Generalized Quantum Search with Parallelism

We generalize Grover's unstructured quantum search algorithm to enable it to use an arbitrary starting superposition and an arbitrary unitary matrix simultaneously. We derive an exact formula for the probability of the generalized Grover's algorithm succeeding after n iterations. We show that the fully generalized formula reduces to the special cases considered by previous authors. We then use the generalized formula to determine the optimal strategy for using the unstructured quantum search algorithm. On average the optimal strategy is about 12% better than the naive use of Grover's algorithm. The speedup obtained is not dramatic but it illustrates that a hybrid use of quantum computing and classical computing techniques can yield a performance that is better than either alone. We extend the analysis to the case of a society of k quantum searches acting in parallel. We derive an analytic formula that connects the degree of parallelism with the optimal strategy for k-parallel quantum search. We then derive the formula for the expected speed of k-parallel quantum search.Comment: 14 pages, 2 figure

Optimal Quantum Circuits for General Two-Qubit Gates

In order to demonstrate non-trivial quantum computations experimentally, such as the synthesis of arbitrary entangled states, it will be useful to understand how to decompose a desired quantum computation into the shortest possible sequence of one-qubit and two-qubit gates. We contribute to this effort by providing a method to construct an optimal quantum circuit for a general two-qubit gate that requires at most 3 CNOT gates and 15 elementary one-qubit gates. Moreover, if the desired two-qubit gate corresponds to a purely real unitary transformation, we provide a construction that requires at most 2 CNOTs and 12 one-qubit gates. We then prove that these constructions are optimal with respect to the family of CNOT, y-rotation, z-rotation, and phase gates.Comment: 6 pages, 8 figures, new title, final journal versio

Non-unitary probabilistic quantum computing circuit and method

A quantum circuit performing quantum computation in a quantum computer. A chosen transformation of an initial n-qubit state is probabilistically obtained. The circuit comprises a unitary quantum operator obtained from a non-unitary quantum operator, operating on an n-qubit state and an ancilla state. When operation on the ancilla state provides a success condition, computation is stopped. When operation on the ancilla state provides a failure condition, computation is performed again on the ancilla state and the n-qubit state obtained in the previous computation, until a success condition is obtained

Quantum interferometric optical lithography:towards arbitrary two-dimensional patterns

As demonstrated by Boto et al. [Phys. Rev. Lett. 85, 2733 (2000)], quantum lithography offers an increase in resolution below the diffraction limit. Here, we generalize this procedure in order to create patterns in one and two dimensions. This renders quantum lithography a potentially useful tool in nanotechnology.Comment: 9 pages, 5 figures Revte

Quantum Clock Synchronization Based on Shared Prior Entanglement

We demonstrate that two spatially separated parties (Alice and Bob) can utilize shared prior quantum entanglement, and classical communications, to establish a synchronized pair of atomic clocks. In contrast to classical synchronization schemes, the accuracy of our protocol is independent of Alice or Bob's knowledge of their relative locations or of the properties of the intervening medium.Comment: 4 page

Efferent projections of C3 adrenergic neurons in the rat central nervous system

C3 neurons constitute one of three known adrenergic nuclei in the rat central nervous system (CNS). While the adrenergic C1 cell group has been extensively characterized both physiologically and anatomically, the C3 nucleus has remained relatively obscure. This study employed a lentiviral tracing technique that expresses green fluorescent protein behind a promoter selective to noradrenergic and adrenergic neurons. Microinjection of this virus into the C3 nucleus enabled the selective tracing of C3 efferents throughout the rat CNS, thus revealing the anatomical framework of C3 projections. C3 terminal fields were observed in over 40 different CNS nuclei, spanning all levels of the spinal cord, as well as various medullary, mesencephalic, hypothalamic, thalamic, and telencephalic nuclei. The highest densities of C3 axon varicosities were observed in Lamina X and the intermediolateral cell column of the thoracic spinal cord, as well as the dorsomedial medulla (both commissural and medial nuclei of the solitary tract, area postrema, and the dorsal motor nucleus of the vagus), ventrolateral periaqueductal gray, dorsal parabrachial nucleus, periventricular and rhomboid nuclei of the thalamus, and paraventricular and periventricular nuclei of the hypothalamus. In addition, moderate and sparse projections were observed in many catecholaminergic and serotonergic nuclei, as well as the area anterior and ventral to the third ventricle, Lamina X of the cervical, lumbar, and sacral spinal cord, and various hypothalamic and telencephalic nuclei. The anatomical map of C3 projections detailed in this survey hopes to lay the first steps toward developing a functional framework for this nucleus

A Random Matrix Model of Adiabatic Quantum Computing

We present an analysis of the quantum adiabatic algorithm for solving hard instances of 3-SAT (an NP-complete problem) in terms of Random Matrix Theory (RMT). We determine the global regularity of the spectral fluctuations of the instantaneous Hamiltonians encountered during the interpolation between the starting Hamiltonians and the ones whose ground states encode the solutions to the computational problems of interest. At each interpolation point, we quantify the degree of regularity of the average spectral distribution via its Brody parameter, a measure that distinguishes regular (i.e., Poissonian) from chaotic (i.e., Wigner-type) distributions of normalized nearest-neighbor spacings. We find that for hard problem instances, i.e., those having a critical ratio of clauses to variables, the spectral fluctuations typically become irregular across a contiguous region of the interpolation parameter, while the spectrum is regular for easy instances. Within the hard region, RMT may be applied to obtain a mathematical model of the probability of avoided level crossings and concomitant failure rate of the adiabatic algorithm due to non-adiabatic Landau-Zener type transitions. Our model predicts that if the interpolation is performed at a uniform rate, the average failure rate of the quantum adiabatic algorithm, when averaged over hard problem instances, scales exponentially with increasing problem size.Comment: 9 pages, 7 figure