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arXiv:2410.11413 · Quantum Science and Technology

Truncation-free quantum simulation of pure-gauge compact QED using Josephson arrays

Guy Pardo, Julian Bender, Nadav Y. Katz, Erez Zohar

Published Apr 17, 2025• 1 citation
arXivHTMLPDF

Abstract

Abstract Quantum simulation is one of the methods that have been proposed and used in practice to bypass computational challenges in the investigation of lattice gauge theories (LGTs). While most of the proposals rely on truncating the infinite dimensional Hilbert spaces that these models feature, we propose a truncation-free method based on the exact analogy between the local Hilbert space of lattice quantum electrodynamics and that of a Josephson junction. We provide several proposals, mostly semi-analog, arranged according to experimental difficulty. Our method can simulate a quasi-2D system of up to 2 × N plaquettes, and we present an approximate method that can simulate the fully-2D theory, but is more demanding experimentally and not immediately feasible. This sets the ground for analog quantum simulation of LGTs with superconducting circuits, in a completely Hilbert space truncation-free procedure, for continuous gauge groups.

DOI
10.1088/2058-9565/adce2a

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Abstract Quantum simulation is one of the methods that have been proposed and used in practice to bypass computational challenges in the investigation of lattice gauge theories (LGTs). While most of the proposals rely on truncating the infinite dimensional Hilbert spaces that these models feature, we propose a truncation-free method based on the exact analogy between the local Hilbert space of lattice quantum electrodynamics and that of a Josephson junction. We provide several proposals, mostly semi-analog, arranged according to experimental difficulty. Our method can simulate a quasi-2D system of up to 2 × N plaquettes, and we present an approximate method that can simulate the fully-2D theory, but is more demanding experimentally and not immediately feasible. This sets the ground for analog quantum simulation of LGTs with superconducting circuits, in a completely Hilbert space truncation-free procedure, for continuous gauge groups.
We propose a superconducting-circuit architecture that realizes a compact U(1) lattice gauge theory using the intrinsic infinite-dimensional Hilbert space of phase and charge variables. The gauge and matter fields are encoded directly in the degrees of freedom of the rotor variables associated with the circuit nodes, and Gauss's law emerges exactly from the conservation of local charge, without auxiliary stabilizers, penalty terms, or Hilbert-space truncation. A minimal gauge-matter coupling arises microscopically from Josephson nonlinearities, whereas the magnetic plaquette interaction is generated perturbatively via virtual matter excitations. Numerical diagonalization confirms the emergence of compact electrodynamics and coherent vortex excitations, underscoring the need for large local Hilbert spaces in the continuum regime. The required circuit parameters are within the current experimental capabilities. Our results establish superconducting circuits as a scalable, continuous-variable platform for analog quantum simulation of non-perturbative gauge dynamics.
Gauge theories establish the standard model of particle physics, and lattice gauge theory (LGT) calculations employing Markov Chain Monte Carlo (MCMC) methods have been pivotal in our understanding of fundamental interactions. The present limitations of MCMC techniques may be overcome by Hamiltonian-based simulations on classical or quantum devices, which further provide the potential to address questions that lay beyond the capabilities of the current approaches. However, for continuous gauge groups, Hamiltonian-based formulations involve infinite-dimensional gauge degrees of freedom that can solely be handled by truncation. Current truncation schemes require dramatically increasing computational resources at small values of the bare couplings, where magnetic field effects become important. Such limitation precludes one from `taking the continuous limit' while working with finite resources. To overcome this limitation, we provide a resource-efficient protocol to simulate LGTs with continuous gauge groups in the Hamiltonian formulation. Our new method allows for calculations at arbitrary values of the bare coupling and lattice spacing. The approach consists of the combination of a Hilbert space truncation with a regularization of the gauge group, which permits an efficient description of the magnetically-dominated regime. We focus here on Abelian gauge theories and use 2+12+12+1 dimensional quantum electrodynamics as a benchmark example to demonstrate this efficient framework to achieve the continuum limit in LGTs. This possibility is a key requirement to make quantitative predictions at the field theory level and offers the long-term perspective to utilise quantum simulations to compute physically meaningful quantities in regimes that are precluded to quantum Monte Carlo.
