Optimizing Variational Quantum Algorithms Using Pontryagin’s Minimum Principle
Abstract
We use Pontryagin’s minimum principle to optimize variational quantum algorithms. We show that for a fixed computation time, the optimal evolution has a bangbang (square pulse) form, both for closed and open quantum systems with Markovian decoherence. Our findings support the choice of evolution ansatz in the recently proposed quantum approximate optimization algorithm. Focusing on the SherringtonKirkpatrick spin glass as an example, we find a systemsize independent distribution of the duration of pulses, with characteristic time scale set by the inverse of the coupling constants in the Hamiltonian. The optimality of the bangbang protocols and the characteristic time scale of the pulses provide an efficient parametrization of the protocol and inform the search for effective hybrid (classical and quantum) schemes for tackling combinatorial optimization problems. Moreover, we find that the success rates of our optimal bangbang protocols remain high even in the presence of weak external noise and coupling to a thermal bath.
 Authors:
 Boston Univ., Boston, MA (United States)
 Univ. of British Columbia, Vancouver, BC (Canada); Western Washington Univ., Bellingham, WA (United States)
 Google Inc., Venice, CA (United States)
 Publication Date:
 Research Org.:
 Boston Univ., MA (United States)
 Sponsoring Org.:
 USDOE
 OSTI Identifier:
 1357856
 Grant/Contract Number:
 FG0206ER46316
 Resource Type:
 Journal Article: Published Article
 Journal Name:
 Physical Review. X
 Additional Journal Information:
 Journal Volume: 7; Journal Issue: 2; Journal ID: ISSN 21603308
 Publisher:
 American Physical Society
 Country of Publication:
 United States
 Language:
 English
 Subject:
 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS
Citation Formats
Yang, Zhi Cheng, Rahmani, Armin, Shabani, Alireza, Neven, Hartmut, and Chamon, Claudio. Optimizing Variational Quantum Algorithms Using Pontryagin’s Minimum Principle. United States: N. p., 2017.
Web. doi:10.1103/PhysRevX.7.021027.
Yang, Zhi Cheng, Rahmani, Armin, Shabani, Alireza, Neven, Hartmut, & Chamon, Claudio. Optimizing Variational Quantum Algorithms Using Pontryagin’s Minimum Principle. United States. doi:10.1103/PhysRevX.7.021027.
Yang, Zhi Cheng, Rahmani, Armin, Shabani, Alireza, Neven, Hartmut, and Chamon, Claudio. 2017.
"Optimizing Variational Quantum Algorithms Using Pontryagin’s Minimum Principle". United States.
doi:10.1103/PhysRevX.7.021027.
@article{osti_1357856,
title = {Optimizing Variational Quantum Algorithms Using Pontryagin’s Minimum Principle},
author = {Yang, Zhi Cheng and Rahmani, Armin and Shabani, Alireza and Neven, Hartmut and Chamon, Claudio},
abstractNote = {We use Pontryagin’s minimum principle to optimize variational quantum algorithms. We show that for a fixed computation time, the optimal evolution has a bangbang (square pulse) form, both for closed and open quantum systems with Markovian decoherence. Our findings support the choice of evolution ansatz in the recently proposed quantum approximate optimization algorithm. Focusing on the SherringtonKirkpatrick spin glass as an example, we find a systemsize independent distribution of the duration of pulses, with characteristic time scale set by the inverse of the coupling constants in the Hamiltonian. The optimality of the bangbang protocols and the characteristic time scale of the pulses provide an efficient parametrization of the protocol and inform the search for effective hybrid (classical and quantum) schemes for tackling combinatorial optimization problems. Moreover, we find that the success rates of our optimal bangbang protocols remain high even in the presence of weak external noise and coupling to a thermal bath.},
doi = {10.1103/PhysRevX.7.021027},
journal = {Physical Review. X},
number = 2,
volume = 7,
place = {United States},
year = 2017,
month = 5
}

We use Pontryagin’s minimum principle to optimize variational quantum algorithms. We show that for a fixed computation time, the optimal evolution has a bangbang (square pulse) form, both for closed and open quantum systems with Markovian decoherence. Our findings support the choice of evolution ansatz in the recently proposed quantum approximate optimization algorithm. Focusing on the SherringtonKirkpatrick spin glass as an example, we find a systemsize independent distribution of the duration of pulses, with characteristic time scale set by the inverse of the coupling constants in the Hamiltonian. The optimality of the bangbang protocols and the characteristic time scale ofmore »

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