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

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.

22 pages, 23 figures

References in corpus (15)

• Novel schemes for measurement-based quantum computation
• Universal resources for measurement-based quantum computation
• Measurement-based quantum computation beyond the one-way model
• Quantum secret sharing with qudit graph states
• Quantum Error Correcting Codes Using Qudit Graph States
• Quantum computational capability of a 2D valence bond solid phase
• Computational Power of Symmetry-Protected Topological Phases
• Fundamentals of universality in one-way quantum computation
• Resource quality of a symmetry-protected topologically ordered phase for quantum computation
• Quantum computation by local measurement
• Symmetry-protected topological phases with uniform computational power in one dimension
• Phase transition of computational power in the resource states for one-way quantum computation
• Qudit quantum computation on matrix product states with global symmetry
• Latent Computational Complexity of Symmetry-Protected Topological Order with Fractional Symmetry
• Universal measurement-based quantum computation in two-dimensional SPT phases

Cited by in corpus (13)

• A computationally universal phase of quantum matter
• Subsystem symmetries, quantum cellular automata, and computational phases of quantum matter
• Symmetry-resolved entanglement in symmetry-protected topological phases
• Pivot Hamiltonians as generators of symmetry and entanglement
• Changing the circuit-depth complexity of measurement-based quantum computation with hypergraph states
• Measurement-Based Quantum Computation
• Full Commuting Projector Hamiltonians of Interacting Symmetry-Protected Topological Phases of Fermions
• Quantum spin systems for measurement-based quantum computation
• Symmetric Finite-Time Preparation of Cluster States via Quantum Pumps
• Measurement-based quantum simulation of Abelian lattice gauge theories
• Pivoting through the chiral-clock family
• $Z_3$ and $(\times Z_3{)}^3$ symmetry protected topological paramagnets
• Two-dimensional topological paramagnets protected by ${\mathbb{Z}}_3$ symmetry: Properties of the boundary Hamiltonian