Deterministic and Cascadable Conditional Phase Gate for Photonic Qubits

Cross-phase modulation (XPM) at the single-photon level, if strong enough, would enable a simple conditional $\pi$-phase gate for photonic qubits. Together with easily realized single-qubit rotations for such qubits, this would provide a universal gate set for quantum computation. However, previous analyses of photonic conditional $\phi$-phase gates that treat XPM in a causal, multimode, quantum field setting suggest that a large ($\sim$$\pi$\,rad) nonlinear phase shift is always accompanied by fidelity-degrading noise. We present a conditional phase gate that, for sufficiently small nonzero $\phi$, has high fidelity. Moreover, our gate is cascadable, in that it preserves the structure of the principal modes used to encode qubit information, and can therefore be cascaded to realize a high-fidelity conditional $\pi$-phase gate. The key components of our gate are: (1) an atomic $\vee$-system to create XPM; (2) a principal-mode restorer that compensates the evolution a principal-mode incurs when the $\vee$-system is driven by a single photon; and (3) a principal-mode projector that exploits the quantum Zeno effect to preclude the accumulation of fidelity-degrading departures from the principal-mode Hilbert space when both control and target photons illuminate the gate.