Measurement-Based Quantum Computation on Symmetry Breaking Thermal States

We consider measurement-based quantum computation (MBQC) on thermal states of the interacting cluster Hamiltonian containing interactions between the cluster stabilizers that undergoes thermal phase transitions. We show that the long-range order of the symmetry breaking thermal states below a critical temperature drastically enhances the robustness of MBQC against thermal excitations. Specifically, we show the enhancement in two-dimensional cases and prove that MBQC is topologically protected below the critical temperature in three-dimensional cases. The interacting cluster Hamiltonian allows us to perform MBQC even at a temperature 1 order of magnitude higher than that of the free cluster Hamiltonian.

Figure 1

(a) The fCH on the square lattice. (b) The iCH on the square lattice. (c) The measurement pattern for the identity and the Hadamard gates of the gate length $l$ on the 2D cluster state. (d) The interacting cluster Hamiltonian on the RHG lattice. (e) The 3D cluster state on the cubic lattice. The gray-shaded qubits are measured in the $Z$ basis to obtain the cluster state on the RHG lattice.

Figure 2

The gate fidelities of the identity gates for various distances $l$. (a) shows the gate fidelity for the fCH, $F(2l)$ for $l=2(\times )$, $l=4(\square )$, $l=6(+)$, and $l=8(\circ )$. (b) shows the gate fidelity for the iCH, $F(2l)$ for $l=2(\times )$, $l=4(\square )$, $l=6(+)$, and $l=8(\circ )$. The vertical dashed line shows the critical temperature for the 2D iCH $T_c/J=2/\ln [1+\sqrt{2}]$, and the horizontal dashed line shows the minimum gate fidelity $F=1/4$.