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A Theory of Trotter Error
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- 1 Intro
- 2 Quantum simulation • Dynamics of a quantum system are given by its Hamiltonian
- 3 Reasons to study quantum simulation
- 4 Product formulas • Also known as Trotterization or the splitting method.
- 5 Higher-order product formulas • A general pth-order product formula takes the form
- 6 Previous analyses of Trotter error • For sufficiently small t. Trotter error can be represented exactly
- 7 Trotter error with commutator scaling Trotter error with commutator scaling A poth-order product formula .(t) can approacimate the evolution
- 8 Analysis of the first-order formula • Altogether, we have the integral representation
- 9 Nearest-neighbor lattice Hamiltonian
- 10 Clustered Hamiltonian Clustered Hamiltonian
- 11 Transverse field Ising model Transverse field Ising model
- 12 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.
- 13 A theory of Trotter error
- 14 Error types • Suppose that we use product formula 3 (t) to approximate the . We consider the active, exponentiated, and multiplicative type
- 15 Error representations
