ⌦ |0ih0|c +m+nX

j=n+1

|�ih�|j ⌦ |0ih0|c,

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

Feynman’s Hamiltonian

hopping term

Universality of non-stoqAQC Aharonov et al ‘04

H(s) = (1� s)Hinitial + sHfinal,

Hinitial = Hin + (Ic � |0ih0|c),

Hfinal = Hin +TX

t=1

1

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

t |t� 1iht|c)],

Hin =nX

i=1

|1ih1|i ⌦ |0ih0|c +m+nX

j=n+1

|�ih�|j ⌦ |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:H(s) = (1� s)Hinitial + sHfinal,

Hinitial = Hin + (Ic � |0ih0|c),

Hfinal = Hin +TX

t=1

1

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

t |t� 1iht|c)],

Hin =nX

i=1

|1ih1|i ⌦ |0ih0|c +m+nX

j=n+1

|�ih�|j ⌦ |0ih0|c,

H(s) = (1� s)Hinitial + sHfinal,

Hinitial = Hin + (Ic � |0ih0|c),

Hfinal = Hin +TX

t=1

1

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

t |t� 1iht|c)],

Hin =nX

i=1

|1ih1|i ⌦ |0ih0|c +m+nX

j=n+1

|�ih�|j ⌦ |0ih0|c,

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

Feynman’s Hamiltonian

H(s) = (1� s)Hinitial + sHfinal,

Hinitial = Hin + (Ic � |0ih0|c),

Hfinal = Hin +TX

t=1

1

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

t |t� 1iht|c)],

Hin =nX

i=1

|1ih1|i ⌦ |0ih0|c +m+nX

j=n+1

|�ih�|j ⌦ |0ih0|c,

• adiabatic quantum computation:

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

| i = 1pT + 1

TX

t=0

Ut · · ·U1|0i⌦n|T ic

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

t

Universality of non-stoqAQC Aharonov et al ‘04

H(s) = (1� s)Hinitial + sHfinal,

Hinitial = Hin + (Ic � |0ih0|c),

Hfinal = Hin +TX

t=1

1

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

t |t� 1iht|c)],

Hin =nX

i=1

|1ih1|i ⌦ |0ih0|c +m+nX

j=n+1

|�ih�|j ⌦ |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:H(s) = (1� s)Hinitial + sHfinal,

Hinitial = Hin + (Ic � |0ih0|c),

Hfinal = Hin +TX

t=1

1

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

t |t� 1iht|c)],

Hin =nX

i=1

|1ih1|i ⌦ |0ih0|c +m+nX

j=n+1

|�ih�|j ⌦ |0ih0|c,

H(s) = (1� s)Hinitial + sHfinal,

Hinitial = Hin + (Ic � |0ih0|c),

Hfinal = Hin +TX

t=1

1

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

t |t� 1iht|c)],

Hin =nX

i=1

|1ih1|i ⌦ |0ih0|c +m+nX

j=n+1

|�ih�|j ⌦ |0ih0|c,

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

Feynman’s Hamiltonian

H(s) = (1� s)Hinitial + sHfinal,

Hinitial = Hin + (Ic � |0ih0|c),

Hfinal = Hin +TX

t=1

1

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

t |t� 1iht|c)],

Hin =nX

i=1

|1ih1|i ⌦ |0ih0|c +m+nX

j=n+1

|�ih�|j ⌦ |0ih0|c,

• adiabatic quantum computation:

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

| i = 1pT + 1

TX

t=0

Ut · · ·U1|0i⌦n|T ic

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

Restriction to stoquastic HamiltoniansH(s) = (1� s)Hinitial + sHfinal,

Hinitial = Hin + (Ic � |0ih0|c),

Hfinal = Hin +TX

t=1

1

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

t |t� 1iht|c)],

Hin =nX

i=1

|1ih1|i ⌦ |0ih0|c +m+nX

j=n+1

|�ih�|j ⌦ |0ih0|c,

off-diagonal terms:

Each element of U should be non negative!

Restriction to stoquastic HamiltoniansH(s) = (1� s)Hinitial + sHfinal,

Hinitial = Hin + (Ic � |0ih0|c),

Hfinal = Hin +TX

t=1

1

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

t |t� 1iht|c)],

Hin =nX

i=1

|1ih1|i ⌦ |0ih0|c +m+nX

j=n+1

|�ih�|j ⌦ |0ih0|c,

off-diagonal terms:

Each element of U should be non negative!

X =

✓0 11 0

◆CNOT =

0

BB@

1 0 0 00 1 0 00 0 0 10 0 1 0

1

CCA

To↵oli

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) ]

Stoquastic AQC as a sampling problem

Arbitrarysingle-qubit

measurements!

Initial Hadamard transformationsare 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.

arbitrarysingle-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 ……

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.

With non-adaptivesingle-qubit measurements?

