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In recent years simulations of chemistry and condensed materials has emerged as one of the preeminent applications of quantum computing, offering an exponential speedup for the solution of the electronic structure for certain strongly correlated electronic systems. To date, most treatments have ignored the question of whether relativistic effects, which are described most generally by quantum electrodynamics (QED), can also be simulated on a quantum computer in polynomial time. Here we show that effective QED, which is equivalent to QED to second order in perturbation theory, can be simulated in polynomial time under reasonable assumptions while properly treating all four components of the wavefunction of the fermionic field. In particular, we provide a detailed analysis of such simulations in position and momentum basis using Trotter-Suzuki formulas. We find that the number of T-gates needed to perform such simulations on a 3D lattice of ns sites scales at worst as O(ns3/ϵ)1+o(1) in the thermodynamic limit for position basis simulations and O(ns4+2/3/ϵ)1+o(1) in momentum basis. We also find that qubitization scales slightly better with a worst case scaling of O~(ns2+2/3/ϵ) for lattice eQED and complications in the prepare circuit leads to a slightly worse scaling in momentum basis of O~(ns5+2/3/ϵ). We further provide concrete gate counts for simulating a relativistic version of the uniform electron gas that show challenging problems can be simulated using fewer than 1013 non-Clifford operations and also provide a detailed discussion of how to prepare multi-reference configuration interaction states in effective QED which can provide a reasonable initial guess for the ground state. Finally, we estimate the planewave cutoffs needed to accurately simulate heavy elements such as gold.

Studying QCD and other gauge theories on quantum hardware requires the preparation of physically interesting states. The Variational Quantum Eigensolver (VQE) provides a way of performing vacuum state preparation on quantum hardware. In this work, VQE is applied to pure SU(3) lattice Yang-Mills on a single plaquette and one dimensional plaquette chains. Bayesian optimization and gradient descent were investigated for performing the classical optimization. Ansatz states for plaquette chains are constructed in a scalable manner from smaller systems using domain decomposition and a stitching procedure analogous to the Density Matrix Renormalization Group (DMRG). Small examples are performed on IBM's superconducting Manila processor. CONTENTSA. Hardware Calculations 14 B. Bayesian Optimization 15 C. Plaquette Chain Tensor Network 16 D. Data from IBM's Manila Processor 17 References 18