  Gauge theories are the most successful theories for describing nature at its
fundamental level, but obtaining analytical or numerical solutions often
remains a challenge. We propose an experimental quantum simulation scheme to
study ground state properties in two-dimensional quantum electrodynamics (2D
QED) using existing quantum technology. The proposal builds on a formulation of
lattice gauge theories as effective spin models in arXiv:2006.14160, which
reduces the number of qubits needed by eliminating redundant degrees of freedom
and by using an efficient truncation scheme for the gauge fields. The latter
endows our proposal with the perspective to take a well-controlled continuum
limit. Our protocols allow in principle scaling up to large lattices and offer
the perspective to connect the lattice simulation to low energy observable
quantities, e.g. the hadron spectrum, in the continuum theory. By including
both dynamical matter and a non-minimal gauge field truncation, we provide the
novel opportunity to observe 2D effects on present-day quantum hardware. More
specifically, we present two Variational Quantum Eigensolver (VQE) based
protocols for the study of magnetic field effects, and for taking an important
first step towards computing the running coupling of QED. For both instances,
we include variational quantum circuits for qubit-based hardware, which we
explicitly apply to trapped ion quantum computers. We simulate the proposed VQE
experiments classically to calculate the required measurement budget under
realistic conditions. While this feasibility analysis is done for trapped ions,
our approach can be easily adapted to other platforms. The techniques presented
here, combined with advancements in quantum hardware pave the way for reaching
beyond the capabilities of classical simulations by extending our framework to
include fermionic potentials or topological terms.