Universal QC with stoqAQC

|0i

|0i

|0i

…

|0i

…

……

H

H

UT · · ·U1

X, CNOT, Toffoli ……

…

|+i|+i

|+i

……

……

…

……

…

Z

Z

Z

…

Ux

e(2⇡i)0.j1j2...jn

e(2⇡i)0.j2...jn

e(2⇡i)0.jn

e2(⇡i)0.j1j2...jn

Kitaev’s Phase estimation and Shor’s algorithm

U†QFT

Inverse QFT j1j2⋮jn

Ux = ∑y

|xy mod N⟩⟨y |

U2x U2n−1

x

modular exponentiation = unitary reversible classical computation

controlled-phase

|0i H X

U2k

x

|0i H

U2k

x

⋱X

…

⋮

classical processing

}⟨X⟩R samples

Quantum-classical hybrid phase estimation

|0i H X

hXi

hY iU2k−1

x

|0i H

U2k−1x

⋱X

…

⋮ ⟨X⟩

|0i H Y

|0i H

⋱Y

⋮

U2k−1x U2k−1

x…

⋮

⋮

classical processing

⟨Y⟩

}}

e2πi0.jk...jn

R samples

jk = 0

jk = 1

|1i

R samples

Quantum-classical hybrid phase estimation

⇥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−1x

sampling & classical processing hXi

hY i

e2πi0.jnjn = 0

jn = 1

jn

hXi

hY ie2πi0.j1...jn

j1 = 1

j1 = 0j1

⋮

⋮

jkhXi

hY ie2πi0.jk...jn

jk = 0

jk = 1

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

Quantum-classical hybrid phase estimation

⇥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−1x

sampling & classical processing hXi

hY i

e2πi0.jnjn = 0

jn = 1

jn

hXi

hY ie2πi0.j1...jn

j1 = 1

j1 = 0j1

⋮

⋮

jkhXi

hY ie2πi0.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.)

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.

Quantum-classical hybrid algorithmclassical computerQuantum 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

“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

model・trial function

tuning・optimization

task

Neural netwrok W, tanh() backpropagation(gradient)

machine learning

Rayleigh-Ritz(Hartree-Fock)

orthogonal functions(Slater determinant)

diagonalization ofHermitian 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]

other?[Nakanishi-KF-Todo ’19]

machine learningground state

dynamics

Variational algorithms

Quantum circuit learning: supervised learning on near-term quantum devices

low depth quantum circuit

・・・

・・・U({✓i})

parameterized quantum circuitinput

data

loss function：

teacher data output from parameterizedquantum circuit

optimization using gradient

xj

L =X

j

[f(xj)� hA(xj , {✓i})i]2

pf(x)|0i+

p1� f(x)|1i

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

Generalization of nonlinear functions

teacher

prediction

nonlinear classification

Quantum circuit learning: supervised learning on near-term quantum devicesK. Mitarai, M. Negoro, M. Kitagawa, and KF Phys. Rev. A 98, 032309 (2018).and many others

Generalization of nonlinear functions

teacher

prediction

nonlinear classification

Quantum circuit learning: supervised learning on near-term quantum devicesK. Mitarai, M. Negoro, M. Kitagawa, and KF Phys. Rev. A 98, 032309 (2018).and many others

Variational quantum algorithms and parameterized quantum circuit

U(�1)

U(�2)

U(�3)

U(�4)

U(�5)

U(�6)

U(�7)

U(�8)

|0i|0i|0i

|0i|0i

| ({�k})i

for example

Rx

(�(1)k

)

Rx

(�(2)k

)

Rz(�(3)k )

Rz(�(4)k )

Rx

(�(5)k

)

Rx

(�(6)k

)

Rz(�(7)k )

Rz(�(8)k )

Rz(�) = e�i(�/2)Z

Rx

(�) = e�i(�/2)X

min{�k}

h ({�k})|H| ({�k})i

trial functionor

ansatz

variational optimization:

How can we obtain the gradient ?@

@�kh ({�k})|H| ({�k})i

Analytical differentiation of quantum circuitsU({�k}) =

Y

k

Wke�i(�k/2)PkParameterized quantum circuit:

hermitian & unitarysuch as Pauli operatorsfixed unitary

Analytical differentiation of quantum circuitsU({�k}) =

Y

k

Wke�i(�k/2)PkParameterized quantum circuit:

hermitian & unitarysuch as Pauli operatorsfixed unitary

Expectation value: hA({�k})i = h ({�k})|A| ({�k})i

where | ({�k})i = U({�k})|0i⌦n

Analytical differentiation of quantum circuitsU({�k}) =

Y

k

Wke�i(�k/2)PkParameterized quantum circuit:

