A Theory of Trotter Error

Explore a comprehensive theory of Trotter error in quantum computing through this seminar by Yuan Su from the University of Maryland. Delve into the limitations of previous approaches based on Baker-Campbell-Hausdorff expansion truncation and discover a new analysis that directly exploits operator summand commutativity. Learn about improved algorithms for digital quantum simulation and quantum Monte Carlo methods, including simulations of various Hamiltonian systems. Examine how product formulas can preserve system locality, leading to simulations of local observables with complexity independent of system size for power-law interacting systems. Gain insights into the accuracy of this new theory in characterizing Trotter error, both in terms of asymptotic scaling and constant prefactor, and understand its implications for quantum simulation and Lieb-Robinson bounds.

Intro
Quantum simulation • Dynamics of a quantum system are given by its Hamiltonian
Reasons to study quantum simulation
Product formulas • Also known as Trotterization or the splitting method.
Higher-order product formulas • A general pth-order product formula takes the form
Previous analyses of Trotter error • For sufficiently small t. Trotter error can be represented exactly
Trotter error with commutator scaling Trotter error with commutator scaling A poth-order product formula .(t) can approacimate the evolution
Analysis of the first-order formula • Altogether, we have the integral representation
Nearest-neighbor lattice Hamiltonian
Clustered Hamiltonian Clustered Hamiltonian
Transverse field Ising model Transverse field Ising model
Simulating local observables • We show that local observables can be simulated with complexity independent of the system size for power-law Hamiltonians, implying a Lieb-Robinson bound as a byproduct.
A theory of Trotter error
Error types • Suppose that we use product formula 3 (t) to approximate the . We consider the active, exponentiated, and multiplicative type
Error representations