Non-Clifford symmetry protected topological hyper-cluster states and multi-qubit universal measurement-based quantum computation

A cluster state is a highly entangled quantum state that serves as a universal resource for measurement-based quantum computation. It is generated by applying controlled-Z (CZ) gates to the product state $\left\vert ++\cdots +\right\rangle $, and its parent Hamiltonian is the ZXZ model. This model exhibits a topological phase protected by the $\mathbb{Z}_{2}^{\text{even}}\times \mathbb{Z}_{2}^{\text{odd}}$\ symmetry. By applying general quantum gates to the state $\left\vert ++\cdots +\right\rangle $, we systematically obtain a general short-range entangled cluster state. If we use the controlled-controlled Z (CCZ) gate instead of the CZ gate, we obtain hyper-cluster states featuring five-body interaction terms. These states enable the implementation of the CZ gate within measurement-based quantum computation. Motivated by this, we propose a generalization to the C$^{N}$Z gate, leading to a hyper-cluster model, in which $(2N+1)$-body entangled states are generated. This model retains the $\mathbb{Z}_{2}^{\text{even}}\times \mathbb{Z}_{2}^{\text{odd}}$\ symmetry, which lies outside the Clifford group for $N\geq 3$. We demonstrate that an open chain in this model exhibits a $2^{2N}$ fold degenerate ground states protected by the symmetry, corresponding to the emergence of $N$ effective free spins at each edge. These boundary degrees of freedom can serve as $N$ qubits for measurement-based quantum computation. In particular, this framework makes it possible to implement the C$^{N-1}$Z gate in measurement-based quantum computation.