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# Quantum algorithm for the Vlasov equation

## Abstract

Typically, the Vlasov-Maxwell system of equations, which describes classical plasma physics, is extremely challenging to solve, even by numerical simulation on powerful computers. By linearizing and assuming a Maxwellian background distribution function, we convert the Vlasov-Maxwell system into a Hamiltonian simulation problem. Then for the limiting case of electrostatic Landau damping, we design and verify a quantum algorithm, appropriate for a future error-corrected universal quantum computer. Although the classical simulation has costs that scale as $$\mathcal{O}(N_v t)$$ for a velocity grid with $$N_v$$ grid points and simulation time $t$, our quantum algorithm scales as $$\mathcal{O}(\text{polylog}(N_v) t/\delta)$$ where $$\delta$$ is the measurement error, and weaker scalings have been dropped. Extensions, including electromagnetics and higher dimensions, are discussed. A quantum computer could efficiently handle a high-resolution, six-dimensional phase-space grid, but the $$1/\delta$$ cost factor to extract an accurate result remains a difficulty. Our work provides insight into the possibility of someday achieving efficient plasma simulation on a quantum computer.

- Authors:

- Univ. of Colorado, Boulder, CO (United States)

- Publication Date:

- Research Org.:
- Univ. of Colorado, Boulder, CO (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC), Fusion Energy Sciences (FES) (SC-24)

- OSTI Identifier:
- 1579923

- Grant/Contract Number:
- SC0020393

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Physical Review A

- Additional Journal Information:
- Journal Volume: 100; Journal Issue: 6; Journal ID: ISSN 2469-9926

- Publisher:
- American Physical Society (APS)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 70 PLASMA PHYSICS AND FUSION TECHNOLOGY; 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS

### Citation Formats

```
Engel, Alexander, Smith, Graeme, and Parker, Scott E. Quantum algorithm for the Vlasov equation. United States: N. p., 2019.
Web. doi:10.1103/PhysRevA.100.062315.
```

```
Engel, Alexander, Smith, Graeme, & Parker, Scott E. Quantum algorithm for the Vlasov equation. United States. doi:10.1103/PhysRevA.100.062315.
```

```
Engel, Alexander, Smith, Graeme, and Parker, Scott E. Wed .
"Quantum algorithm for the Vlasov equation". United States. doi:10.1103/PhysRevA.100.062315.
```

```
@article{osti_1579923,
```

title = {Quantum algorithm for the Vlasov equation},

author = {Engel, Alexander and Smith, Graeme and Parker, Scott E.},

abstractNote = {Typically, the Vlasov-Maxwell system of equations, which describes classical plasma physics, is extremely challenging to solve, even by numerical simulation on powerful computers. By linearizing and assuming a Maxwellian background distribution function, we convert the Vlasov-Maxwell system into a Hamiltonian simulation problem. Then for the limiting case of electrostatic Landau damping, we design and verify a quantum algorithm, appropriate for a future error-corrected universal quantum computer. Although the classical simulation has costs that scale as $\mathcal{O}(N_v t)$ for a velocity grid with $N_v$ grid points and simulation time $t$, our quantum algorithm scales as $\mathcal{O}(\text{polylog}(N_v) t/\delta)$ where $\delta$ is the measurement error, and weaker scalings have been dropped. Extensions, including electromagnetics and higher dimensions, are discussed. A quantum computer could efficiently handle a high-resolution, six-dimensional phase-space grid, but the $1/\delta$ cost factor to extract an accurate result remains a difficulty. Our work provides insight into the possibility of someday achieving efficient plasma simulation on a quantum computer.},

doi = {10.1103/PhysRevA.100.062315},

journal = {Physical Review A},

number = 6,

volume = 100,

place = {United States},

year = {2019},

month = {12}

}

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