# Electron-Phonon Systems on a Universal Quantum Computer

## Abstract

We present an algorithm that extends existing quantum algorithms for simulating fermion systems in quantum chemistry and condensed matter physics to include bosons in general and phonons in particular. We introduce a qubit representation for the low-energy subspace of phonons which allows an efficient simulation of the evolution operator of the electron-phonon systems. As a consequence of the Nyquist-Shannon sampling theorem, the phonons are represented with exponential accuracy on a discretized Hilbert space with a size that increases linearly with the cutoff of the maximum phonon number. The additional number of qubits required by the presence of phonons scales linearly with the size of the system. The additional circuit depth is constant for systems with finite-range electron-phonon and phonon-phonon interactions and linear for long-range electron-phonon interactions. Our algorithm for a Holstein polaron problem was implemented on an Atos quantum learning machine quantum simulator employing the quantum phase estimation method. In conclusion, the energy and the phonon number distribution of the polaron state agree with exact diagonalization results for weak, intermediate, and strong electron-phonon coupling regimes.

- Authors:

- Fermi National Accelerator Lab. (FNAL), Batavia, IL (United States)

- Publication Date:

- Research Org.:
- Fermi National Accelerator Lab. (FNAL), Batavia, IL (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC), High Energy Physics (HEP) (SC-25)

- OSTI Identifier:
- 1438542

- Alternate Identifier(s):
- OSTI ID: 1469746

- Report Number(s):
- arXiv:1802.07347; FERMILAB-PUB-18-050-CD-T

Journal ID: ISSN 0031-9007; PRLTAO; 1657311

- Grant/Contract Number:
- AC02-07CH11359

- Resource Type:
- Journal Article: Accepted Manuscript

- Journal Name:
- Physical Review Letters

- Additional Journal Information:
- Journal Volume: 121; Journal Issue: 11; Journal ID: ISSN 0031-9007

- Publisher:
- American Physical Society (APS)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICS AND COMPUTING

### Citation Formats

```
Macridin, Alexandru, Spentzouris, Panagiotis, Amundson, James, and Harnik, Roni.
```*Electron-Phonon Systems on a Universal Quantum Computer*. United States: N. p., 2018.
Web. doi:10.1103/PhysRevLett.121.110504.

```
Macridin, Alexandru, Spentzouris, Panagiotis, Amundson, James, & Harnik, Roni.
```*Electron-Phonon Systems on a Universal Quantum Computer*. United States. doi:10.1103/PhysRevLett.121.110504.

```
Macridin, Alexandru, Spentzouris, Panagiotis, Amundson, James, and Harnik, Roni. Wed .
"Electron-Phonon Systems on a Universal Quantum Computer". United States.
doi:10.1103/PhysRevLett.121.110504.
```

```
@article{osti_1438542,
```

title = {Electron-Phonon Systems on a Universal Quantum Computer},

author = {Macridin, Alexandru and Spentzouris, Panagiotis and Amundson, James and Harnik, Roni},

abstractNote = {We present an algorithm that extends existing quantum algorithms for simulating fermion systems in quantum chemistry and condensed matter physics to include bosons in general and phonons in particular. We introduce a qubit representation for the low-energy subspace of phonons which allows an efficient simulation of the evolution operator of the electron-phonon systems. As a consequence of the Nyquist-Shannon sampling theorem, the phonons are represented with exponential accuracy on a discretized Hilbert space with a size that increases linearly with the cutoff of the maximum phonon number. The additional number of qubits required by the presence of phonons scales linearly with the size of the system. The additional circuit depth is constant for systems with finite-range electron-phonon and phonon-phonon interactions and linear for long-range electron-phonon interactions. Our algorithm for a Holstein polaron problem was implemented on an Atos quantum learning machine quantum simulator employing the quantum phase estimation method. In conclusion, the energy and the phonon number distribution of the polaron state agree with exact diagonalization results for weak, intermediate, and strong electron-phonon coupling regimes.},

doi = {10.1103/PhysRevLett.121.110504},

journal = {Physical Review Letters},

number = 11,

volume = 121,

place = {United States},

year = {Wed Sep 12 00:00:00 EDT 2018},

month = {Wed Sep 12 00:00:00 EDT 2018}

}