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 $\left\vert ++\cdots +\right\rangle $. 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 $ \left\vert ++\cdots +\right\rangle $, we systematically obtain a general short-range entangled cluster state. If we use a non-Clifford gate such as the controlled phase-shift gate, we obtain a non-Clifford cluster state. Furthermore, if we use the controlled-controlled Z (CCZ) gate instead of the CZ gate, we obtain non-Clifford cluster states with five-body entanglement. We generalize it to the C$^{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 $N\geq 3$. 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.

Figures (11)

• Figure 1: ZXZ model. (a) Quantum circuit generating the ZXZ cluster state. (b) Its graph representation. The vertices represent the state $\left\vert +\right\rangle$ and the edges represent the CZ gates.
• Figure 2: Energy spectrum of the ZXZ model. (a) Closed chain, where there is a single gapped ground state as indicated by a closed red circle. (b) Open chain, where there are four-fold degenerate gapped ground state as indicated by a closed blue oval. We have set $L=3$. The vertical axis is the energy, while the horizontal axis is the index of the energy.
• Figure 3: (a1) Energy spectrum of the ZXZ model for a closed chain. The energy is symmetric at $\alpha =1/2$ reflecting the $\mathbb{Z}_{2}^{\text{CZ}}$ symmetry. (b1) That for an open chain. The ground states are four-fold degenerate at $\alpha =0$ and not degenerate at $\alpha =1$. The degeneracy splits at $\alpha$ slightly smaller than $1/2$. (c1) String order parameter. There is a jump at $\alpha =1/2$, implying a topological phase transition at $\alpha =1/2$. (a2) Energy spectrum of the CCZ model for a closed chain. (b2) That for an open chain. The ground states are 16-fold degenerate at $\alpha =0$ and not degenerate at $\alpha =1$. The degeneracy splits for $\alpha >0$. (c2) String order parameter for the CCZ model. There is no jump at $\alpha =1/2$, implying a topological phase transition at $\alpha =0$. The horizontal axis is $\alpha$. We have set $L=11$
• Figure 4: Quantum circuit for the class of the instantaneous quantum polynomial time (IQP).
• Figure 5: XZX model. (a) Quantum circuit representation. (b) Equivalent quantum circuit using the CX gates.