Simple and high-precision Hamiltonian simulation by compensating Trotter errors with linear combination of unitary operations

Trotter and linear-combination-of-unitary (LCU) methods are two popular Hamiltonian simulation algorithms. The Trotter method is easy to implement and enjoys good system-size dependence endowed by commutator scaling, while the LCU method admits high accuracy simulation with smaller gate cost.

We study the Hamiltonian simulation algorithms which combine the Trotter and LCU method --- In each time segment, we first implement Trotter method, and then apply LCU to compensate the residue error. We provide versatile realizations based on either random-sampling or coherent implementation of LCU formulas.

We propose two types of composite algorithms with unique advantages in various regions. The first algorithm is generic, with the random-sampling implementation, we prove that with only one ancillary qubit, one can achieve a gate cost almost linear to the time and logarithmically dependent on the inversed accuracy. In our second algorithm, we consider the detailed structure of Hamiltonians and propose a modified composite algorithms with commutator scaling utilizing nested-commutator analysis in the Trotter algorithms. Consequently, for the lattice Hamiltonians, our algorithm enjoys almost linear time and system-size dependence, and achieves a higher accuracy than 2Kth-order Trotter only with Kth-order Trotter's gate complexity.