We consider a generic elementary gate sequence which is needed to implement a general quantum gate acting on n qubits-a unitary transformation with 4(n) degrees of freedom. For synthesizing the gate sequence, a method based on the so-called cosine-sine matrix decomposition is presented. The result is optimal in the number of elementary one-qubit gates, 4(n), and scales more favorably than the previously reported decompositions requiring 4(n)-2(n+1) controlled NOT gates.

The basis of synchronous manipulation of individual electrons in solid-state devices was laid by the rise of single electronics about two decades ago 1-3 . Ultrasmall structures in a low-temperature environment form an ideal domain for addressing electrons one by one. In the so-called metrological triangle, voltage from the Josephson effect and resistance from the quantum Hall effect would be tested against current via Ohm's law for a consistency check of the fundamental constants of nature,h and e (ref. 4). Several attempts to create a metrological current source that would comply with the demanding criteria of extreme accuracy, high yield and implementation with not too many control parameters have been reported [5][6][7][8][9][10][11] . Here, we propose and prove the unexpected concept of a hybrid normal-metalsuperconductor turnstile in the form of a one-island singleelectron transistor with one gate, which demonstrates robust current plateaux at multiple levels of e f at frequency f .Synchronized sources, where current I is related to frequency by I = N ef and N is the integer number of electrons injected in one period, are the prime candidates for the devices to define one ampere in quantum metrology. The accuracy of these devices is based on the discreteness of the electron charge and the high accuracy of frequency determined from atomic clocks. Modern methods are replacing classical definitions of electrical quantities; voltage can be derived on the basis of the a.c. Josephson effect of superconductivity 12 and resistance by the quantum Hall effect 13,14 , but one ampere still needs to be determined via the mutual force exerted by the leads carrying the current. Early proposals of current pumps for quantum metrology were based on arrays of mesoscopic metallic tunnel junctions 5,6 , in which small currents could eventually be pumped at very low error rates 7 . However, these multijunction devices are hard to control and relatively slow 15 . Thus, the quest for feasible implementation with a possibility of parallel architecture for higher yield have led to alternative solutions such as surface-acoustic-wave-driven one-dimensional channels 8 , superconducting devices 11,16-21 and semiconducting quantum dots 22 . These do produce large currents in the nano-ampere range but their accuracy is still limited.Surprisingly, a simple hybrid single-electron transistor, with a small normal-metal (N) island and superconducting (S) leads, has been overlooked in this context. As demonstrated here, an SNS transistor, or alternatively an NSN transistor, see Fig. 1, presents a robust turnstile for electrons showing current plateaux at multiples of ef . We emphasize here that a one-island turnstile does not work even in principle without the hybrid design. An important feature in the present system is that hybrid tunnel junctions suppress tunnelling in an energy range determined by the gap ∆ in the density of states of the superconductor, see Fig. 1d bottom inset; current through a junction vanishes as long as |V J | ∼ < ∆/e...

We consider a unitary transformation which maps any given pure state of an $n$-qubit quantum register into another one. This transformation has applications in the initialization of a quantum computer, and also in some quantum algorithms. Employing uniformly controlled rotations, we present a quantum circuit of $2^{n+2}-4n-4$ CNOT gates and $2^{n+2}-5$ one-qubit elementary rotations that effects the state transformation. The complexity of the circuit is noticeably lower than the previously published results. Moreover, we present an analytic expression for the rotation angles needed for the transformation.

We present the first measurements of the Berry phase in a superconducting Cooper pair pump. A fixed amount of Berry phase is accumulated to the quantum-mechanical ground state in each adiabatic pumping cycle, which is determined by measuring the charge passing through the device. The dynamic and geometric phases are identified and measured quantitatively from their different response when pumping in opposite directions. Our observations, in particular, the dependencies of the dynamic and geometric effects on the superconducting phase bias across the pump, agree with the basic theoretical model of coherent Cooper pair pumping.

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