# $\sigma $ Models on Quantum Computers

## Abstract

We formulate a discretization of σ models suitable for simulation by quantum computers. Space is substituted with a lattice, as usually done in lattice field theory, while the target space (a sphere) is replaced by the “fuzzy sphere”, a construction well known from noncommutative geometry. Contrary to more naive discretizations of the sphere, in this construction the exact O(3) symmetry is maintained, which suggests that the discretized model is in the same universality class as the continuum model. That would allow for continuum results to be obtained for very rough discretizations of the target space as long as the space discretization is made fine enough. The cost of performing time evolution, measured as the number of controlled-not operations necessary, is 12LT/Δt, where L is the number of spatial sites, T the maximum time extent, and Δt the time spacing.

- Authors:

- Publication Date:

- Research Org.:
- Univ. of Maryland, College Park, MD (United States); George Washington Univ., Washington, DC (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC)

- Contributing Org.:
- NuQS Collaboration

- OSTI Identifier:
- 1559079

- Alternate Identifier(s):
- OSTI ID: 1610232

- Grant/Contract Number:
- FG02-95ER40907; FG02-93ER40762

- Resource Type:
- Published Article

- Journal Name:
- Physical Review Letters

- Additional Journal Information:
- Journal Name: Physical Review Letters Journal Volume: 123 Journal Issue: 9; Journal ID: ISSN 0031-9007

- Publisher:
- American Physical Society (APS)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICS AND COMPUTING; physics; lattice field theory; lattice gauge theory; quantum computation; quantum simulation

### Citation Formats

```
Alexandru, Andrei, Bedaque, Paulo F., Lamm, Henry, Lawrence, Scott, and NuQS Collaboration. σ Models on Quantum Computers. United States: N. p., 2019.
Web. doi:10.1103/PhysRevLett.123.090501.
```

```
Alexandru, Andrei, Bedaque, Paulo F., Lamm, Henry, Lawrence, Scott, & NuQS Collaboration. σ Models on Quantum Computers. United States. doi:10.1103/PhysRevLett.123.090501.
```

```
Alexandru, Andrei, Bedaque, Paulo F., Lamm, Henry, Lawrence, Scott, and NuQS Collaboration. Wed .
"σ Models on Quantum Computers". United States. doi:10.1103/PhysRevLett.123.090501.
```

```
@article{osti_1559079,
```

title = {σ Models on Quantum Computers},

author = {Alexandru, Andrei and Bedaque, Paulo F. and Lamm, Henry and Lawrence, Scott and NuQS Collaboration},

abstractNote = {We formulate a discretization of σ models suitable for simulation by quantum computers. Space is substituted with a lattice, as usually done in lattice field theory, while the target space (a sphere) is replaced by the “fuzzy sphere”, a construction well known from noncommutative geometry. Contrary to more naive discretizations of the sphere, in this construction the exact O(3) symmetry is maintained, which suggests that the discretized model is in the same universality class as the continuum model. That would allow for continuum results to be obtained for very rough discretizations of the target space as long as the space discretization is made fine enough. The cost of performing time evolution, measured as the number of controlled-not operations necessary, is 12LT/Δt, where L is the number of spatial sites, T the maximum time extent, and Δt the time spacing.},

doi = {10.1103/PhysRevLett.123.090501},

journal = {Physical Review Letters},

number = 9,

volume = 123,

place = {United States},

year = {2019},

month = {8}

}

DOI: 10.1103/PhysRevLett.123.090501

*Citation information provided by*

Web of Science

Web of Science

Works referenced in this record:

##
Monte Carlo simulations of lattice models with finite subgroups of SU(3) as gauge groups

journal, December 1981

- Bhanot, G.; Rebbi, C.
- Physical Review D, Vol. 24, Issue 12

##
Asymptotically free models and discrete non-Abelian groups

journal, July 2001

- Caracciolo, Sergio; Montanari, Andrea; Pelissetto, Andrea
- Physics Letters B, Vol. 513, Issue 1-2

##
Scalar quantum field theories as a benchmark for near-term quantum computers

journal, March 2019

- Yeter-Aydeniz, Kübra; Dumitrescu, Eugene F.; McCaskey, Alex J.
- Physical Review A, Vol. 99, Issue 3

##
Dual variables for lattice gauge theories and the phase structure of $Z\left(N\right)\mathrm{}$ systems

journal, February 1980

- Ukawa, Akira; Windey, Paul; Guth, Alan H.
- Physical Review D, Vol. 21, Issue 4

##
Noncommutative field theory

journal, November 2001

- Douglas, Michael; Nekrasov, Nikita
- Reviews of Modern Physics, Vol. 73, Issue 4

##
Multi-spin coding: A very efficient technique for Monte Carlo simulations of spin systems

journal, May 1981

- Jacobs, Laurence; Rebbi, Claudio
- Journal of Computational Physics, Vol. 41, Issue 1

##
Internal space decimation for lattice gauge theories

journal, November 1984

- Flyvbjerg, H.
- Nuclear Physics B, Vol. 240, Issue 4

##
Real-time dynamics of lattice gauge theories with a few-qubit quantum computer

journal, June 2016

- Martinez, Esteban A.; Muschik, Christine A.; Schindler, Philipp
- Nature, Vol. 534, Issue 7608

