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# Digitizing gauge fields: Lattice Monte Carlo results for future quantum computers

## Abstract

We report that in the near-future noisy intermediate-scale quantum (NISQ) era of quantum computing technology, applications of quantum computing will be limited to calculations of very modest scales in terms of the number of qubits used. The need to represent numeric quantities using limited resources leads to digitization errors which must be taken into account. As a first step towards quantum simulations of realistic high-energy physics problems, we explore classically the effects of digitizing elements of the $$\mathrm{SU}(2)$$ gauge group to a finite set. We consider several methods for digitizing the group, finding the best performance from an action-preserving projection onto a mesh. Working in (3+1) dimensions, we find that using $$\sim 7$$ (qu)bits to represent each $$\mathrm{SU}(2)$$ gauge link induces a digitization error on the order of $$10\%$$ in short-distance observables and $$2\%$$ in long-distance observables. Promisingly, our results indicate that each $$\mathrm{SU}(2)$$ gauge link can be represented by $$\mathcal{O}(10)$$ (qu)bits, from which we estimate that a $16^3$ $$\mathrm{SU}(2)$$ lattice could be simulated with no more than $$\mathcal{O}(10^5)$$ (qu)bits. Lastly, our results on digitization are also of interest as a form of lossy compression that could be used in high-performance classical computing to alleviate communications bottlenecks.

- Authors:

- Univ. of Colorado, Boulder, CO (United States)
- Fermi National Accelerator Lab. (FNAL), Batavia, IL (United States)
- Univ. of Colorado, Boulder, CO (United States); Fermi National Accelerator Lab. (FNAL), Batavia, IL (United States)
- Univ. of Colorado, Boulder, CO (United States); Brookhaven National Lab. (BNL), Upton, NY (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:
- 1488593

- Alternate Identifier(s):
- OSTI ID: 1546279

- Report Number(s):
- arXiv:1811.03629; FERMILAB-PUB-18-615-T

Journal ID: ISSN 2469-9926; PLRAAN; 1702968

- Grant/Contract Number:
- AC02-07CH11359; SC0010005; SC0012704

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Physical Review A

- Additional Journal Information:
- Journal Volume: 99; Journal Issue: 6; Journal ID: ISSN 2469-9926

- Publisher:
- American Physical Society (APS)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS

### Citation Formats

```
Hackett, Daniel C., Howe, Kiel, Hughes, Ciaran, Jay, William, Neil, Ethan T., and Simone, James N. Digitizing gauge fields: Lattice Monte Carlo results for future quantum computers. United States: N. p., 2019.
Web. doi:10.1103/PhysRevA.99.062341.
```

```
Hackett, Daniel C., Howe, Kiel, Hughes, Ciaran, Jay, William, Neil, Ethan T., & Simone, James N. Digitizing gauge fields: Lattice Monte Carlo results for future quantum computers. United States. doi:10.1103/PhysRevA.99.062341.
```

```
Hackett, Daniel C., Howe, Kiel, Hughes, Ciaran, Jay, William, Neil, Ethan T., and Simone, James N. Fri .
"Digitizing gauge fields: Lattice Monte Carlo results for future quantum computers". United States. doi:10.1103/PhysRevA.99.062341.
```

```
@article{osti_1488593,
```

title = {Digitizing gauge fields: Lattice Monte Carlo results for future quantum computers},

author = {Hackett, Daniel C. and Howe, Kiel and Hughes, Ciaran and Jay, William and Neil, Ethan T. and Simone, James N.},

abstractNote = {We report that in the near-future noisy intermediate-scale quantum (NISQ) era of quantum computing technology, applications of quantum computing will be limited to calculations of very modest scales in terms of the number of qubits used. The need to represent numeric quantities using limited resources leads to digitization errors which must be taken into account. As a first step towards quantum simulations of realistic high-energy physics problems, we explore classically the effects of digitizing elements of the $\mathrm{SU}(2)$ gauge group to a finite set. We consider several methods for digitizing the group, finding the best performance from an action-preserving projection onto a mesh. Working in (3+1) dimensions, we find that using $\sim 7$ (qu)bits to represent each $\mathrm{SU}(2)$ gauge link induces a digitization error on the order of $10\%$ in short-distance observables and $2\%$ in long-distance observables. Promisingly, our results indicate that each $\mathrm{SU}(2)$ gauge link can be represented by $\mathcal{O}(10)$ (qu)bits, from which we estimate that a $16^3$ $\mathrm{SU}(2)$ lattice could be simulated with no more than $\mathcal{O}(10^5)$ (qu)bits. Lastly, our results on digitization are also of interest as a form of lossy compression that could be used in high-performance classical computing to alleviate communications bottlenecks.},

doi = {10.1103/PhysRevA.99.062341},

journal = {Physical Review A},

number = 6,

volume = 99,

place = {United States},

year = {2019},

month = {6}

}

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