Some Notes on Parallel Quantum Computation

We exhibit some simple gadgets useful in designing shallow parallel circuits for quantum algorithms. We prove that any quantum circuit composed entirely of controlled-not gates or of diagonal gates can be parallelized to logarithmic depth, while circuits composed of both cannot. Finally, while we note the Quantum Fourier Transform can be parallelized to linear depth, we exhibit a simple quantum circuit related to it that we believe cannot be parallelized to less than linear depth, and therefore might be used to prove that QNC < QP

Extreme UV QSOs

We present a sample of spectroscopically confirmed QSOs with FUV-NUV color (as measured by GALEX photometry) bluer than canonical QSO templates and than the majority of known QSOs. We analyze their FUV to NIR colors, luminosities and optical spectra. The sample includes a group of 150 objects at low redshift (z $<$ 0.5), and a group of 21 objects with redshift 1.7$<$z$<$2.6. For the low redshift objects, the "blue" FUV-NUV color may be caused by enhanced Ly$\alpha$ emission, since Ly$\alpha$ transits the GALEX FUV band from z=0.1 to z=0.47. Synthetic QSO templates constructed with Ly$\alpha$ up to 3 times stronger than in standard templates match the observed UV colors of our low redshift sample. The H$\alpha$ emission increases, and the optical spectra become bluer, with increasing absolute UV luminosity. The UV-blue QSOs at redshift about 2, where the GALEX bands sample restframe about 450-590A (FUV) and about 590-940A(NUV), are fainter than the average of UV-normal QSOs at similar redshift in NUV, while they have comparable luminosities in other bands. Therefore we speculate that their observed FUV-NUV color may be explained by a combination of steep flux rise towards short wavelengths and dust absorption below the Lyman limit, such as from small grains or crystalline carbon. The ratio of Ly$\alpha$ to CIV could be measured in 10 objects; it is higher (30% on average) than for UV-normal QSOs, and close to the value expected for shock or collisional ionization. FULL VERSION AVAILABLE FROM AUTHOR'S WEB SITE: http://dolomiti.pha.jhu.edu/papers/2009_AJ_Extreme_UV_QSOs.pdfComment: Astronomical Journal, in pres

Predicting Non-linear Cellular Automata Quickly by Decomposing Them into Linear Ones

We show that a wide variety of non-linear cellular automata (CAs) can be decomposed into a quasidirect product of linear ones. These CAs can be predicted by parallel circuits of depth O(log^2 t) using gates with binary inputs, or O(log t) depth if ``sum mod p'' gates with an unbounded number of inputs are allowed. Thus these CAs can be predicted by (idealized) parallel computers much faster than by explicit simulation, even though they are non-linear. This class includes any CA whose rule, when written as an algebra, is a solvable group. We also show that CAs based on nilpotent groups can be predicted in depth O(log t) or O(1) by circuits with binary or ``sum mod p'' gates respectively. We use these techniques to give an efficient algorithm for a CA rule which, like elementary CA rule 18, has diffusing defects that annihilate in pairs. This can be used to predict the motion of defects in rule 18 in O(log^2 t) parallel time

Glassy dynamics and aging in an exactly solvable spin model

We introduce a simple two-dimensional spin model with short-range interactions which shows glassy behavior despite a Hamiltonian which is completely homogeneous and possesses no randomness. We solve exactly for both the static partition function of the model and the distribution of energy barriers, giving us the equilibration time-scales at low temperature. Simulations of instantaneous quenches and of annealing of the model are in good agreement with the analytic calculations. We also measure the two-time spin correlation as a function of waiting time, and show that the model has aging behavior consistent with the distribution of barrier heights. The model appears to have no sharp glass transition. Instead, it falls out of equilibrium at a temperature which decreases logarithmically as a function of the cooling time.Comment: 16 pages, 4 postscript figures, typeset in LaTeX using the RevTeX macro packag

On Computational Power of Quantum Read-Once Branching Programs

In this paper we review our current results concerning the computational power of quantum read-once branching programs. First of all, based on the circuit presentation of quantum branching programs and our variant of quantum fingerprinting technique, we show that any Boolean function with linear polynomial presentation can be computed by a quantum read-once branching program using a relatively small (usually logarithmic in the size of input) number of qubits. Then we show that the described class of Boolean functions is closed under the polynomial projections.Comment: In Proceedings HPC 2010, arXiv:1103.226