Approximate real-time evolution operator for potential with one ancillary qubit and application to first-quantized Hamiltonian simulation

TL;DR

This paper compares methods for implementing real-time evolution operators generated by diagonal unitaries, proposing efficient approximate techniques for large systems and applying them to first-quantized Hamiltonian simulation.

Contribution

It introduces polynomial approximation methods for diagonal unitaries, analyzing resource overheads and applying these to potential evolution in quantum simulations.

Findings

Polynomial approximation reduces gate count for large systems.

Resource estimates depend on error bounds and grid parameters.

Approximate methods are effective for potential evolution in quantum simulations.

Abstract

In this article, we compare the methods implementing the real-time evolution operator generated by a unitary diagonal matrix where its entries obey a known underlying real function. When the size of the unitary diagonal matrix is small, a well-known method based on Walsh operators gives a good and precise implementation. In contrast, as the number of qubits grows, the precise one uses exponentially increasing resources, and we need an efficient implementation based on suitable approximate functions. Using piecewise polynomial approximation of the function, we summarize the methods with different polynomial degrees. Moreover, we obtain the overheads of gate count for different methods concerning the error bound and grid parameter (number of qubits). This enables us to analytically find a relatively good method as long as the underlying function, the error bound, and the grid parameter are given. This study contributes to the problem of encoding a known function in the phase factor, which plays a crucial role in many quantum algorithms/subroutines. In particular, we apply our methods to implement the real-time evolution operator for the potential part in the first-quantized Hamiltonian simulation and estimate the resources (gate count and ancillary qubits) regarding the error bound, which indicates that the error coming from the approximation of the potential function is not negligible compared to the error from the Trotter-Suzuki formula.