We introduce quantum hybrid gates that act on qudits of different dimensions. In particular, we develop two representative two-qudit hybrid gates (SUM and SWAP) and many-qudit hybrid Toffoli and Fredkin gates. We apply the hybrid SUM gate to generating entanglement, and find that operator entanglement of the SUM gate is equal to the entanglement generated by it for certain initial states. We also show that the hybrid SUM gate acts as an automorphism on the Pauli group for two qudits of different dimension under certain conditions. Finally, we describe a physical realization of these hybrid gates for spin systems.

After reviewing the general properties of zero-energy quantum states, we give the explicit solutions of the Schrödinger equation with E = 0 for the class of potentials V = −|γ|/r ν , where −∞ < ν < ∞. For ν > 2, these solutions are normalizable and correspond to bound states, if the angular momentum quantum number l > 0. [These states are normalizable, even for l = 0, if we increase the space dimension, D, beyond 4; i.e. for D > 4.] For ν < −2 the above solutions, although unbound, are normalizable. This is true even though the corresponding potentials are repulsive for all r. We discuss the physics of these unusual effects.

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We define a special matrix multiplication among a special subset of 2N × 2N matrices, and study the resulting (non-associative) algebras and their subalgebras. We derive the conditions under which these algebras become alternative non-associative and when they become associative. In particular, these algebras yield special matrix representations of octonions and complex numbers; they naturally lead to the Cayley-Dickson doubling process. Our matrix representation of octonions also yields elegant insights into Dirac's equation for a free particle. A few other results and remarks arise as byproducts.