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

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.

💡 Research Summary

The paper investigates how to generate higher‑dimensional Affleck‑Kennedy‑Lieb‑Tasaki (AKLT) states using measurement‑based techniques that run in constant depth, and it evaluates the computational power of the resulting states for measurement‑based quantum computation (MBQC).
AKLT states are defined on arbitrary graphs by placing virtual spin‑½ qubits on each vertex, connecting neighboring virtual qubits with singlet bonds, and projecting the set of virtual qubits at each vertex onto the symmetric subspace, which yields a physical spin‑S particle (S = degree/2). In one dimension the AKLT state is an MPS and can be prepared with a circuit of depth Ω(log N) or linear depth, but recent work showed that mid‑circuit measurements together with classical feed‑forward can reduce the depth to a constant (typically two or three layers).
The authors extend this idea by introducing a “fusion measurement” protocol. They first prepare a small four‑qubit building block |Ψ_B⟩ = P_{23}(|Ψ⁻⟩{12}⊗|Ψ⁻⟩{34}) using a modest gate sequence. Two such blocks are then fused by measuring a pair of dangling virtual qubits in the Bell basis (BSM) or by a Hadamard‑test (HT). If the BSM outcome is the singlet, the blocks merge perfectly; if a triplet is obtained, a Pauli correction (σ_x, σ_y, or σ_z) is applied to one side and propagated through the symmetric structure, exactly as in quantum teleportation. Repeating this operation builds arbitrarily large AKLT chains in constant depth.
For loop‑free graphs such as Bethe lattices (regular trees) the same protocol yields a deterministic, constant‑time preparation of AKLT states, even for deformed versions where each edge may carry an arbitrary Bell state (singlet or one of the three triplet states). The authors prove that after applying the same POVM used in earlier MBQC studies (the three‑outcome spin‑1 or spin‑3/2 POVM), the post‑measurement state is an encoded graph state regardless of which Bell state was used on each edge. Consequently, if the original singlet‑bond AKLT on a trivalent lattice is a universal MBQC resource, the random‑bond AKLT is also universal.
In two dimensions, loops introduce non‑correctable defects when one tries to prepare an exact AKLT state deterministically. To circumvent this, the authors relax the target: they allow each edge to be randomly “decorated” with one or more spin‑1 vertices (degree‑2 nodes) that arise as by‑products of failed fusions. The resulting “randomly decorated AKLT state” is a statistical ensemble of AKLT states on slightly altered graphs. Using percolation theory, they show that for any non‑zero decoration probability the resulting random graph remains in the super‑critical regime, guaranteeing a giant connected component. Applying the standard POVM to every site converts each member of the ensemble into an encoded random graph state, which can be further reduced to a regular 2‑D cluster state. Hence the ensemble possesses at least the same computational power as the ideal, undecorated AKLT state.
The paper also discusses symmetry‑protected topological (SPT) order: 1‑D AKLT exhibits Haldane SPT, 2‑D AKLT on the honeycomb lattice shows weak SPT, while the square‑lattice AKLT displays strong SPT. The POVM construction works straightforwardly for spin‑3/2 and spin‑2 AKLT, but for spins larger than 2 the POVM elements no longer sum to the identity, requiring additional correction terms.
Overall, the work makes three major contributions: (1) a deterministic, constant‑depth preparation scheme for AKLT states on loop‑free Bethe lattices and their deformed variants; (2) a proof that randomly decorated AKLT states on trivalent 2‑D lattices form a universal MBQC resource despite being prepared probabilistically; and (3) the introduction of random‑bond AKLT states, showing that arbitrary Bell‑state bonds do not diminish computational universality. These results provide a practical pathway for generating complex many‑body entangled states on near‑term quantum hardware using only shallow circuits and adaptive measurements.