Hamiltonian Quantum Cellular Automata in 1D

TL;DR

This paper presents a translationally invariant 1D Hamiltonian that enables universal quantum computation autonomously, with results indicating classical simulation of such systems is likely infeasible unless quantum computers are classically simulatable.

Contribution

It introduces a simple, translationally invariant Hamiltonian model for universal quantum computing without external control, advancing quantum computational theory.

Findings

Achieves universal quantum computation with a 1D Hamiltonian

Computational results are obtained after polynomial time evolution

Implicates classical intractability of simulating such Hamiltonians

Abstract

We construct a simple translationally invariant, nearest-neighbor Hamiltonian on a chain of 10-dimensional qudits that makes it possible to realize universal quantum computing without any external control during the computational process. We only require the ability to prepare an initial computational basis state which encodes both the quantum circuit and its input. The computational process is then carried out by the autonomous Hamiltonian time evolution. After a time polynomially long in the size of the quantum circuit has passed, the result of the computation is obtained with high probability by measuring a few qudits in the computational basis. This result also implies that there cannot exist efficient classical simulation methods for generic translationally invariant nearest-neighbor Hamiltonians on qudit chains, unless quantum computers can be efficiently simulated by classical computers (or, put in complexity theoretic terms, unless BPP=BQP).