From quantum link models to D-theory: a resource efficient framework for the quantum simulation and computation of gauge theories

Abstract

Quantum link models provide an extension of Wilson’s lattice gauge theory in which the link Hilbert space is finite-dimensional and corresponds to a representation of an embedding algebra. In contrast to Wilson’s parallel transporters, quantum links are intrinsically quantum degrees of freedom. In D-theory, these discrete variables undergo dimensional reduction, thus giving rise to asymptotically free theories. In this way ( 1 + 1 ) -d C P ( N − 1 ) models emerge by dimensional reduction from ( 2 + 1 ) -d S U ( N ) quantum spin ladders, the ( 2 + 1 ) -d confining U ( 1 ) gauge theory emerges from the Abelian Coulomb phase of a ( 3 + 1 ) -d quantum link model, and ( 3 + 1 ) -d QCD arises from a non-Abelian Coulomb phase of a ( 4 + 1 ) -d S U ( 3 ) quantum link model, with chiral quarks arising naturally as domain wall fermions. Thanks to their finite-dimensional Hilbert space and their economical mechanism of reaching the continuum limit by dimensional reduction, quantum link models provide a resource efficient framework for the quantum simulation and computation of gauge theories. This article is part of the theme issue ‘Quantum technologies in particle physics’.

Ground-state phase diagram of quantum link electrodynamics in $(2+1)$-d

“…As mentioned before, within the link formulation, one can consider spin representations with increasing S to recover the Wilson-Kogut-Susskind limit. In that case, as is well known, the gauge-field operators become the corresponding raising and lowering operators operating in the infinite-dimensional local Hilbert space of a quantum rotor and recover the commutation of U and U † [65].…”

Introducing Fermionic Link Models

“…A theoretically appealing approach is to regulate this local infinity by gauge-invariant finite-dimensional operators, called quantum links [13][14][15]. For the simpler case of an (1) Abelian lattice gauge theory, these operators are quantum spins [16], while for non-Abelian gauge theories, such as QCD, they can be chosen to be appropriate finite-dimensional operators [17,18]. One particular representation using rishons (fermion bilinears) [17] is particularly suited for realizing these models in analog quantum simulators [19].…”

Introducing Fermionic Link Models

“…A theoretically appealing approach is to regulate this local infinity by gauge-invariant finite-dimensional operators, called quantum links [13][14][15]. For the simpler case of an 𝑈 (1) Abelian lattice gauge theory, these operators are quantum spins [16], while for non-Abelian gauge theories, such as QCD, they can be chosen to be appropriate finite-dimensional operators [17,18]. One particular representation using rishons (fermion bilinears) [17] is particularly suited for realizing these models in analog quantum simulators [19].…”