To date, all proposed quantum algorithms for simulating quantum field theory (QFT) simulate (continuous-time) Hamiltonian lattice QFT as a stepping stone. Two overlooked issues are how large we can take the timestep in these simulations while getting the right physics and whether we can go beyond the standard recipe that relies on Hamiltonian lattice QFT. The first issue is crucial in practice for, e.g., trapped-ion experiments which actually have a lower bound on the possible ratio of timestep to lattice spacing. To this end, we show that a timestep equal to or going to zero faster than the spatial lattice spacing is necessary for quantum simulations of QFT, but far more importantly a timestep equal to the lattice spacing is actually sufficient. To do this, first for
ϕ4 theory, we give a quantum circuit exactly equivalent to the real-time path integral from the discrete-time Lagrangian formulation of lattice QFT. Next we give another circuit with no lattice QFT analogue, but, by using Feynman rules applied to the circuit, we see that it also reproduces the correct continuum behaviour. Finally, we look at non-abelian gauge fields, showing that the discrete-time lattice QFT path-integral is exactly equivalent to a finite-depth local circuit. All of these circuits have an analogue of a lightcone on the lattice and therefore are examples of quantum cellular automata. Aside from the potential practical benefit of these circuits, this all suggests that the path-integral approach to lattice QFT need not be overlooked in quantum simulations of physics and has a simple quantum information interpretation.
Simulating the real-time dynamics of quantum field theories (QFTs) is one of the most promising applications of quantum simulators. Regularizing a bosonic QFT for quantum simulation purposes typically involves a truncation in Hilbert space in addition to a discretization of space. Here, we discuss how to perform such a regularization of scalar QFTs by explicitly constructing suitable many-body lattice Hamiltonians using multilevel or qudit systems and show that this enables quantitative predictions in the continuum limit by extrapolating results obtained for large-spin models. With extensive matrix-product-state simulations, we numerically demonstrate the sequence of extrapolations that leads to quantitative agreement of observables for the integrable sine-Gordon (sG) QFT. We further show how to prepare static and moving-soliton excitations and we analyze their scattering dynamics in the continuum limit, in agreement with a semiclassical model and with quantitative analytical predictions. Finally, we illustrate how a nonintegrable perturbation of the sG model gives rise to dynamics reminiscent of string breaking and plasma oscillations in gauge theories. Our methods are directly applicable in state-of-the-art analog quantum simulators, opening the door to quantitatively investigating a wide variety of scalar-field theories and tackling long-standing questions in nonequilibrium QFT such as the fate of the false vacuum.
Quantum Fourier transform (QFT) is a widely used building block for quantum algorithms, whose scalable implementation is challenging in experiments. Here, we propose a protocol of quadratic QFT (QQFT) considering cold atoms confined in an optical lattice. This QQFT is equivalent to QFT in the single-particle subspace and generalizes to a many-body setting through the second quantization formalism. We show this QQFT protocol can be implemented using a programmable laser potential with the digital-micromirror-device techniques recently developed in experiments. The QQFT protocol enables programmable Hamiltonian engineering and allows quantum simulations of Hamiltonian models, which are difficult to realize with conventional approaches. The flexibility of our approach is demonstrated by performing quantum simulations of one-dimensional Poincar\'e crystal physics and two-dimensional topological flat bands, where the QQFT protocol effectively generates the required long-range tunnelings despite the locality of the cold atom system. We find the discrete Poincar\'e symmetry and topological properties in the two examples, respectively, have robustness against a certain degree of noise that is potentially existent in the experimental realization. We expect this approach would open wide opportunities for optical lattice-based programmable quantum simulations.