hermitian & unitarysuch as Pauli operatorsfixed unitary

Expectation value: hA({�k})i = h ({�k})|A| ({�k})i

where | ({�k})i = U({�k})|0i⌦n

Analytical differentiation:@

@�lhA({�k})i =

hA({�1, ...,�l + ✏,�l+1, ...}i � hA({�1, ...,�l � ✏,�l+1, ...}i2 sin ✏

✏ = ⇡/2

Analytical differentiation of quantum circuitsU({�k}) =

Y

k

Wke�i(�k/2)PkParameterized quantum circuit:

hermitian & unitarysuch as Pauli operatorsfixed unitary

Expectation value: hA({�k})i = h ({�k})|A| ({�k})i

where | ({�k})i = U({�k})|0i⌦n

Analytical differentiation:@

@�lhA({�k})i =

hA({�1, ...,�l + ✏,�l+1, ...}i � hA({�1, ...,�l � ✏,�l+1, ...}i2 sin ✏

✏ = ⇡/2

hA(✓)i = h |e�i(✓/2)PAei(✓/2)P | i@

@✓hA(✓)i = h |(�iP/2)A| i+ h |A(iP/2)| i

= �i(1/2)(h |PA i � h |AP | i)

A = e�i(✓/2)PAei(✓/2)P( (hA(✓ + ✏)i =cos

2(✏/2)h ˜Ai+ sin

2(✏/2)hP ˜AP i

� i sin(✏/2) cos(✏/2)hP ˜Ai+ i cos(✏/2) sin(✏/2)h ˜AP i

Analytical differentiation of quantum circuits

・・・

RZ(�1) RX(�2) RZ(�3)

RZ(�4) RX(�5) RZ(�6)・・・

Analytical differentiation of quantum circuits

・・・

RZ(�1) RX(�2) RZ(�3)

RZ(�4) RX(�5) RZ(�6)・・・

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

e�i(�l/2)Pl@

@�l

= e�i�l+✏

2 Pl ー1

2 sin ✏( (e�i�l�✏

2 Pl

✏ = ⇡/2

Analytical differentiation of quantum circuits

See alsoM. Schuld, et al. (Xanadu) "Evaluating analytic gradients on quantum hardware." Physical Review A 99, 032331 (2019) → PennyLaneZ. 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(�1) RX(�2) RZ(�3)

RZ(�4) RX(�5) RZ(�6)・・・

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

e�i(�l/2)Pl@

@�l

= e�i�l+✏

2 Pl ー1

2 sin ✏( (e�i�l�✏

2 Pl

✏ = ⇡/2

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.

Sequential minimal optimization for quantum-classical hybrid algorithmsKen M. Nakanishi, Keisuke Fujii, Synge Todo, arXiv:1903.12166

・・・

・・・

U({✓i})parameterized quantum circuit

fix except for{✓i} ✓j

hA(✓)i = h |e�i(✓/2)PAei(✓/2)P | i= cos

2(✓/2)hAi+ sin

2(✓/2)hPAP i+ cos(✓/2) sin(✓/2)ih[A,P ]i

= ↵ sin(✓ + �) + �

Unknown parameters are only three.

Sequential minimal optimization for quantum-classical hybrid algorithmsKen M. Nakanishi, Keisuke Fujii, Synge Todo, arXiv:1903.12166

-1

-0.5

0

0.5

1

0.5π 1.0π 1.5π 2.0π

Estimation of the cost function at three independent points

exactminimum

・・・

・・・

U({✓i})parameterized quantum circuit

fix except for{✓i} ✓j

hA(✓)i = h |e�i(✓/2)PAei(✓/2)P | i= cos

2(✓/2)hAi+ sin

2(✓/2)hPAP i+ cos(✓/2) sin(✓/2)ih[A,P ]i

= ↵ sin(✓ + �) + �

Unknown parameters are only three.

Benchmark task：5qubit, 100 parameters

Comparison between gradient-based and -free optimizations

・・・

・・・

・・・

randomly chosen fixed unitary

U †(✓⇤)U(✓)

parameterizedquantumcircuit

|0i|0i|0i

|0i

fidelity

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

Optimization methods:

• 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 optimizations

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

SMO

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

• 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 optimizations

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

SMO

# of samples to estimate an observable is changed.

Variations of variational approaches:model・

trial functiontuning・

optimizationtask

Neural netwrok W, tanh() backpropagation(gradient)

machine learning

Rayleigh-Ritz(Hartree-Fock)

orthogonal functions(Slater determinant)

diagonalization ofHermitian matrix

(HF equation)ground state

Tensor network(MPS, PEPS,

MERA)tensor network singular value decomposition ground state

(dynamics)

Variational quantum

algorithms

parameterized quantum circuitwhich kind?

gradient?[Mitarai-Negoro-Kitagawa-KF ‘18]

other?[Nakanishi-KF-Todo ’19]

machine learningground state

dynamicsadvantage?

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: