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arXiv:2206.14679 · EPJ Quantum Technology

Dihedral lattice gauge theories on a quantum annealer

Michael Fromm, Owe Philipsen, Christopher Winterowd

Published Aug 25, 2023• 9 citations
arXivHTMLPDF

Abstract

Abstract We study lattice gauge theory with discrete, non-Abelian gauge groups. We extend the formalism of previous studies on D-Wave’s quantum annealer as a computing platform to finite, simply reducible gauge groups. As an example, we use the dihedral group $D_{n}$ D n with $n=3,4$ n = 3 , 4 on a two plaquette ladder for which we provide proof-of-principle calculations of the ground-state and employ the known time evolution formalism with Feynman clock states.

DOI
10.1140/epjqt/s40507-023-00188-9

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•Aug 30, 2021•1 citation
Abstract We study lattice gauge theory with discrete, non-Abelian gauge groups. We extend the formalism of previous studies on D-Wave’s quantum annealer as a computing platform to finite, simply reducible gauge groups. As an example, we use the dihedral group DnD_{n}Dn​ D n with n=3,4n=3,4n=3,4 n = 3 , 4 on a two plaquette ladder for which we provide proof-of-principle calculations of the ground-state and employ the known time evolution formalism with Feynman clock states.
Lattice gauge theory is an essential tool for strongly interacting non-Abelian fields, such as those in quantum chromodynamics where lattice results have been of central importance for several decades. Recent studies suggest that quantum computers could extend the reach of lattice gauge theory in dramatic ways, but the usefulness of quantum annealing hardware for lattice gauge theory has not yet been explored. In this work, we implement SU(2) pure gauge theory on a quantum annealer for lattices comprising a few plaquettes in a row with a periodic boundary condition. These plaquettes are in two spatial dimensions and calculations use the Hamiltonian formulation where time is not discretized. Numerical results are obtained from calculations on D-Wave Advantage hardware for eigenvalues, eigenvectors, vacuum expectation values, and time evolution. The success of this initial exploration indicates that the quantum annealer might become a useful hardware platform for some aspects of lattice gauge theories.
Lattice gauge theories provide a first-principles, non-perturbative formulation of the fundamental interactions of the Standard Model, and their Hamiltonian reformulation on a quantum simulator offers a promising route to regimes---real-time dynamics, finite baryon density---that lie beyond the reach of Euclidean Monte Carlo methods. With the ultimate goal of overcoming these shortcomings, over the last decade, quantum simulations of Abelian lattice gauge theories have expanded from the first proofs of principle to the experimental study of confinement and string breaking by pair production. One of the next frontiers lies in the simulation of non-Abelian theories, which are significantly more demanding both in local Hilbert space dimension and in symmetry-protection requirements, and ultimately essential for addressing open problems in high-energy physics. In this thesis, we investigate dihedral non-Abelian lattice gauge theories DND_NDN​---whose small order and finite local dimension make D3D_3D3​ and D4D_4D4​ natural targets for near-term quantum platforms---as a minimal yet physically rich arena for the study of non-Abelian gauge dynamics. After introducing dihedral groups and the associated lattice gauge theories, we propose a gauge-fixed ladder model that reduces the effective dimensionality while preserving the essential physics. We use it to show that the D3D_3D3​ theory on a ladder displays a smooth crossover---rather than a sharp phase transition---between electric- and magnetic-dominated regimes, with the nonstabilizerness peaking in the crossover region. Furthermore, we introduce a formulation of the model compatible with the analysis of the Eigenstate Thermalization Hypothesis. We then study string breaking in pure-gauge DND_NDN​ ladders in the presence of static charges through tensor-network DMRG simulations in the rishon formalism. We show that for odd NNN static charges are screened by gluelumps---charge-glueball bound states enabled by the trivial centre of the group---while for even NNN a fundamental flux string persists. Turning to the quantum-simulation side, we introduce a dynamical post-selection (DPS) protocol in which the local gauge charges are tested by mid-circuit measurements on auxiliary registers, and we show that gauge protection emerges through a sharp quantum Zeno transition in the spectrum of an associated Liouvillian. We verify the generality of this picture on Z2\mathbb{Z}_2Z2​, Z3\mathbb{Z}_3Z3​, and truncated U(1)U(1)U(1) gauge groups, identifying an optimal window of measurement frequencies that survives also in the presence of noisy measurements. Finally, we benchmark DPS against an alternative post-processed symmetry verification scheme on a two-plaquette D3D_3D3​ model designed for state-of-the-art qudit hardware. We find that both protocols significantly improve the reconstructed dynamics, with DPS tracking the exact evolution over longer times. Taken together, the results of this thesis position dihedral gauge theories as a concrete stepping stone from Abelian proofs of principle toward genuinely non-Abelian quantum simulation on near-term devices.
We describe the simulation of dihedral gauge theories on digital quantum computers. The non-Abelian discrete gauge group DN{D}_{N}DN​---the dihedral group---serves as an approximation to $U(1)\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ lattice gauge theory. In order to carry out such a lattice simulation, we detail the construction of efficient quantum circuits to realize basic primitives including the non-Abelian Fourier transform over DN{D}_{N}DN​, the trace operation, and the group multiplication and inversion operations. For each case the required quantum resources scale linearly or as low-degree polynomials in n=logNn=\mathrm{log}Nn=logN. We experimentally benchmark our gates on the Rigetti Aspen-9 quantum processor for the case of D4{D}_{4}D4​. The estimated fidelity of all D4{D}_{4}D4​ gates was found to exceed 80%.
We explore the potential of D-Wave's quantum annealers for computing some of the basic components required for quantum simulations of standard model physics. By implementing a basic multigrid (including ``zooming'') and specializing Feynman-clock algorithms, D-Wave's Advantage is used to study harmonic and anharmonic oscillators relevant for lattice scalar field theories and effective field theories, the time evolution of a single plaquette of SU(3) Yang-Mills lattice gauge field theory, and the dynamics of flavor entanglement in four-neutrino systems.
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 describe the simulation of dihedral gauge theories on digital quantum computers. The nonabelian discrete gauge group DND_NDN​ -- the dihedral group -- serves as an approximation to U(1)×Z2U(1)\times\mathbb{Z}_2U(1)×Z2​ lattice gauge theory. In order to carry out such a lattice simulation, we detail the construction of efficient quantum circuits to realize basic primitives including the nonabelian Fourier transform over DND_NDN​, the trace operation, and the group multiplication and inversion operations. For each case the required quantum resources scale linearly or as low-degree polynomials in n=log⁡Nn=\log Nn=logN. We experimentally benchmark our gates on the Rigetti Aspen-9 quantum processor for the case of D4D_4D4​. The fidelity of all D4D_4D4​ gates was found to exceed 80%80\%80%.