Non-Clifford symmetry protected topological higher-order cluster states in multi-qubit measurement-based quantum computation

Abstract

A cluster state is a strongly entangled state, which is a source of measurement-based quantum computation. It is generated by applying controlled-Z (CZ) gates to the state $|++\cdots +⟩$. It is protected by the ${\mathbb{Z}}_2^{\text{even}}\times {\mathbb{Z}}_2^{\text{odd}}$ symmetry. By applying general quantum gates to the state $|++\cdots +⟩$, we systematically obtain a general short-range entangled cluster state. If we use a non-Clifford gate such as the controlled phase-shift gate, we obtain a non-Clifford cluster state. Furthermore, if we use the controlled-controlled Z (CCZ) gate instead of the CZ gate, we obtain non-Clifford cluster states with five-body entanglement. We generalize it to the C${}^N$Z gate, where $(2N+1)$-body entangled states are generated. The ${\mathbb{Z}}_2^{\text{even}}\times {\mathbb{Z}}_2^{\text{odd}}$ symmetry is non-Clifford for $N\ge 3$. We demonstrate that there emerge $2^{2N}$ fold degenerate ground states for an open chain, indicating the emergence of $N$ free spins at each edge. They can be used as an $N$-qubit input and an $N$-qubit output in measurement-based quantum computation. We also study the non-invertible symmetry, the Kennedy-Tasaki transformation and the string-order parameter in addition to the ${\mathbb{Z}}_2^{\text{even}}\times {\mathbb{Z}}_2^{\text{odd}}$ symmetry in these models.