# A quantum algorithm for simulating non-sparse Hamiltonians

## TL;DR

This paper introduces a quantum algorithm capable of simulating non-sparse Hamiltonians efficiently by leveraging qRAM and quantum walks, achieving poly-logarithmic precision dependence and broad applicability.

## Contribution

The paper presents a novel quantum algorithm for simulating general Hamiltonians using qRAM and quantum walks, extending beyond sparse cases with improved efficiency.

## Key findings

- Achieves poly-logarithmic dependence on precision.
- Provides a complexity of $O(t ext{sqrt}(N) ext{polylog}(N, t ext{||}H ext{||}, 1/ ext{epsilon}))$.
- Applicable as a subroutine for unitary implementation and quantum linear systems.

## Abstract

We present a quantum algorithm for simulating the dynamics of Hamiltonians that are not necessarily sparse. Our algorithm is based on the input model where the entries of the Hamiltonian are stored in a data structure in a quantum random access memory (qRAM) which allows for the efficient preparation of states that encode the rows of the Hamiltonian. We use a linear combination of quantum walks to achieve poly-logarithmic dependence on precision. The time complexity of our algorithm, measured in terms of the circuit depth, is $O(t\sqrt{N}\|H\|\,\mathrm{polylog}(N, t\|H\|, 1/\epsilon))$, where $t$ is the evolution time, $N$ is the dimension of the system, and $\epsilon$ is the error in the final state, which we call precision. Our algorithm can be directly applied as a subroutine for unitary implementation and quantum linear systems solvers, achieving $\widetilde{O}(\sqrt{N})$ dependence for both applications.