With the long term perspective of using quantum computers and tensor networks for lattice gauge theory simulations, an efficient method of digitizing gauge group elements is needed. We thus present our results for a handful of discretization approaches for the non-trivial example of SU(2), such as its finite subgroups, as well as different classes of finite subsets. We focus our attention on a freezing transition observed towards weak couplings. A generalized version of the Fibonacci spiral appears to be particularly efficient and close to optimal.With the long term perspective of using quantum computers and tensor networks for lattice gauge theory simulations, an efficient method of digitizing gauge group elements is needed. We thus present our results for a handful of discretization approaches for the non-trivial example of SU(2), such as its finite subgroups, as well as different classes of finite subsets. We focus our attention on a freezing transition observed towards weak couplings. A generalized version of the Fibonacci spiral appears to be particularly efficient and close to optimal.

arXiv:2212.09496Dec 19, 20220 citations

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

Daniel C. Hackett, Kiel Howe, Ciaran Hughes +3 more

In the near-future noisy intermediate-scale quantum 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 use ensembles of SU(2) lattice gauge fields generated with standard classical computing to explore the effects of digitizing elements of the 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 $\ensuremath{\sim}7$ (qu)bits to represent each $\mathrm{SU}(2)$ gauge link induces a digitization error of 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 $O(10)$ (qu)bits, from which we estimate that a ${16}^{3}\phantom{\rule{4pt}{0ex}}\mathrm{SU}(2)$ lattice could be simulated with no more than $O({10}^{5})$ (qu)bits. 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.In the near-future noisy intermediate-scale quantum 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 use ensembles of SU(2) lattice gauge fields generated with standard classical computing to explore the effects of digitizing elements of the 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 $\ensuremath{\sim}7$ (qu)bits to represent each $\mathrm{SU}(2)$ gauge link induces a digitization error of 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 $O(10)$ (qu)bits, from which we estimate that a ${16}^{3}\phantom{\rule{4pt}{0ex}}\mathrm{SU}(2)$ lattice could be simulated with no more than $O({10}^{5})$ (qu)bits. 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.

arXiv:1811.03629Physical Review AJun 28, 201961 citations

Gauge field digitization in the Hamiltonian limit

Attila Pásztor, Dávid Pesznyák

Quantum computers can circumvent the numerical sign problem in gauge theories at finite density or in real time. Quantum simulations of gauge theories require a finite-dimensional representation of continuous gauge fields. Replacing a continuous gauge group by a finite subgroup can substantially reduce the required quantum resources, but introduces digitization errors that must be controlled in the Hamiltonian, or continuous-time, limit. Previous studies, using the isotropic Euclidean lattices showed that the freezing transition of the discrete subgroup can make it a bad approximation for the continuous group at large Euclidean couplings. Here, we study the digitization of U(1) by its Z( $N$ ) subgroups in 2+1 dimensions using anisotropic Euclidean lattices. We derive the trajectories of the spatial and temporal gauge couplings along which the Hamiltonian limit is approached at fixed Hamiltonian coupling. While the temporal coupling exhibits power-law scaling in the continuous U(1) theory, it grows only logarithmically for finite Z( $N$ ). Using classical lattice simulations and exact diagonalization, we verify that these trajectories reproduce the corresponding Hamiltonian theories. We find that the freezing transition persists in the Hamiltonian limit of discrete gauge groups and that finite- $N$ theories can differ substantially from U(1) even outside the frozen regime, in contrast to the behavior on isotropic Euclidean lattices, where for small couplings, the discrete group provides a very accurate approximation of the continuous group. Our results provide a classical benchmark for quantifying the systematic errors due to gauge-field digitization in quantum simulations.Quantum computers can circumvent the numerical sign problem in gauge theories at finite density or in real time. Quantum simulations of gauge theories require a finite-dimensional representation of continuous gauge fields. Replacing a continuous gauge group by a finite subgroup can substantially reduce the required quantum resources, but introduces digitization errors that must be controlled in the Hamiltonian, or continuous-time, limit. Previous studies, using the isotropic Euclidean lattices showed that the freezing transition of the discrete subgroup can make it a bad approximation for the continuous group at large Euclidean couplings. Here, we study the digitization of U(1) by its Z($N$) subgroups in 2+1 dimensions using anisotropic Euclidean lattices. We derive the trajectories of the spatial and temporal gauge couplings along which the Hamiltonian limit is approached at fixed Hamiltonian coupling. While the temporal coupling exhibits power-law scaling in the continuous U(1) theory, it grows only logarithmically for finite Z($N$). Using classical lattice simulations and exact diagonalization, we verify that these trajectories reproduce the corresponding Hamiltonian theories. We find that the freezing transition persists in the Hamiltonian limit of discrete gauge groups and that finite-$N$ theories can differ substantially from U(1) even outside the frozen regime, in contrast to the behavior on isotropic Euclidean lattices, where for small couplings, the discrete group provides a very accurate approximation of the continuous group. Our results provide a classical benchmark for quantifying the systematic errors due to gauge-field digitization in quantum simulations.

