Universal quantum computing using $(\mathbb{Z}_d)^3$ symmetry-protected topologically ordered states

TL;DR

This paper demonstrates that certain symmetry-protected topologically ordered states of qudits on specific lattices are universal resources for measurement-based quantum computation, extending previous qubit-based results to higher-dimensional systems.

Contribution

It generalizes the construction of universal resource states from qubits to qudits on various lattices, proving universality for prime dimensions greater than 2.

Findings

Universal quantum computation is achievable with $(d-1)$-qudit SPT states on triangular and other 3-colorable lattices.

The construction is valid for prime $d > 2$, extending previous qubit-based results.

The states are proven to be universal resources for measurement-based quantum computation.

Abstract

Measurement-based quantum computation describes a scheme where entanglement of resource states is utilized to simulate arbitrary quantum gates via local measurements. Recent works suggest that symmetry-protected topologically non-trivial, short-ranged entanged states are promising candidates for such a resource. Miller and Miyake [NPJ Quantum Information 2, 16036 (2016)] recently constructed a particular $\mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_2$ symmetry-protected topological state on the Union-Jack lattice and established its quantum computational universality. However, they suggested that the same construction on the triangular lattice might not lead to a universal resource. Instead of qubits, we generalize the construction to qudits and show that the resulting $(d-1)$ qudit nontrivial $\mathbb{Z}_d \times \mathbb{Z}_d \times \mathbb{Z}_d$ symmetry-protected topological states are universal on the triangular lattice, for $d$ being a prime number greater than 2. The same construction also holds for other 3-colorable lattices, including the Union-Jack lattice.