Optimizing Variational Quantum Algorithms Using Pontryagin’s Minimum Principle

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 an...

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Published in:Physical review. X Vol. 7; no. 2; p. 021027
Main Authors: Yang, Zhi-Cheng, Rahmani, Armin, Shabani, Alireza, Neven, Hartmut, Chamon, Claudio
Format: Journal Article
Language:English
Published: College Park American Physical Society 18.05.2017
Subjects:
Algorithms
CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS
Combinatorial analysis
Computing costs
Control systems
Control theory
Coupling
Data processing
Digital computers
Evolutionary algorithms
Feedback control
Optimization
Parameter identification
Parameterization
Quantum computers
Quantum computing
Quantum phenomena
Quantum theory
Spin glasses
Switches
Thermal baths
Time
ISSN:2160-3308, 2160-3308
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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. Furthermore, 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.
AbstractList 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. Furthermore, 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.
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.
ArticleNumber 021027
Author Neven, Hartmut
Chamon, Claudio
Rahmani, Armin
Yang, Zhi-Cheng
Shabani, Alireza
Author_xml – sequence: 1
  givenname: Zhi-Cheng
  surname: Yang
  fullname: Yang, Zhi-Cheng
– sequence: 2
  givenname: Armin
  surname: Rahmani
  fullname: Rahmani, Armin
– sequence: 3
  givenname: Alireza
  surname: Shabani
  fullname: Shabani, Alireza
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  givenname: Hartmut
  surname: Neven
  fullname: Neven, Hartmut
– sequence: 5
  givenname: Claudio
  surname: Chamon
  fullname: Chamon, Claudio
BackLink https://www.osti.gov/biblio/1357856$$D View this record in Osti.gov
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Snippet We use Pontryagin’s minimum principle to optimize variational quantum algorithms. We show that for a fixed computation time, the optimal evolution has a...
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StartPage 021027
SubjectTerms Algorithms
CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS
Combinatorial analysis
Computing costs
Control systems
Control theory
Coupling
Data processing
Digital computers
Evolutionary algorithms
Feedback control
Optimization
Parameter identification
Parameterization
Quantum computers
Quantum computing
Quantum phenomena
Quantum theory
Spin glasses
Switches
Thermal baths
Time
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Title Optimizing Variational Quantum Algorithms Using Pontryagin’s Minimum Principle
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https://www.osti.gov/biblio/1357856
https://doaj.org/article/6e9dd5632b0945c2bfaa3bd85b0b6222
Volume 7
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