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arXiv:cond-mat/0302104 · Physical Review B

Discrete non-Abelian gauge theories in Josephson-junction arrays and quantum computation

Benoît Douçot, L. B. Ioffe, Julien Vidal

Published Jun 3, 2004• 98 citations
arXivHTMLPDF

Abstract

We discuss real-space lattice models equivalent to gauge theories with a discrete non-Abelian gauge group. We construct the Hamiltonian formalism which is appropriate for their solid-state physics implementation and outline their basic properties. The unusual physics of these systems is due to local constraints on the degrees of freedom which are variables localized on the links of the lattice. We discuss two types of constraints that become equivalent after a duality transformation for Abelian theories but are qualitatively different for non-Abelian ones. We emphasize highly nontrivial topological properties of the excitations (fluxons and charges) in these non-Abelian discrete lattice gauge theories. We show that an implementation of these models may provide one with the realization of an ideal quantum computer, that is, the computer that is protected from the noise and allows a full set of precise manipulations required for quantum computations. We suggest a few designs of the Josephson-junction arrays that provide the experimental implementations of these models and discuss the physical restrictions on the parameters of their junctions.

DOI
10.1103/physrevb.69.214501

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•Aug 11, 2014•2 citations
We discuss real-space lattice models equivalent to gauge theories with a discrete non-Abelian gauge group. We construct the Hamiltonian formalism which is appropriate for their solid-state physics implementation and outline their basic properties. The unusual physics of these systems is due to local constraints on the degrees of freedom which are variables localized on the links of the lattice. We discuss two types of constraints that become equivalent after a duality transformation for Abelian theories but are qualitatively different for non-Abelian ones. We emphasize highly nontrivial topological properties of the excitations (fluxons and charges) in these non-Abelian discrete lattice gauge theories. We show that an implementation of these models may provide one with the realization of an ideal quantum computer, that is, the computer that is protected from the noise and allows a full set of precise manipulations required for quantum computations. We suggest a few designs of the Josephson-junction arrays that provide the experimental implementations of these models and discuss the physical restrictions on the parameters of their junctions.
We construct the local Hamiltonian description of the Chern-Simons theory with discrete non-Abelian gauge group on a lattice. We show that the theory is fully determined by the phase factors associated with gauge transformations and classify all possible non-equivalent phase factors. We also construct the gauge invariant electric field operators that move fluxons around and create/anihilate them. We compute the resulting braiding properties of the fluxons. We apply our general results to the simplest class of non-Abelian groups, dihedral groups D_n.
The development of quantum technology and the advent of quantum computation have renewed interest in Hamiltonian formulations of lattice gauge theories. In contrast to Euclidean Monte Carlo calculations, Hamiltonian simulation requires a direct representation of the physical Hilbert space, an efficient treatment of enforcing Gauss' law as constraints, and a formulation of real-time dynamics in terms of degrees of freedom that can be encoded on quantum or quantum-inspired classical platforms. The starting point corresponds to finding a suitable basis for Hamiltonian lattice gauge theories. The existing literature focuses on several complementary approaches: electric basis, magnetic basis, gauge-fixed, quantum-link, qubit-regularised, finite-group, and orbifold formulations. This article primarily focuses on the prepotential and loop--string--hadron formulations, where non-Abelian Gauss laws are solved locally and the dynamics is written directly in terms of gauge-invariant loop, string and hadron excitations. The discussion covers the SU(2) and SU(3) constructions in one spatial dimension, the role of residual Abelian Gauss laws, and the extension to higher dimensions through point splitting. We conclude by highlighting its direct applications: eliminating redundant gauge degrees of freedom makes exact diagonalisation and tensor network calculations feasible for studying real-time dynamics, provides insights into thermalisation properties, enables mapping of physical degrees of freedom onto analog and digital quantum hardware, and enables quantum simulation of dynamics.
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 Zl\mathbb{Z}_lZl​ 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.
We study a lattice gauge theory in Wilson's Hamiltonian formalism. In view of the realization of a quantum simulator for QED in one dimension, we introduce an Abelian model with a discrete gauge symmetry , approximating the U (1) theory for large n . We analyze the role of the finiteness of the gauge fields and the properties of physical states, that satisfy a generalized Gauss's law. We finally discuss a possible implementation strategy, that involves an effective dynamics in physical space.
Abstract The design of quantum many body systems, which have to fulfill an extensive number of constraints, appears as a formidable challenge within the field of quantum simulation. Lattice gauge theories are a particular important class of quantum systems with an extensive number of local constraints and play a central role in high energy physics, condensed matter and quantum information. Whereas recent experimental progress points towards the feasibility of large-scale quantum simulation of abelian gauge theories, the quantum simulation of non-abelian gauge theories appears still elusive. In this paper we present minimal non-abelian lattice gauge theories, whereby we introduce the necessary formalism in well-known abelian gauge theories, such as the Jaynes–Cumming model. In particular, we show that certain minimal non-abelian lattice gauge theories can be mapped to three or four level systems, for which the design of a quantum simulator is standard with current technologies. Further we give an upper bound for the Hilbert space dimension of a one dimensional SU(2) lattice gauge theory, and argue that the implementation with current digital quantum computer appears feasible.
We construct two dimensional lattice models from transfer matrices of lattice gauge theories with discrete gauge groups. These transfer matrices are built out of local operators acting on links, vertices and plaquettes and are parametrized by the center of the gauge group algebra and its dual. They contain the familiar 2D quantum double models for a particular choice of parameters, which includes the well studied example of the toric code. However for more general choices of parameters the transfer matrix contains operators acting on links which can also be thought of as perturbations to the quantum double model driving it out of its topological phase towards a paramagnetic phase. These perturbations can be thought of as magnetic fields added to the system which destroy the exact solvability of the quantum double model. Nevertheless from the same transfer matrix with perturbations we exhibit exactly solvable models which remain in a quantum phase, thus nullifying the effect of the perturbation. The Abelian cases are shown to be in the quantum double phase whereas the non-Abelian phases are shown to be in a modified phase of the corresponding quantum double phase. This is shown by working with the groups Zn and S3 for the Abelian and non-Abelian cases respectively. The quantum phases are found by studying the excitations of these systems. The fusion rules and the statistics of these anyons indicate the quantum phases of these models. The implementation of these models can possibly improve the use of quantum double models for fault tolerant quantum computation. We then construct theories which arise from transfer matrices that are not the transfer matrices of lattice gauge theories. In particular we show that for the Z2 case this contains the double semion model. More generally for other discrete groups these transfer matrices contain the twisted quantum double models. These transfer matrices can be thought of as being obtained by introducing extra parameters into the transfer matrix of lattice gauge theories. These parameters are central elements belonging to the tensor products of the algebra and its dual and are associated to vertices and volumes of the three dimensional lattice. As in the case of the lattice gauge theories we construct the operators creating the excitations in this case and study their braiding and fusion properties.