# Large time-step discretisation of adiabatic quantum dynamics

## TL;DR

This paper demonstrates that large, uniform time steps can be used in discretizing adiabatic quantum dynamics, significantly reducing complexity while maintaining accuracy, especially with boundary cancellation conditions.

## Contribution

It introduces a novel analysis showing large, uniform time steps are feasible in adiabatic quantum simulation, improving efficiency without sacrificing accuracy.

## Key findings

- Large time steps reduce simulation complexity.
- Uniform time steps are effective regardless of convergence order.
- Boundary cancellation exponentially suppresses diabatic errors.

## Abstract

Adiabatic quantum computing is a general framework for preparing eigenstates of Hamiltonians on quantum devices. However, its digital implementation requires an efficient Hamiltonian simulation subroutine, which may introduce extra computational overhead or complicated quantum control logic. In this work, we show that the time step sizes in time discretization can be much larger than expected, and the overall complexity is greatly reduced. Remarkably, regardless of the general convergence order of the numerical method, we can choose a uniform time step size independent of tolerated error and evolution time for sufficiently accurate simulation. Furthermore, with the boundary cancellation condition where the continuous diabatic errors are exponentially suppressed, we provide strong evidence on an exponential convergence of even first-order Trotter with uniform time step size. We apply our analysis to the example of adiabatic unstructured search and show several preferable features of the Trotterized adiabatic approach: it can match the Grover lower bound, it does not require a priori knowledge on the number of marked states, and its performance can be asymptotically comparable with that of the quantum approximate optimization algorithm.