Abstract. The quantum Fourier transform (QFT), a quantum analog of the classical Fourier transform, has been shown to be a powerful tool in developing quantum algorithms. However, in classical computing there is another class of unitary transforms, the wavelet transforms, which are every bit as useful as the Fourier transform. Wavelet transforms are used to expose the multi-scale structure of a signal and are likely to be useful for quantum image processing and quantum data compression. In this paper, we derive efficient, complete, quantum circuits for two representative quantum wavelet transforms, the quantum Haar and quantum Daubechies D (4) transforms. Our approach is to factor the classical operators for these transforms into direct sums, direct products and dot products of unitary matrices. In so doing, we find that permutation matrices, a particular class of unitary matrices, play a pivotal role. Surprisingly, we find that operations that are easy and inexpensive to implement classically are not always easy and inexpensive to implement quantum mechanically, and vice versa. In particular, the computational cost of performing certain permutation matrices is ignored classically because they can be avoided explicitly. However, quantum mechanically, these permutation operations must be performed explicitly and hence their cost enters into the full complexity measure of the quantum transform. We consider the particular set of permutation matrices arising in quantum wavelet transforms and develop efficient quantum circuits that implement them. This allows us to design efficient, complete quantum circuits for the quantum wavelet transform.

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Abstract.To date most of the microelectromechanical system (MEMS) devices have been based on silicon. This is due to the technological know-how accumulated on the manipulation, machining and manufacturing of silicon. However, only very few devices involve moving parts. This is because of the rapid wear arising from high friction in these silicon-based systems. Recent tribometric experiments carried out by Gardos on silicon and polycrystalline diamond (PCD) show that this rapid wear is caused by a variety of factors, related both to surface chemistry and cohesive energy density of these likely MEMS bearing materials. In particular, the 1.8-times stronger C-C bond in diamond as opposed to the Si-Si bond in the bulk translates into a more than 10 4 -times difference in wear rates, even though the difference in flexural strength is only 20-times, in hardness 10-times and in fracture toughness 5-times. It also has been shown that the wear rates of silicon and PCD are controlled by high-friction-induced surface cracking, and the friction is controlled by the number of dangling, reconstructed or adsorbate-passivated surface bonds. Therefore, theoretical and tribological characterization of Si and PCD surfaces is essential prior to device fabrication to assure reliable MEMS operation under various atmospheric environments, especially at elevated temperatures.As a part of the rational design and manufacturing of MEMS devices containing moving mechanical assemblies (MEMS-MMA) and especially nanoelectromechanical devices (NEMS), theory and simulation can play an important role. Predicting system properties such as friction and wear, and materials properties such as thermal conductivity is of critical importance for materials and components to be used in MEMS-MMAs. In this paper, we present theoretical studies of friction and wear processes on diamond surfaces using a steady state molecular dynamics method. We studied the atomic friction of the diamond-(100) surface using an extended bond-order-dependent potential for hydrocarbon systems. Unlike traditional empirical potentials, bond order potentials can simulate bond breaking and formation processes. Therefore, it is a natural choice to study surface dynamics under friction and wear. In order to calculate the material properties correctly, we have established a consistent approach to incorporate non-bond interactions into the bond order potentials. We have also developed an easy-to-use software to evaluate the atomic-level friction coefficient for an arbitrary system, and interfaced it into a third-party graphical software.

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