arXiv:2609.07886Sep 7, 20260 citations

Gluon field digitization via group space decimation for quantum computers

Ji Yao, Henry Lamm, Shuchen Zhu

Efficient digitization is required for quantum simulations of gauge theories. Schemes based on discrete subgroups use fewer qubits at the cost of systematic errors. We systematize this approach by deriving a single plaquette action for approximating general continuous gauge groups through integrating out field fluctuations. This provides insight into the effectiveness of these approximations, and how they could be improved. We accompany the scheme by simulations of pure gauge over the largest discrete subgroup of $SU(3)$ up to the third order.Efficient digitization is required for quantum simulations of gauge theories. Schemes based on discrete subgroups use fewer qubits at the cost of systematic errors. We systematize this approach by deriving a single plaquette action for approximating general continuous gauge groups through integrating out field fluctuations. This provides insight into the effectiveness of these approximations, and how they could be improved. We accompany the scheme by simulations of pure gauge over the largest discrete subgroup of $SU(3)$ up to the third order.

arXiv:2005.14221Physical review. D/Physical review. D.Dec 21, 202070 citations

Magic of discrete lattice gauge theories

Gianluca Esposito, Simone Cepollaro, Luigi Cappiello +1 more

Simulation of quantum field theories and fundamental interactions are one of the most challenging tasks in modern particle physics. Classical computers generally fail to reproduce accurate results when it comes to strongly coupled theories such as QCD. Recent developments in quantum technologies open up the possibility of simulating such physical regimes by using quantum computers. In this paper, we study the quantum resource related to the simulability of a quantum theory, i.e. non-stabilizerness for Lattice Gauge Theory (LGT) with discrete symmetry gauge groups. We show that enforcing gauge constraints for $\mathbb{Z}_l$ LGTs has no cost in terms of this resource and discuss the relation between non-abelianity of the gauge group with the average non-stabilizerness of the gauge invariant Hilbert space.Simulation of quantum field theories and fundamental interactions are one of the most challenging tasks in modern particle physics. Classical computers generally fail to reproduce accurate results when it comes to strongly coupled theories such as QCD. Recent developments in quantum technologies open up the possibility of simulating such physical regimes by using quantum computers. In this paper, we study the quantum resource related to the simulability of a quantum theory, i.e. non-stabilizerness for Lattice Gauge Theory (LGT) with discrete symmetry gauge groups. We show that enforcing gauge constraints for $\mathbb{Z}_l$ LGTs has no cost in terms of this resource and discuss the relation between non-abelianity of the gauge group with the average non-stabilizerness of the gauge invariant Hilbert space.

hep-latCross-listedarXiv:2601.15842International Journal of Geometric Methods in Modern PhysicsJan 23, 20262 citations

Gluon digitization via character expansion for quantum computers

Ji Yao, Henry Lamm, Shuchen Zhu

Efficient digitization is required for quantum simulations of gauge theories. Schemes based on discrete subgroups use a smaller, fixed number of qubits at the cost of systematic errors. We systematize this approach by deriving the single plaquette action through matching the continuous group action to that of a discrete one via group character expansions modulo the field fluctuation contributions. We accompany this scheme by simulations of pure gauge over the largest discrete crystal-like subgroup of $SU(3)$ up to the fifth order in the coupling constant.Efficient digitization is required for quantum simulations of gauge theories. Schemes based on discrete subgroups use a smaller, fixed number of qubits at the cost of systematic errors. We systematize this approach by deriving the single plaquette action through matching the continuous group action to that of a discrete one via group character expansions modulo the field fluctuation contributions. We accompany this scheme by simulations of pure gauge over the largest discrete crystal-like subgroup of $SU(3)$ up to the fifth order in the coupling constant.

arXiv:2203.02330Physical review. D/Physical review. D.Jun 8, 202318 citations

(3+1)

dimensions, we find that using $\ensuremath{\sim}7$ (qu)bits to represent each

\mathrm{SU}(2)

gauge link induces a digitization error of 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

O(10)

(qu)bits, from which we estimate that a

{16}^{3}\phantom{\rule{4pt}{0ex}}\mathrm{SU}(2)

lattice could be simulated with no more than

O({10}^{5})

(qu)bits. 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.

N

) subgroups in 2+1 dimensions using anisotropic Euclidean lattices. We derive the trajectories of the spatial and temporal gauge couplings along which the Hamiltonian limit is approached at fixed Hamiltonian coupling. While the temporal coupling exhibits power-law scaling in the continuous U(1) theory, it grows only logarithmically for finite Z(

N

). Using classical lattice simulations and exact diagonalization, we verify that these trajectories reproduce the corresponding Hamiltonian theories. We find that the freezing transition persists in the Hamiltonian limit of discrete gauge groups and that finite-

N

theories can differ substantially from U(1) even outside the frozen regime, in contrast to the behavior on isotropic Euclidean lattices, where for small couplings, the discrete group provides a very accurate approximation of the continuous group. Our results provide a classical benchmark for quantifying the systematic errors due to gauge-field digitization in quantum simulations.

SU(3)

up to the third order.

\mathbb{Z}_l

LGTs has no cost in terms of this resource and discuss the relation between non-abelianity of the gauge group with the average non-stabilizerness of the gauge invariant Hilbert space.

SU(3)

up to the fifth order in the coupling constant.