Tight Bounds on the Synthesis of 3-bit Reversible Circuits: NFT Library

The reversible circuit synthesis problem can be reduced to permutation group. This allows Schreier-Sims Algorithm for the strong generating set-finding problem to be used to find tight bounds on the synthesis of 3-bit reversible circuits using the NFT library. The tight bounds include the maximum and minimum length of 3-bit reversible circuits, the maximum and minimum cost of 3-bit reversible circuits. The analysis shows better results than that found in the literature for the lower bound of the cost. The analysis also shows that there are 1960 universal reversible sub-libraries from the main NFT library.Comment: 18 pages. arXiv admin note: text overlap with arXiv:1101.438

Reading a Single Qubit System Using Weak Measurement with Variable Strength

Acquiring information about an unknown qubit in a superposition of two states is essential in any computation process. Quantum measurement, or sharp measurement, is usually used to read the information contents of that unknown qubit system. Quantum measurement is an irreversible operation that makes the superposition collapses to one of the two possible states in a probabilistic way. In this paper, a quantum algorithm will be proposed to read the information in an unknown qubit without applying sharp measurement on that qubit. The proposed algorithm will use a quantum feedback control scheme by applying sharp measurement iteratively on an auxiliary qubit weakly entangled with the unknown qubit. The information contents of the unknown qubit can be read by counting the outcomes from the sharp measurement on the auxiliary qubit. Iterative measurements on the auxiliary qubit will make the amplitudes of the superposition move in a random walk manner where a weak measurement is applied on the unknown qubit which can be reversed when the random walk takes opposite steps to decrease the disturbance introduced to the system. The proposed algorithm will define the strength of the weak measurement so that it can be controlled by adding an arbitrary number of dummy qubits $\mu$ to the system. This will make the measurement process slowdown to an arbitrary scale so that the effect of the sharp measurement on the unknown qubit is reached after $O(\mu^2)$ measurements on the auxiliary qubit.Comment: 15 pages, 4 figure