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Lithium niobate photonic circuits have the salutary property of permittingthe generation, transmission, and processing of photons to be accommodated on asingle chip. Compact photonic circuits such as these, with multiple componentsintegrated on a single chip, are crucial for efficiently implementing quantuminformation processing schemes. We present a set of basic transformations thatare useful for manipulating modal qubits in Ti:LiNbO$_3$ photonic quantumcircuits. These include the mode analyzer, a device that separates the even andodd components of a state into two separate spatial paths; the mode rotator,which rotates the state by an angle in mode space; and modal Pauli spinoperators that effect related operations. We also describe the design of adeterministic, two-qubit, single-photon, CNOT gate, a key element in certainsets of universal quantum logic gates. It is implemented as a Ti:LiNbO$_3$photonic quantum circuit in which the polarization and mode number of a singlephoton serve as the control and target qubits, respectively. It is shown thatthe effects of dispersion in the CNOT circuit can be mitigated by augmenting itwith an additional path. The performance of all of these components areconfirmed by numerical simulations. The implementation of these transformationsrelies on selective and controllable power coupling among single- and two-modewaveguides, as well as the polarization sensitivity of the Pockels coefficientsin LiNbO$_3$.

Modal and Polarization Qubits in Ti:LiNbO$_3$ Photonic Circuits for a Universal Quantum Logic Gate