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Title: 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 bang-bang (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 Sherrington-Kirkpatrick spin glass as an example, we find a system-size 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 bang-bang 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 bang-bang protocols remain high even in the presence of weak external noise and coupling to a thermal bath.

Authors:
 [1];  [2];  [3];  [3];  [1]
  1. Boston Univ., Boston, MA (United States)
  2. Univ. of British Columbia, Vancouver, BC (Canada); Western Washington Univ., Bellingham, WA (United States)
  3. Google Inc., Venice, CA (United States)
Publication Date:
Research Org.:
Boston Univ., MA (United States)
Sponsoring Org.:
USDOE
OSTI Identifier:
1357856
Grant/Contract Number:
FG02-06ER46316
Resource Type:
Journal Article: Published Article
Journal Name:
Physical Review. X
Additional Journal Information:
Journal Volume: 7; Journal Issue: 2; Journal ID: ISSN 2160-3308
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 bang-bang (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 Sherrington-Kirkpatrick spin glass as an example, we find a system-size 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 bang-bang 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 bang-bang 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
}

Journal Article:
Free Publicly Available Full Text
Publisher's Version of Record at 10.1103/PhysRevX.7.021027

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  • In spirit of the principle of least action, which means that when a perturbation is applied to a physical system, its reaction is such that it modifies its state to “agree” with the perturbation by “minimal” change of its initial state. In particular, the electron field emission should produce the minimum current consistent with boundary conditions. It can be found theoretically by solving corresponding equations using different techniques. We apply here the variational method for the current calculation, which can be quite effective even when involving a short set of trial functions. The approach to a better result can bemore » monitored by the total current that should decrease when we on the right track. Here, we present only an illustration for simple geometries of devices with the electron flow. The development of these methods can be useful when the emitter and/or anode shapes make difficult the use of standard approaches. Though direct numerical calculations including particle-in-cell technique are very effective, but theoretical calculations can provide an important insight for understanding general features of flow formation and even sometimes be realized by simpler routines.« less
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