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

TL;DR

This paper develops constant-time, fusion-based measurement schemes to prepare higher-dimensional AKLT states on general graphs, including randomly decorated edges and random-bond variants, and proves these ensembles remain universal resources for measurement-based quantum computation. It introduces building-block preparation and fusion via Bell-state measurements or Hadamard tests, and extends from 1D to arbitrary graphs, with deterministic preparation on Bethe lattices and percolation-based arguments for universality in decorated and random-bond cases. By mapping post-POVM states to encoded graph states, the authors show decorated and random-bond AKLT states retain MBQC power on lattices where the original AKLT state is universal, and they discuss distributed implementations and potential 3D extensions. The results broaden the practical utility of AKLT-inspired states as scalable, low-depth quantum resources for universal computation, leveraging symmetry, percolation, and local corrections to manage decorations and bond randomness.

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.

Figures (11)

• Figure 1: 1D AKLT state and a scheme to create it by fusing elementary blocks. Each dot represents a qubit, and two dots connected by a dashed line is a two-qubit single state: $|\Psi^-\rangle=(|01\rangle-|10\rangle)/\sqrt{2}$. A circle represents a physical site, and its actual physical degree of freedom corresponds to the symmetric subspace of the qubits inside the circle. (a) Schematic definition of 1D spin-1 AKLT state. The blue circles here represent the mapping from virtual qubits to physical spins. (b) Several elementary blocks to be used in the fusion procedure, shown in (c), (d) & (e), which uses Bell-state measurement (BSM) in (c) and Hadamard test (HT) in (d), indicated by yellow trapezoids, on two dangling qubits to fuse the two blocks into a larger part of the AKLT state. The measurement in (e) can be BSM or HT. Illustration for creating (c) 1D and (e) 2D AKLT states.
• Figure 2: Illustration of 2D AKLT states. (a) Bethe lattice with coordination number $z=3$, (b) Hexagonal lattice, (c) square lattice, and (d) (uniformly) decorated square lattice. Each dot inside a circle represents a qubit or a spin-1/2 entity. A circle represents a physical site, which corresponds to spin magnitude $S=z_v/2$ (equivalently, the symmetric subspace of the qubits inside the circle), where $z_v$ is the number of dots inside the circle.
• Figure 3: Example circuit that creates $|B(2)\rangle$ using Eq. (\ref{['eq:PsiB1']}).
• Figure 4: Examples of symmetry in AKLT states. (a) The $Z_2\times Z_2$ symmetry of the 1D AKLT state can be summarized using a local MPS. (b) The diagram in (a) can be interpreted as moving the defect operator $\sigma_\alpha$ from l.h.s. to r.h.s. by acting on the physical degree of freedom by a $\pi$ rotation w.r.t. to $\alpha$-axis. (c) & (d) represent the $Z_2\times Z_2$ symmetry for hexagonal- and square-lattice AKLT states. The diagrams hold for the specific choice of deformation that preserves the symmetry.
• Figure 5: (a) AKLT quasichain that consists of spin-3/2 and spin-1/2 sites. (b) Stack of two such quasichains. This can be used to build a 2D system. There can be several alternatives to merge the neighboring two dangling spin-1/2 sites.