# 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:

- Publication Date:

- Research Org.:
- Boston Univ., MA (United States)

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 1357856

- Alternate Identifier(s):
- OSTI ID: 1389776

- Grant/Contract Number:
- FG02-06ER46316

- Resource Type:
- Published Article

- Journal Name:
- Physical Review X

- Additional Journal Information:
- Journal Name: Physical Review X 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. Thu .
"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}

}

DOI: 10.1103/PhysRevX.7.021027

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