##
Group space decimation: A way to simulate QCD by the 1080 element subgroup of SU(3)?

journal, September 1984

- Flyvbjerg, Henrik
- Nuclear Physics B, Vol. 243, Issue 2

##
Low-energy effective theories of quantum spin and quantum link models

journal, March 2001

- Schlittgen, B.; Wiese, U. -J.
- Physical Review D, Vol. 63, Issue 8

##
Oracles for Gauss's law on digital quantum computers

journal, April 2019

- Stryker, Jesse R.
- Physical Review A, Vol. 99, Issue 4

##
Lattice simulations of real-time quantum fields

journal, February 2007

- Berges, J.; Borsányi, Sz.; Sexty, D.
- Physical Review D, Vol. 75, Issue 4

##
Phase structure of lattice gauge theories for non-abelian subgroups of SU(3)

journal, April 1981

- Grosse, H.; Kühnelt, H.
- Physics Letters B, Vol. 101, Issue 1-2

##
Quantum-classical computation of Schwinger model dynamics using quantum computers

journal, September 2018

- Klco, N.; Dumitrescu, E. F.; McCaskey, A. J.
- Physical Review A, Vol. 98, Issue 3

##
Quantum field theory on noncommutative spaces

journal, May 2003

- Szabo, R.
- Physics Reports, Vol. 378, Issue 4

##
Gauge-invariant rotor Hamiltonian from dual variables of 3D $U\mathbf{(}1\mathbf{)}$ gauge theory

journal, April 2019

- Unmuth-Yockey, Judah F.
- Physical Review D, Vol. 99, Issue 7

##
O(3) nonlinear sigma model in $1+1$ dimensions with matrix product states

journal, April 2019

- Bruckmann, Falk; Jansen, Karl; Kühn, Stefan
- Physical Review D, Vol. 99, Issue 7

##
Simulation of Nonequilibrium Dynamics on a Quantum Computer

journal, October 2018

- Lamm, Henry; Lawrence, Scott
- Physical Review Letters, Vol. 121, Issue 17

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

journal, June 2019

- Hackett, Daniel C.; Howe, Kiel; Hughes, Ciaran
- Physical Review A, Vol. 99, Issue 6

##
Monte Carlo study of Abelian lattice gauge theories

journal, October 1979

- Creutz, Michael; Jacobs, Laurence; Rebbi, Claudio
- Physical Review D, Vol. 20, Issue 8

##
Universal features of the Abelian Polyakov loop in $1+1$ dimensions

journal, November 2018

- Unmuth-Yockey, J.; Zhang, Jin; Bazavov, A.
- Physical Review D, Vol. 98, Issue 9

##
Phase diagram of a truncated tetrahedral model

journal, August 2016

- Krcmar, Roman; Gendiar, Andrej; Nishino, Tomotoshi
- Physical Review E, Vol. 94, Issue 2

##
Schwinger-Keldysh formalism on the lattice: A faster algorithm and its application to field theory

journal, June 2017

- Alexandru, Andrei; Başar, Gökçe; Bedaque, Paulo F.
- Physical Review D, Vol. 95, Issue 11

##
Gauge-invariant implementation of the Abelian-Higgs model on optical lattices

journal, October 2015

- Bazavov, A.; Meurice, Y.; Tsai, S. -W.
- Physical Review D, Vol. 92, Issue 7

##
Numerical simulations of a non-commutative theory: the scalar model on the fuzzy sphere

journal, May 2007

- Panero, Marco
- Journal of High Energy Physics, Vol. 2007, Issue 05

##
Continuum limit of two-dimensional spin models with continuous symmetry and conformal quantum field theory

journal, January 1998

- Patrascioiu, Adrian; Seiler, Erhard
- Physical Review E, Vol. 57, Issue 1

##
The Fermion Doubling Problem and Noncommutative Geometry

journal, June 2000

- Balachandran, A. P.; Govindarajan, T. R.; Ydri, B.
- Modern Physics Letters A, Vol. 15, Issue 19

##
Quantum simulation of lattice gauge theories using Wilson fermions

journal, June 2018

- Zache, T. V.; Hebenstreit, F.; Jendrzejewski, F.
- Quantum Science and Technology, Vol. 3, Issue 3

##
Digitization of scalar fields for quantum computing

journal, May 2019

- Klco, Natalie; Savage, Martin J.
- Physical Review A, Vol. 99, Issue 5

##
Asymptotically free theories based on discrete subgroups

journal, March 2001

- Hasenfratz, Peter; Niedermayer, Ferenc
- Nuclear Physics B - Proceedings Supplements, Vol. 94, Issue 1-3

##
Simulating Nonequilibrium Quantum Fields with Stochastic Quantization Techniques

journal, November 2005

- Berges, J.; Stamatescu, I. -O.
- Physical Review Letters, Vol. 95, Issue 20

##
Quantum Simulation of the Universal Features of the Polyakov Loop

journal, November 2018

- Zhang, Jin; Unmuth-Yockey, J.; Zeiher, J.
- Physical Review Letters, Vol. 121, Issue 22

##
QCD as a quantum link model

journal, September 1999

- Brower, R.; Chandrasekharan, S.; Wiese, U. -J.
- Physical Review D, Vol. 60, Issue 9

##
Electron-Phonon Systems on a Universal Quantum Computer

journal, September 2018

- Macridin, Alexandru; Spentzouris, Panagiotis; Amundson, James
- Physical Review Letters, Vol. 121, Issue 11

##
Quantum link models: A discrete approach to gauge theories

journal, May 1997

- Chandrasekharan, S.; Wiese, U. -J
- Nuclear Physics B, Vol. 492, Issue 1-2

##
Discrete subsets of SU(3) for lattice gauge theory

journal, June 1982

- Lisboa, P.; Michael, C.
- Physics Letters B, Vol. 113, Issue 4