Probabilistic implementation of Hadamard and unitary gates

Abstract

We show that the Hadamard and Unitary gates could be implemented by a unitary evolution together with a measurement for any unknown state chosen from a set A = |Ψ i , Ψ i (i = 1, 2) if and only if |Ψ 1 , |Ψ 2 , Ψ 1 , Ψ 2 are linearly independent. We also derive the best transformation efficiencies. [3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22] which operate either in a deterministic or probabilistic way. Corresponding to the quantum no-cloning theorem, Pati and Braunstein [23] demonstrated that the linearity of quantum mechanics also forbids one to delete one unknown state ideally against a copy [23], which is called the quantum no-deleting principle. Notably, Zurek [24] further verified the existence of limitations on cloning and deleting completely an unknown state and pointed out the importance of studying approximate and probabilistic deletion corresponding to cloners [3,4,10]. And some probabilistic and state-dependent deleting machines have

Hadamard Type Operations for Qubits

“…al. [11] have tried to implement the Hadamard gate in a probabilistic manner for any unknown state chosen from a set of linearly independent states.…”

Impossibility of Partial Swapping of Quantum Information

“…A universal deterministic quantum deletion machine is designed in an unconventional way that improves the fidelity of deletion from 0.5 and takes it to 0.75 in the limiting sense [19]. Many other impossible operations generally referred as 'General Impossible operations' [10] can not be achieved successfully with certainty , but one can carry out these operations at least probabilistically with certain probability of success [11].…”