<!--HTML-->Quantum link models provide an extension of Wilson's lattice gauge theory in which the link variables have operator-valued entries. For example, in a U(1) quantum link model the link variables are raising and lowering operators of quantum spins that belong to a link-based SU(2) embedding algebra. For non-Abelian SU(N), Spin(N), or Sp(N) quantum link models, the embedding algebras are SU(2N), Spin(2N), and Sp(2N), respectively. In contrast to Wilson's framework, quantum link models can be realized in a finite-dimensional link Hilbert space corresponding to a representation of the embedding algebra. This is well suited for a resource efficient implementation of quantum link models in quantum simulation experiments. The quantum link dynamics can be embodied with a finite number of well-controlled states of ultra-cold matter, including atoms in an optical lattice, ions in a trap, or quantum circuits. For example, using dual variables, a densely encoded quantum circuit has recently been constructed for a (2+1)-d U(1) quantum link model on a triangular lattice that shows qualitatively new nematic phases with rich confining dynamics. Quantum spins and quantum links are discrete quantum variables which give rise to collective low-energy excitations. In (2+1)-d, SU(N) quantum spins give rise to massless Goldstone bosons which undergo dimensional reduction to (1+1)-d asymptotically free CP(N-1) models. This can be realized in quantum simulation experiments with alkaline-earth atoms in an optical lattice. In (3+1)-d, Abelian quantum links give rise to massless photons which undergo dimensional reduction to (2+1)-d confining U(1) gauge theory, which can be embodied with spin-ice. In (4+1)-d, non-Abelian quantum links give rise to massless gluons which undergo dimensional reduction to (3+1)-d Yang-Mills theory. Quarks are naturally included as domain wall fermions. The dimensional reduction of discrete quantum variables defines D-theory, in which quantum field theories emerge from a minimal set of degrees of freedom, thus enabling resource efficient implementations in quantum simulation experiments. D-theory provides a concrete vision how to realize the ultimate long-term goal of quantum simulating QCD.
<!--HTML-->Quantum link models provide an extension of Wilson's lattice gauge theory in which the link variables have operator-valued entries. For example, in a U(1) quantum link model the link variables are raising and lowering operators of quantum spins that belong to a link-based SU(2) embedding algebra. For non-Abelian SU(N), Spin(N), or Sp(N) quantum link models, the embedding algebras are SU(2N), Spin(2N), and Sp(2N), respectively. In contrast to Wilson's framework, quantum link models can be realized in a finite-dimensional link Hilbert space corresponding to a representation of the embedding algebra. This is well suited for a resource efficient implementation of quantum link models in quantum simulation experiments. The quantum link dynamics can be embodied with a finite number of well-controlled states of ultra-cold matter, including atoms in an optical lattice, ions in a trap, or quantum circuits. For example, using dual variables, a densely encoded quantum circuit has recently been constructed for a (2+1)-d U(1) quantum link model on a triangular lattice that shows qualitatively new nematic phases with rich confining dynamics. Quantum spins and quantum links are discrete quantum variables which give rise to collective low-energy excitations. In (2+1)-d, SU(N) quantum spins give rise to massless Goldstone bosons which undergo dimensional reduction to (1+1)-d asymptotically free CP(N-1) models. This can be realized in quantum simulation experiments with alkaline-earth atoms in an optical lattice. In (3+1)-d, Abelian quantum links give rise to massless photons which undergo dimensional reduction to (2+1)-d confining U(1) gauge theory, which can be embodied with spin-ice. In (4+1)-d, non-Abelian quantum links give rise to massless gluons which undergo dimensional reduction to (3+1)-d Yang-Mills theory. Quarks are naturally included as domain wall fermions. The dimensional reduction of discrete quantum variables defines D-theory, in which quantum field theories emerge from a minimal set of degrees of freedom, thus enabling resource efficient implementations in quantum simulation experiments. D-theory provides a concrete vision how to realize the ultimate long-term goal of quantum simulating QCD.
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 [Formula: see text] [Formula: see text] models emerge by dimensional reduction from [Formula: see text] [Formula: see text] quantum spin ladders, the [Formula: see text] confining [Formula: see text] gauge theory emerges from the Abelian Coulomb phase of a [Formula: see text] quantum link model, and [Formula: see text] QCD arises from a non-Abelian Coulomb phase of a [Formula: see text] [Formula: see text] 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'.
We use quantum link models to construct a quantum simulator for U(N) and SU(N) lattice gauge theories. These models replace Wilson’s classical link variables by quantum link operators, reducing the link Hilbert space to a finite number of dimensions. We show how to embody these quantum link models with fermionic matter with ultracold alkaline-earth atoms using optical lattices. Unlike classical simulations, a quantum simulator does not suffer from sign problems and can thus address the corresponding dynamics in real time. Using exact diagonalization results we show that these systems share qualitative features with QCD, including chiral symmetry breaking and we study the expansion of a chirally restored region in space in real time.
A new quantum link microstructure was proposed for the lattice quantum chromodynamics (QCD) Hamiltonian, replacing the Wilson gauge links with a bilinear of fermionic qubits, later generalized to D-theory. This formalism provides a general framework for building lattice field theory algorithms for quantum computing. We focus mostly on the simplest case of a quantum rotor for a single compact U(1) field. We also make some progress for non-Abelian setups, making it clear that the ideas developed in the U(1) case extend to other groups. These in turn are building blocks for 1 + 0 -dimensional ( 1 + 0 -D) matrix models, 1 + 1 -D sigma models and non-Abelian gauge theories in 2 + 1 and 3 + 1 dimensions. By introducing multiple flavors for the U(1) field, where the flavor symmetry is gauged, we can efficiently approach the infinite-dimensional Hilbert space of the quantum O(2) rotor with increasing flavors. The emphasis of the method is on preserving the symplectic algebra exchanging fermionic qubits by sigma matrices (or hard bosons) and developing a formal strategy capable of generalization to a SU ( 3 ) field for lattice QCD and other non-Abelian 1 + 1 -D sigma models or 3 + 1 -D gauge theories. For U(1), we discuss briefly the qubit algorithms for the study of the discrete 1 + 1 -D sine-Gordon equation. Published by the American Physical Society 2024
We use quantum link models to construct a quantum simulator for U(N) and SU(N) lattice gauge theories. These models replace Wilson's classical link variables by quantum link operators, reducing the link Hilbert space to a finite number of dimensions. We show how to embody these quantum link models with fermionic matter with ultracold alkaline-earth atoms using optical lattices. Unlike classical simulations, a quantum simulator does not suffer from sign problems and can thus address the corresponding dynamics in real time. Using exact diagonalization results we show that these systems share qualitative features with QCD, including chiral symmetry breaking and we study the expansion of a chirally restored region in space in real time.
Quantum link models (QLMs) are extensions of Wilson-type lattice gauge theories which realize exact gauge invariance with finite-dimensional Hilbert spaces. QLMs not only reproduce standard features of Wilson lattice gauge theories in equilibrium, but can also host new phenomena such as crystalline confined phases. The local constraints due to gauge invariance also provide kinetic restrictions that can influence substantially the real-time dynamics in these systems. We aim to characterize the nonequilibrium evolution in lattice gauge theories through the lens of dynamical quantum phase transitions, which provide general principles for real-time dynamics in quantum many-body systems. Specifically, we study quantum quenches for two representative cases, U(1) QLMs in (1+1)D and (2+1)D, for initial conditions exhibiting long-range order. Finally, we discuss the connection to the high-energy perspective and the experimental feasibility to observe the discussed phenomena in recent quantum simulator settings such as trapped ions, ultracold atoms, and Rydberg atoms.