Measurement-Based Preparation of Higher-Dimensional AKLT States and Their Quantum Computational Power

TL;DR

This paper presents a measurement-based scheme to efficiently prepare higher-dimensional AKLT states with random decorations, demonstrating their utility in quantum computation and establishing their equivalence in computational power to known models.

Contribution

It introduces a constant-time, fusion measurement scheme for creating higher-dimensional AKLT states with random decorations and analyzes their quantum computational capabilities.

Findings

Randomly decorated AKLT states have at least the same computational power as non-random ones.

Deterministic, constant-time preparation schemes exist for AKLT states on Bethe lattices.

Random-bond AKLT states can be converted to encoded graph states with similar computational power.

Abstract

We investigate a constant-time, fusion measurement-based scheme to create AKLT states beyond one dimension. We show that it is possible to prepare such states on a given graph up to random spin-1 `decorations', each corresponding to a probabilistic insertion of a vertex along an edge. In investigating their utility in measurement-based quantum computation, we demonstrate that any such randomly decorated AKLT state possesses at least the same computational power as non-random ones, such as those on trivalent planar lattices. For AKLT states on Bethe lattices and their decorated versions we show that there exists a deterministic, constant-time scheme for their preparation. In addition to randomly decorated AKLT states, we also consider random-bond AKLT states, whose construction involves any of the canonical Bell states in the bond degrees of freedom instead of just the singlet in the original construction. Such states naturally emerge upon measuring all the decorative spin-1 sites in the randomly decorated AKLT states. We show that those random-bond AKLT states on trivalent lattices can be converted to encoded random graph states after acting with the same POVM on all sites. We also argue that these random-bond AKLT states possess similar quantum computational power as the original singlet-bond AKLT states via the percolation perspective.