Unitary Operations with Five and More Qubits: Roadmaps and Effective Quantum Circuits

This article presents the general method of QR decomposition of r-qubit operations, r≥ 3, by means of quantum signal-induced heap transformations (QsiHT). These are quantum analogues of discrete signal-induced heap transformations, which are generated by given signals and paths, or orders, of processing the data. The case of the 5-qubit operations is described in detail, and a recurrent form of calculation of all 5-qubit QsiHTs from the 4-qubit QsiHTs is given. For that, roadmaps and quantum circuits are presented for all 31 5-point QsiHTs that are used in the QR decomposition. New roadmaps, namely the schemes with paths for performing basic operations on qubits, and corresponding quantum circuits, are also described for the 4- and 3-qubit transformations. All QsiHTs use fast paths, which allow us to calculate the QR decomposition only on disjoint bit planes. As a result, we build the quantum circuits for 5- and more-qubit operations without permutations, only elementary rotations. Unitary operations with real numbers are considered. In the general case, the method of compositing all the roadmaps and quantum schemes for the calculation of any r-qubit operation by only (2r−1)2r−1 elementary rotations is described.