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