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) Nakanishi-KF-Todo, arXiv:1903.12166 Keisuke Fujii Graduate School of Science and Engineering, Osaka University JST PRESTO
Overview Quantum computer Classical computer quantum easy task classical easy task sampling update
Overview Quantum computer Classical computer quantum easy task classical easy task sampling update ・ Is there any situation where a quantum-classical hybrid approach provides a complexity theoretic advantage? → adiabatic quantum computation with stoquastic Hamiltonian
Overview Quantum computer Classical computer quantum easy task classical easy task sampling update ・ 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
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 non-universals model of quantum computation IQP DQC1 Boson Sampling (commuting circuits) (one-clean qubit model) Aaronson-Arkhipov ‘13 Bremner-Jozsa-Shepherd ‘11 Knill-Laflamme ‘98 Universal linear optics Morimae-KF-Fitzsimons ’14 Science (2015) | + i T KF et al, ‘18 | + i | 0 i H H | + i T | + i } … … U T I/ 2 n | + i Linear optical quantum computation Ising type interaction Experimental demonstrations J. B. Spring et al. Science 339 , 798 (2013) KF-Morimae ‘13 M. A. Broome, Science 339 , 794 (2013) Bremner-Montanaro-Shepherd ‘15 M. Tillmann et al., Nature Photo. 7 , 540 (2013) A. Crespi et al., Nature Photo. 7 , 545 (2013) NMR spin ensemble Gao-Wang-Duan ‘15 N. Spagnolo et al., Nature Photo. 8 , 615 (2014) J. Carolan et al., Science 349 , 711 (2015) Farhi-Harrow ‘16
Quantum computational supremacy non-universals model of quantum computation IQP DQC1 Boson Sampling (commuting circuits) (one-clean qubit model) Aaronson-Arkhipov ‘13 Bremner-Jozsa-Shepherd ‘11 Knill-Laflamme ‘98 Universal linear optics Morimae-KF-Fitzsimons ’14 Science (2015) | + i T KF et al, ‘18 | + i | 0 i H H | + i T | + i } … … U T I/ 2 n | + i Linear optical quantum computation Ising type interaction Experimental demonstrations J. B. Spring et al. Science 339 , 798 (2013) KF-Morimae ‘13 M. A. Broome, Science 339 , 794 (2013) Bremner-Montanaro-Shepherd ‘15 M. Tillmann et al., Nature Photo. 7 , 540 (2013) A. Crespi et al., Nature Photo. 7 , 545 (2013) NMR spin ensemble Gao-Wang-Duan ‘15 N. Spagnolo et al., Nature Photo. 8 , 615 (2014) J. Carolan et al., Science 349 , 711 (2015) Farhi-Harrow ‘16 → 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: X X H = J ij Z i Z J − h X i ij i
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: X X H = J ij Z i Z J − h X i ij i ・ Bose-Hubbard model with negative hopping: ( a † i a j + a † X X X H = − ω j a i ) − µ n i + U n i ( n i − 1) ij i i
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: X X H = J ij Z i Z J − h X i ij i ・ Bose-Hubbard model with negative hopping: ( a † i a j + a † X X X H = − ω j a i ) − µ n i + U n i ( n i − 1) ij i i ・ Heisernberg anti-ferro magnetic on a bipartite graph: X H = J ( X i X j + Y i Y j + Z i Z j ) − ij
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: X X H = J ij Z i Z J − h X i ij i ・ Bose-Hubbard model with negative hopping: ( a † i a j + a † X X X H = − ω j a i ) − µ n i + U n i ( n i − 1) ij i i ・ Heisernberg anti-ferro magnetic on a bipartite graph: X H = J ( X i X j + Y i Y j + Z i Z j ) − ij 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.
H T Circuit model and adiabatic model universal set of gate Circuit model:
T H Circuit model and adiabatic model universal set of gate Circuit model: Adiabatic quantum computation: a ( t ) b ( t ) H ( t ) = a ( t ) H initial + b ( t ) H final trivial state solution →→→→→ ↓ ↓ ↑ ↑↓ ↑ adiabatic theorem
Feynman’s seminal idea ‘84 Mapping each step of quantum computation to each site! H work ⊗ H clock working system for clock to track the step quantum computation
・ ・ ・ Feynman’s seminal idea ‘84 Mapping each step of quantum computation to each site! H work ⊗ H clock working system for clock to track the step quantum computation | 0 i ⊗ n | 0 i c U 1 | 0 i ⊗ n | 1 i c U t · · · U 1 | 0 i ⊗ n | t i c Clock makes all theses intermediate states orthogonal!
・ ・ ・ Feynman’s seminal idea ‘84 Mapping each step of quantum computation to each site! H work ⊗ H clock working system for clock to track the step quantum computation | 0 i ⊗ n | 0 i c U 1 | 0 i ⊗ n | 1 i c U t · · · U 1 | 0 i ⊗ n | t i c Clock makes all theses intermediate states orthogonal! H = 1 X [( | i ih i | + | i � 1 ih i � 1 | ) � ( | i ih i � 1 | + h . c)] tight-binding Hamiltonian: 2 i 1 X → ground state: p | Ψ i = | i i N i
Universality of non-stoqAQC Aharonov et al ‘04 � • energy penalty for the initial clock state: H initial = H in + ( I c � | 0 ih 0 | c ) , X • energy penalty for the initial state of the working system: n X H in = | 1 ih 1 | i ⌦ | 0 ih 0 | c + i =1 imposing initial state should be all 0s
Universality of non-stoqAQC Aharonov et al ‘04 � • energy penalty for the initial clock state: H initial = H in + ( I c � | 0 ih 0 | c ) , X • energy penalty for the initial state of the working system: n X H in = | 1 ih 1 | i ⌦ | 0 ih 0 | c + i =1 imposing initial state should be all 0s • tight-binding Hamiltonian (Kitaev-Shen-Vyalyi ’02) : � | ih | site energy hopping term T 1 2[ | t ih t | c + | t � 1 ih t � 1 | c � ( U t | t ih t � 1 | c + U † X H final = H in + t | t � 1 ih t | c )] , t =1 Feynman’s Hamiltonian n m + n
Universality of non-stoqAQC Aharonov et al ‘04 � • energy penalty for the initial clock state: H initial = H in + ( I c � | 0 ih 0 | c ) , X • energy penalty for the initial state of the working system: n X H in = | 1 ih 1 | i ⌦ | 0 ih 0 | c + i =1 imposing initial state should be all 0s • tight-binding Hamiltonian (Kitaev-Shen-Vyalyi ’02) : � | ih | site energy hopping term T 1 2[ | t ih t | c + | t � 1 ih t � 1 | c � ( U t | t ih t � 1 | c + U † X H final = H in + t | t � 1 ih t | c )] , t =1 Feynman’s Hamiltonian n m + n • adiabatic quantum computation: H ( s ) = (1 � s ) H initial + sH final , The lowest energy gap is always lower bounded by an inverse of polynomial.
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