Product formulas for exponentials of commutators
Abstract
We provide a recursive method for constructing product formula approximations to exponentials of commutators, giving the first approximations that are accurate to arbitrarily high order. Using these formulas, we show how to approximate unitary exponentials of (possibly nested) commutators using exponentials of the elementary operators, and we upper bound the number of elementary exponentials needed to implement the desired operation within a given error tolerance. By presenting an algorithm for quantum search using evolution according to a commutator, we show that the scaling of the number of exponentials in our product formulas with the evolution time is nearly optimal. Finally, we discuss applications of our product formulas to quantum control and to implementing anticommutators, providing new methods for simulating manybody interaction Hamiltonians.
 Publication:

Journal of Mathematical Physics
 Pub Date:
 June 2013
 DOI:
 10.1063/1.4811386
 arXiv:
 arXiv:1211.4945
 Bibcode:
 2013JMP....54f2202C
 Keywords:

 03.67.Ac;
 02.30.Jr;
 02.60.Lj;
 Quantum algorithms protocols and simulations;
 Partial differential equations;
 Ordinary and partial differential equations;
 boundary value problems;
 Quantum Physics;
 Mathematical Physics
 EPrint:
 22 pages, 4 figures. Revised to include discussion of existing approximation building methods and an improved version of the approximation building method of Jean and Koseleff has been added