Hamiltonian quantum computer in one dimension

TL;DR

This paper explores the design of one-dimensional Hamiltonian quantum computers, demonstrating how increasing locality reduces the required local Hilbert space dimension for universal quantum computation, with implications for simulating spin chains.

Contribution

It constructs 1D Hamiltonian quantum computers with specific locality and dimension parameters, showing the trade-offs and requirements for universality and translation invariance.

Findings

A 3-local Hamiltonian with d=5 suffices for universal computation in 1D.

Simulating 1D spin-2 chains is BQP-complete.

Translation invariance increases the minimum required local dimension to 8.

Abstract

Quantum computation can be achieved by preparing an appropriate initial product state of qudits and then letting it evolve under a fixed Hamiltonian. The readout is made by measurement on individual qudits at some later time. This approach is called the Hamiltonian quantum computation and it includes, for example, the continuous-time quantum cellular automata and the universal quantum walk. We consider one spatial dimension and study the compromise between the locality $k$ and the local Hilbert space dimension $d$. For geometrically 2-local (i.e., $k=2$), it is known that $d=8$ is already sufficient for universal quantum computation but the Hamiltonian is not translationally invariant. As the locality $k$ increases, it is expected that the minimum required $d$ should decrease. We provide a construction of Hamiltonian quantum computer for $k=3$ with $d=5$. One implication is that simulating 1D chains of spin-2 particles is BQP-complete. Imposing translation invariance will increase the required $d$. For this we also construct another 3-local ($k=3$) Hamiltonian that is invariant under translation of a unit cell of two sites but that requires $d$ to be 8.