Truncation uncertainties for accurate quantum simulations of lattice gauge theories

Abstract

The encoding of lattice gauge theories onto quantum computers requires a discretization of the gauge field's Hilbert space on each link, which presents errors with respect to the Kogut--Susskind limit. In the electric basis, Hilbert space fragmentation has recently been shown to limit the excitation of large electric fields. Here, we leverage this to develop a formalism for estimating the size of truncation errors in the electric basis. Generically, the truncation error falls off as a factorial of the field truncation. Examples of this formalism are applied to the Schwinger model and a pure U(1) lattice gauge theory. For reasonable choices of parameters, we improve on previous error estimates by a factor of 10^{306}.

Figures (8)

• Figure 1: Maximum error in the expectation of $\hat{E}^2$ as a function of time on a single plaquette with $g=0.5$ and a max evolution time of $T=30$. The blue, green, and purple lines correspond to using different electric basis states as the initial state. The red curve is the error bound in Eq. \ref{['eq:OnePlaqE2Bound']}, computed using $\Lambda_0 = 4$.
• Figure 2: Evolution of the electric vacuum state on a single plaquette. Numerical simulations were performed with a maximum electric field of $20$ and $g=0.5$. The left panel shows the expectation of $\hat{\Pi}_\Lambda$ for various values of $\Lambda$ as a function of time $t$. The solid curves are the exact time evolution, and the dashed lines are the rigorous long-time bounds. The right panel shows the maximum of the expectation of $\hat{\Pi}_\Lambda$ for the simulated time evolution. The blue curve is the exact result, the green curve is the bound from energy conservation, and the red curve is the leading contribution to the expectation obtained by calculating $L(g,\Lambda,\Lambda_0,T)$ with $\Lambda_0=4$.
• Figure 3: Evolution of the electric vacuum state on a single plaquette. Numerical simulations were performed with a maximum electric field of $20$, $g=\sqrt{3}$, and a maximum evolution time of $t=8$. The blue curve is the maximum of the expectation of $\hat{\Pi}_\Lambda$ during this evolution. The green points are the leading contribution to the expectation obtained by calculating $L(g,\Lambda,\Lambda_0,T)$ with $\Lambda_0=0$.
• Figure 4: Simulation of an infinite plaquette chain with $g=0.8$. The left panel shows the electric energy as a function of time for different truncations of the electric field. The solid curves in the right panel show the difference in electric energy between the $\Lambda=5$ truncation and the lower truncations. The dashed lines show the predicted error in the electric energy from truncating the Hamiltonian, computed using Eq. \ref{['eq:PlaqChainElError']} with $\Lambda_0=1$. See \ref{['app:iMPS']} for details regarding the iMPS implementation.
• Figure 5: Simulation of an infinite plaquette chain with four different values of $g = 1.0, 0.9, 0.8$, and $0.7$ and maximum electric fields of $5,6,6,$ and $7$ respectively. The solid curves in each panel show the expectation value of the projection operator $\hat{{\Pi}}_{\Lambda,\vec{x}}$ with respect to the states obtained by time evolving the electric vacuum for various choices of $\Lambda$. See \ref{['app:iMPS']} for details regarding the iMPS time evolution. The dashed lines in each panel correspond to the upper bound on the probability given by \ref{['eq:prob_upper_bound']} with $\Lambda_0 = 0,1, 1$, and $2$ respectively for each choice of $g$. The dash-dotted lines in each panel show the upper bound based on energy conservation, computed using \ref{['eq:energy_conservation_prob_bound']} for the largest value of $\Lambda$ considered for each value of g.