We propose an architecture for a high-fidelity deterministic controlled-phase gate between two photonic qubits using bulk optical nonlinearities in near-term feasible photonic integrated circuits. The gate is enabled by converting travelling continuous-mode photons into stationary cavity modes using strong classical control fields that dynamically change the cavity-waveguide coupling rate. This process limits the fidelity degrading noise pointed out by Shapiro [J. Shapiro, Phys. Rev. A, 73, 2006] and Gea-Banacloche [J. Gea-Banacloche, Phys. Rev. A, 81, 2010]. We show that high-fidelity gates can be achieved with self-phase modulation in $chi^{scriptscriptstyle(3)}$ materials as well as second-harmonic generation in $chi^{scriptscriptstyle(2)}$ materials. The gate fidelity asymptotically approaches unity with increasing storage time for a fixed duration of the incident photon wave packet. Further, dynamically coupled cavities enable a trade-off between errors due to loss and wave packet distortions since loss does not affect the ability to emit wave packets with the same shape as the incoming photons. Our numerical results show that gates with $99%$ fidelity are feasible with near-term improvements in cavity loss using LiNbO$_3$ or GaAs.
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