Optimal mode transformations for linear-optical cluster-state generation

We analyze the generation of linear-optical cluster states (LOCSs) via sequential addition of one and two qubits. Existing approaches employ the stochastic linear-optical two-qubit controlled-Z (cz) gate with success rate of $\frac{1}{9}$ per operation. The question of optimality of the cz gate with respect to LOCS generation has remained open. We report that there are alternative schemes to the cz gate that are exponentially more efficient and show that sequential LOCS growth is indeed globally optimal. We find that the optimal cluster growth operation is a state transformation on a subspace of the full Hilbert space. We show that the maximal success rate of postselected entangling $n$ photonic qubits or $m$ Bell pairs into a cluster is ${\left(\frac{1}{2}\right)}^{n-1}$ and ${\left(\frac{1}{4}\right)}^{m-1}$, respectively, with no ancilla photons, and we give an explicit optical description of the optimal mode transformations.

Figure 1

Adding a single qubit to a $C_n$ cluster.

Figure 2

Joining a Bell pair to a $C_n$ cluster.

Figure 3

Entanglement swapping by the stretch gate.