PNNL

Abstract

In recent years simulations of chemistry and condensed materials has emerged as one of the preeminent applications of quantum computing oering 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 eects, which are described most generally by quantum electrodynamics (QED), can also be simulated on a quantum computer in polynomial time. Here we show that eective 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 eld. In particular, we provide a detailed analysis of such simulations in position and momentum basis using Trotter-Suzuki formulas. We nd that the number of T-gates needed to perform such simulations on a 3D lattice of ns sites scales at worst as O(n3s=)1+o(1) in the thermodynamic limit for position basis simulations and O(n4+2=3s =)1+o(1) in momentum basis. We also nd that qubitization scales slightly better with a worst case scaling of eO(n2+2=3s =) for lattice eQED and complications in the prepare circuit leads to a slightly worse scaling in momentum basis of eO(n5+2=3s =). 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-Cliord operations and also provide a detailed discussion of how to prepare MRCISD states in eective QED which provide a reasonable initial guess for the groundstate. Finally, we estimate the planewave cutos needed to accurately simulate heavy elements such as gold.