Optics has served as the expermental basis and model for much of science since Galielo. The minds at work in mechanics were so too in optics, in its early geometric aspects through Snell up to Newton, and its wave aspects since Huygens through Fourier and including Sommerfeld. Hamilton’s original application of generating functions for canonical transformations was in optics; only later were they recognized as a sharp tool for abstract mechanics through the introduction of the concept of a phase space with a symplectic metric [ 1 ]. This concept in embrionary form, served Heisenberg [ 2 ] and Weyl [ 3 ] for the introduction of quantum mechanics, where the Hamiltonian is quadratic in momentum. The history and scope of Sophus Lie’s work is well known to this audience, as is the fact that its manifold applications have been most successful when applied to systems with quadratic Hamiltonians [ 4 ]. Almost invariably, the kinetic energy term is p 2 /2m. This Hamiltonian is indeed also useful in optics, albeit in its Gaussian approximation, i.e., first-order optics which deals with linear transformations of phase space. The associated methods have been under research since the early seventies from the perspective of quantum mechanics [ 5 ] and groups of integral transforms [ 6 , 7 ].
We present a general scheme for a family of linear-optical quantum control gates, including controlled-not and controlled-swap gates for two or three qubits. Our approach utilizes polarization-path-entangled pairs of photons and encodes qubits in mixed degrees of freedom with the control qubit specifically occupying the polarization degree of freedom. By exploiting multiple degrees of freedom and initial state entanglement of the two photons, the proposed control gates do not require any ancilla photons or measurement-induced nonlinearities. Since our gates are purely linear and we implicitly use nonlinearities in a standard manner to create entanglement via parametric nonlinear process, our work demonstrates that a need to have nonlinearities in the photonic gates can be shifted to the state preparation stage; the cost of such shift is that the construction of a certain class of single-qubit operations, such as Hadamard, needs to be probabilistic. In particular, we focus on a deterministic linear-optical quantum Fredkin (controlled-swap) gate and perform a full characterization of the gate performance with a high fidelity typically well above
99 under realistic conditions. The proposed control gates rely on simple linear-optical elements and polarization-entangled photon pairs readily generated from ubiquitous sources, making the gates experimentally feasible with current technologies.