Gauge theories establish the standard model of particle physics, and lattice gauge theory (LGT) calculations employing Markov Chain Monte Carlo (MCMC) methods have been pivotal in our understanding of fundamental interactions. The present limitations of MCMC techniques may be overcome by Hamiltonian-based simulations on classical or quantum devices, which further provide the potential to address questions that lay beyond the capabilities of the current approaches. However, for continuous gauge groups, Hamiltonian-based formulations involve infinite-dimensional gauge degrees of freedom that can solely be handled by truncation. Current truncation schemes require dramatically increasing computational resources at small values of the bare couplings, where magnetic field effects become important. Such limitation precludes one from `taking the continuous limit' while working with finite resources. To overcome this limitation, we provide a resource-efficient protocol to simulate LGTs with continuous gauge groups in the Hamiltonian formulation. Our new method allows for calculations at arbitrary values of the bare coupling and lattice spacing. The approach consists of the combination of a Hilbert space truncation with a regularization of the gauge group, which permits an efficient description of the magnetically-dominated regime. We focus here on Abelian gauge theories and use 2 + 1 dimensional quantum electrodynamics as a benchmark example to demonstrate this efficient framework to achieve the continuum limit in LGTs. This possibility is a key requirement to make quantitative predictions at the field theory level and offers the long-term perspective to utilise quantum simulations to compute physically meaningful quantities in regimes that are precluded to quantum Monte Carlo.
Due to rapidly improving quantum computing hardware, Hamiltonian simulations of relativistic lattice field theories have seen a resurgence of attention. This computational tool requires turning the formally infinite-dimensional Hilbert space of the full theory into a finite-dimensional one. For gauge theories, a widely-used basis for the Hilbert space relies on the representations induced by the underlying gauge group, with a truncation that keeps only a set of the lowest dimensional representations. This works well at large bare gauge coupling, but becomes less efficient at small coupling, which is required for the continuum limit of the lattice theory. In this work, we develop a new basis suitable for the simulation of an SU(2) lattice gauge theory in the maximal tree gauge. In particular, we show how to perform a Hamiltonian truncation so that the eigenvalues of both the magnetic and electric gauge-fixed Hamiltonian are mostly preserved, which allows for this basis to be used at all values of the coupling. Little prior knowledge is assumed, so this may also be used as an introduction to the subject of Hamiltonian formulations of lattice gauge theories.
Quantum simulations of lattice gauge theories offer the potential to directly study the nonperturbative dynamics of quantum chromodynamics, but naive analyses suggest that they require large computational resources. Large N c expansions are performed to order 1 / N c to simplify the Hamiltonian of pure SU ( N c ) lattice gauge theories. A reformulation of the electric basis is introduced with a truncation strategy based on the construction of local Krylov subspaces with plaquette operators. Numerical simulations show that these truncated Hamiltonians are consistent with traditional lattice calculations at relatively small couplings. It is shown that the computational resources required for quantum simulation of time evolution generated by these Hamiltonians is 17–19 orders of magnitude smaller than previous approaches, provided that the truncations in this work can reach lattice spacings in three-dimensional simulations comparable to the two-dimensional simulations performed.
One of the methods proposed in recent years for studying nonperturbative gauge theory physics is quantum simulation, where lattice gauge theories are mapped onto quantum devices which can be built in the laboratory, or quantum computers. While being very promising and already showing some experimental results, these methods still face several challenges related to the interface between the technological capabilities and the demands of the simulated models; in particular, one such challenge is the need to simulate infinitely dimensional local Hilbert spaces, describing the gauge fields on the links in the case of compact Lie gauge groups, requiring some truncations and approximations which are not completely understood or controllable in the general case. This work proposes a way to obtain arbitrarily large such local Hilbert spaces by using coarse graining of simple, low-dimensional qubit systems, made of components available on most quantum simulation platforms, and thus opening the way to new types of lattice gauge theory quantum simulations. Published by the American Physical Society 2024