Reduced Hamiltonian technique for designing quantum logic gates and data channels

Abstract

A scheme for realizing arbitrary controlled-unitary operations in a two qubit system is presented here. If the 2 × 2 unitary matrix is special unitary (has unit determinant), the controlled-unitary gate operation can be realized in a single pulse operation. The pulse in this scheme constitutes varying one of the parameters of the system between an arbitrary maximum and a “calculated” minimum value. This parameter will constitute the variable parameter of the system while the other parameters, which include the coupling between the two qubits, will be treated as fixed parameters. The values of the parameters are what are solved for to realize an arbitrary controlled-unitary operation where the computational complexity of the operation is no greater than that required for a Controlled-NOT (CNOT) gate. Since conventional schemes realize a controlled-unitary operation by breaking it into a sequence of single-qubit and CNOT gate operations, the method presented here is an improvement because not only does it require lesser time duration, but also fewer control lines, to implement the same operation. Furthermore, the method can be applied to a wide range of coupling schemes and can be used to realize gate operations between two qubits coupled via Ising, Heisenberg and anisotropic interactions. Next, a general scheme for implementing bi-directional quantum state transfer in a linear architecture involving un-tunable nearest-neighbor interactions is presented. Unlike quantum spin networks, the scheme allows transmission of several quantum states at a time, requiring only a two qubit separation between quantum states. Moreover, it is shown that only eight control lines are required to achieve state transfer along a channel of arbitrary length, making the scheme efficient.