The power of quantum systems on a line

TL;DR

This paper demonstrates that one-dimensional quantum systems can perform universal adiabatic quantum computation and that approximating their ground state energy is QMA-complete, highlighting complex computational properties unlike classical counterparts.

Contribution

It proves universal adiabatic quantum computation in 1D systems and establishes QMA-completeness for ground state energy approximation with novel techniques.

Findings

Universal adiabatic quantum computation is possible in 1D systems with 9 states per particle.

Ground state energy approximation in 1D quantum systems is QMA-complete.

Some 1D quantum systems take exponential time to relax to their ground states, resembling spin glasses.

Abstract

We study the computational strength of quantum particles (each of finite dimensionality) arranged on a line. First, we prove that it is possible to perform universal adiabatic quantum computation using a one-dimensional quantum system (with 9 states per particle). This might have practical implications for experimentalists interested in constructing an adiabatic quantum computer. Building on the same construction, but with some additional technical effort and 12 states per particle, we show that the problem of approximating the ground state energy of a system composed of a line of quantum particles is QMA-complete; QMA is a quantum analogue of NP. This is in striking contrast to the fact that the analogous classical problem, namely, one-dimensional MAX-2-SAT with nearest neighbor constraints, is in P. The proof of the QMA-completeness result requires an additional idea beyond the usual techniques in the area: Not all illegal configurations can be ruled out by local checks, so instead we rule out such illegal configurations because they would, in the future, evolve into a state which can be seen locally to be illegal. Our construction implies (assuming the quantum Church-Turing thesis and that quantum computers cannot efficiently solve QMA-complete problems) that there are one-dimensional systems which take an exponential time to relax to their ground states at any temperature, making them candidates for being one-dimensional spin glasses.