Limitations on the simulation of non-sparse Hamiltonians

TL;DR

This paper explores the fundamental limitations of simulating dense, non-sparse Hamiltonians on quantum computers, extending existing theorems and identifying cases where efficient simulation is possible.

Contribution

It generalizes the no-fast-forwarding theorem to dense Hamiltonians and establishes new lower bounds on simulation time, highlighting the challenges and potential for specific cases.

Findings

Dense Hamiltonians cannot be simulated faster than linear in ||Ht||

Known quantum walk simulations cannot be significantly improved for general dense Hamiltonians

Certain non-sparse Hamiltonians with small graph arboricity can be simulated efficiently

Abstract

The problem of simulating sparse Hamiltonians on quantum computers is well studied. The evolution of a sparse N x N Hamiltonian H for time t can be simulated using O(||Ht||poly(log N)) operations, which is essentially optimal due to a no--fast-forwarding theorem. Here, we consider non-sparse Hamiltonians and show significant limitations on their simulation. We generalize the no--fast-forwarding theorem to dense Hamiltonians, ruling out generic simulations taking time o(||Ht||), even though ||H|| is not a unique measure of the size of a dense Hamiltonian $H$. We also present a stronger limitation ruling out the possibility of generic simulations taking time poly(||Ht||,log N), showing that known simulations based on discrete-time quantum walk cannot be dramatically improved in general. On the positive side, we show that some non-sparse Hamiltonians can be simulated efficiently, such as those with graphs of small arboricity.