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arXiv · 1711.00094

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

Yanzhu Chen·Abhishodh Prakash·Tzu-Chieh Wei
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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.

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Yanzhu Chen, Abhishodh Prakash, Tzu-Chieh Wei. 2018-01-24. Universal quantum computing using $(\mathbb{Z}_d)^3$ symmetry-protected topologically ordered states. https://doi.org/10.1103/physreva.97.022305

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