Universal adiabatic quantum computation via the space-time circuit-to-Hamiltonian construction

TL;DR

This paper demonstrates how to perform universal adiabatic quantum computation using a 2D local Hamiltonian, leveraging mappings to exactly solvable models to analyze the spectral gap and computational universality.

Contribution

It introduces a new Hamiltonian construction for universal adiabatic quantum computation on a 2D grid, with a detailed spectral gap analysis via mappings to known solvable models.

Findings

Eigenvalue gap bounded using ferromagnetic XXZ chain mapping

Time evolution corresponds to a quantum walk on Young's lattice

Related time-independent Hamiltonian also capable of universal computation

Abstract

We show how to perform universal adiabatic quantum computation using a Hamiltonian which describes a set of particles with local interactions on a two-dimensional grid. A single parameter in the Hamiltonian is adiabatically changed as a function of time to simulate the quantum circuit. We bound the eigenvalue gap above the unique groundstate by mapping our model onto the ferromagnetic XXZ chain with kink boundary conditions; the gap of this spin chain was computed exactly by Koma and Nachtergaele using its $q$-deformed version of SU(2) symmetry. We also discuss a related time-independent Hamiltonian which was shown by Janzing to be capable of universal computation. We observe that in the limit of large system size, the time evolution is equivalent to the exactly solvable quantum walk on Young's lattice.