Optimisation of Quantum Hamiltonian Evolution: Two Projection Operators
Apoorva D. Patel, Anjani Priyadarsini · arXiv (Cornell University) · 2015
Given a quantum Hamiltonian and its evolution time, the corresponding unitary evolution operator can be constructed in many different ways, corresponding to different trajectories between the desired end-points. A choice among these trajectories can then be made to obtain the best computational complexity and control over errors. It is shown how a construction based on Grover's algorithm scales linearly in time and logarithmically in error bound, and is exponentially superior in error complexity to the scheme based on straightforward application of the Lie-Trotter formula. The strategy is then extended to simulation of any Hamiltonian that is a linear combination of two projection operators. The key feature is to construct an evolution in terms of reflection operators instead of taking small time steps. The reflection operation amounts to taking the largest possible step consistent with unitarity in the specified direction. Though that makes discretisation error for individual steps large, the total error on the overall evolution can be efficiently controlled.