The Complexity of Translationally-Invariant Spin Chains with Low Local Dimension

TL;DR

This paper proves that estimating the ground state energy of translationally-invariant 1D spin chains with low local dimension is computationally very hard (QMAEXP-complete), using a novel encoding of quantum computation into such systems.

Contribution

It introduces a new model for encoding quantum computation into translationally-invariant systems and extends existing techniques to allow more complex computational paths, enabling the proof of QMAEXP-completeness.

Findings

Ground state energy estimation is QMAEXP-complete for low-dimensional translationally-invariant chains.

New encoding techniques allow complex computational paths including branching and cycles.

Spectral analysis of Hamiltonians is achieved via graph Laplacians with unitary edge labels.

Abstract

We prove that estimating the ground state energy of a translationally-invariant, nearest-neighbour Hamiltonian on a 1D spin chain is QMAEXP-complete, even for systems of low local dimension (roughly 40). This is an improvement over the best previously-known result by several orders of magnitude, and it shows that spin-glass-like frustration can occur in translationally-invariant quantum systems with a local dimension comparable to the smallest-known non-translationally-invariant systems with similar behaviour. While previous constructions of such systems rely on standard models of quantum computation, we construct a new model that is particularly well-suited for encoding quantum computation into the ground state of a translationally-invariant system. This allows us to shift the proof burden from optimizing the Hamiltonian encoding a standard computational model to proving universality of a simple model. Previous techniques for encoding quantum computation into the ground state of a local Hamiltonian allow only a linear sequence of gates, hence only a linear (or nearly linear) path in the graph of all computational states. We extend these techniques by allowing significantly more general paths, including branching and cycles, thus enabling a highly efficient encoding of our computational model. However, this requires more sophisticated techniques for analysing the spectrum of the resulting Hamiltonian. To address this, we introduce a framework of graphs with unitary edge labels. After relating our Hamiltonian to the Laplacian of such a unitary labelled graph, we analyse its spectrum by combining matrix analysis and spectral graph theory techniques.