Approximately preparing an N-fold rotationally symmetric cat state with the phase space instruction set requires circuit depth Omega(phi(N)) = Omega(N / log log N), and for prime N a depth-4N protocol saturates this bound with optimal runtime Theta(alpha).
Coherent and Incoherent States of the Radiation Field,
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Preparing approximate $N$-fold cat states with the phase space instruction set
Approximately preparing an N-fold rotationally symmetric cat state with the phase space instruction set requires circuit depth Omega(phi(N)) = Omega(N / log log N), and for prime N a depth-4N protocol saturates this bound with optimal runtime Theta(alpha).