Universal Quantum Computation by Scattering in the Fermi-Hubbard Model

TL;DR

This paper demonstrates that the dynamics of the fermionic Hubbard model can encode universal quantum computation through wave packet scattering on planar graphs, establishing the model's computational universality.

Contribution

It introduces a method to encode universal quantum computation into the Hubbard model via wave packet scattering, showing the model's computational complexity.

Findings

Simulation of the Hubbard model is as hard as quantum computation.

Wave packets can encode universal quantum gates.

Scattering in the t-J model exhibits purely reflective behavior.

Abstract

The Hubbard model may be the simplest model of particles interacting on a lattice, but simulation of its dynamics remains beyond the reach of current numerical methods. In this article, we show that general quantum computations can be encoded into the physics of wave packets propagating through a planar graph, with scattering interactions governed by the fermionic Hubbard model. Therefore, simulating the model on planar graphs is as hard as simulating quantum computation. We give two different arguments, demonstrating that the simulation is difficult both for wave packets prepared as excitations of the fermionic vacuum, and for hole wave packets at filling fraction one-half in the limit of strong coupling. In the latter case, which is described by the t-J model, there is only reflection and no transmission in the scattering events, as would be the case for classical hard spheres. In that sense, the construction provides a quantum mechanical analog of the Fredkin-Toffoli billiard ball computer.