Hybridized Methods for Quantum Simulation in the Interaction Picture

Abstract

Conventional methods of quantum simulation involve trade-offs that limit their applicability to specific contexts where their use is optimal. In particular, the interaction picture simulation has been found to provide substantial asymptotic advantages for some Hamiltonians but incurs prohibitive constant factors and is incompatible with methods like qubitization. We provide a framework that allows different simulation methods to be hybridized and thereby improve performance for interaction picture simulations over known algorithms. These approaches show asymptotic improvements over the individual methods that comprise them and further make interaction picture simulation methods practical in the near term. Physical applications of these hybridized methods yield a gate complexity scaling as log 2 Λ in the electric cutoff Λ for the Schwinger Model and independent of the electron density for collective neutrino oscillations, outperforming the scaling for all current algorithms with these parameters. For the general problem of Hamiltonian simulation subject to dynamical constraints, these methods yield a query complexity independent of the penalty parameter λ used to impose an energy cost on time-evolution into an unphysical subspace.

Digital Quantum Simulation of the Schwinger Model and Symmetry Protection with Trapped Ions

“…To make implementations of more complex gauge theories possible, including non-Abelian and higherdimensional models, unifying physics insights, algorithm optimization, hardware implementation, and postprocessing is required, as demonstrated in this work. In this context, it would be interesting to investigate whether more resource-efficient encodings of such theories exist, if optimal Trotter decompositions and term ordering schemes can be found, to what degree these preserve local gauge symmetries, whether information regarding the initial state and the symmetries can be incorporated to further tighten the algorithmic error bounds [70,[91][92][93][94][95][96], how to balance these errors with experimental errors, and whether symmetry-protection schemes are advantageous in suppressing algorithmic and experimental errors. While progress along these lines is already being made [23,24,27,30,32,36,67,[97][98][99][100][101][102][103][104][105], further technological advances in quantum hardware are essential to enable advanced gauge-theory simulations in the upcoming years.…”

“…The Hamiltonian formulation of the lattice Schwinger model, i.e., the Schwinger model defined in a discretized space and with continuous time, has served as a testbed for numerous computational techniques in recent years, including quantum simulation and computation. In particular, there have been several theoretical proposals for analog quantum simulation [56][57][58][59][60][61][62] and gate-based quantum algorithms [22,24,[63][64][65][66][67][68][69][70] of the Schwinger model, along with experimental implementations on various quantum platforms such as trapped ions [71,72], Rydberg atoms [60,73], ultracold atoms [74,75], and superconducting qubits [64,76].…”

Classical and Quantum Evolution in a Simple Coherent Neutrino Problem

“…This problem can be circumvented using algorithms to perform simulations in the interaction picture (see e.g. [26]) and we plan to leverage this technology in future work. We assume that the presence of any other charged leptons is negligible.…”

Time-dependent Hamiltonian Simulation of Highly Oscillatory Dynamics and Superconvergence for Schrödinger Equation

“…For example, highly oscillatory wave functions or Hamiltonian with large spectral norm are commonly observed in adiabatic quantum computation [23,1], k-local Hamiltonians (with a large number of sites) [35,36,19], and the electronic structure problem (with a real space discretization) [32,31,2,47]. On the other hand, in quantum control problems with ultrafast lasers [41,42,22], and interaction picture Hamiltonian simulation [37,43], the Hamiltonian itself typically contains highly oscillatory component. We remark that under certain assumptions, a Hamiltonian of large spectral norm can be more effectively simulated in terms of a fast oscillatory Hamiltonian in the interaction picture, which will be further discussed later.…”