Provably accurate simulation of gauge theories and bosonic systems

Abstract

Quantum many-body systems involving bosonic modes or gauge fields have infinite-dimensional local Hilbert spaces which must be truncated to perform simulations of real-time dynamics on classical or quantum computers. To analyze the truncation error, we develop methods for bounding the rate of growth of local quantum numbers such as the occupation number of a mode at a lattice site, or the electric field at a lattice link. Our approach applies to various models of bosons interacting with spins or fermions, and also to both abelian and non-abelian gauge theories. We show that if states in these models are truncated by imposing an upper limit $\Lambda$ on each local quantum number, and if the initial state has low local quantum numbers, then an error at most $\epsilon$ can be achieved by choosing $\Lambda$ to scale polylogarithmically with ${\epsilon }^{-1}$, an exponential improvement over previous bounds based on energy conservation. For the Hubbard-Holstein model, we numerically compute a bound on $\Lambda$ that achieves accuracy $\epsilon$, obtaining significantly improved estimates in various parameter regimes. We also establish a criterion for truncating the Hamiltonian with a provable guarantee on the accuracy of time evolution. Building on that result, we formulate quantum algorithms for dynamical simulation of lattice gauge theories and of models with bosonic modes; the gate complexity depends almost linearly on spacetime volume in the former case, and almost quadratically on time in the latter case. We establish a lower bound showing that there are systems involving bosons for which this quadratic scaling with time cannot be improved. By applying our result on the truncation error in time evolution, we also prove that spectrally isolated energy eigenstates can be approximated with accuracy $\epsilon$ by truncating local quantum numbers at $\Lambda =\text{polylog}({\epsilon }^{-1})$.

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• On the complexity of implementing Trotter steps
• Hybrid Oscillator-Qubit Quantum Processors: Instruction Set Architectures, Abstract Machine Models, and Applications
• Strong Error Bounds for Trotter & Strang-Splittings and Their Implications for Quantum Chemistry
• Real-time scattering in the lattice Schwinger model
• Holomorphic representation of quantum computations
• Exponential improvements in the simulation of lattice gauge theories using near-optimal techniques
• Tensor Networks for Lattice Gauge Theories beyond one dimension: a Roadmap
• Gluon Digitization via Character Expansion for Quantum Computers
• Effective light cone and digital quantum simulation of interacting bosons
• Average-case Speedup for Product Formulas
• Towards a Quantum Simulation of Nonlinear Sigma Models with a Topological Term
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• A Comprehensive Cross-Model Framework for Benchmarking the Performance of Quantum Hamiltonian Simulations
• Time-dependent Hamiltonian Simulation via Magnus Expansion: Algorithm and Superconvergence
• End-to-end complexity for simulating the Schwinger model on quantum computers
• Simulating Quantum Mechanics with a $\theta$-term and an 't Hooft Anomaly on a Synthetic Dimension
• Thermal Area Law for Lattice Bosons
• Solving lattice gauge theories using the quantum Krylov algorithm and qubitization
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• Quantum simulating continuum field theories with large-spin lattice models
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• On the Trotter Error in Many-body Quantum Dynamics with Coulomb Potentials
• Discrete Superconvergence Analysis for Quantum Magnus Algorithms of Unbounded Hamiltonian Simulation
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• Creation of Wave Packets for Quantum Chromodynamics on Quantum Computers
• A posteriori error estimates for the Lindblad master equation