Pebbling, Entropy and Branching Program Size Lower Bounds

We contribute to the program of proving lower bounds on the size of branching programs solving the Tree Evaluation Problem introduced by Cook et. al. (2012). Proving a super-polynomial lower bound for the size of nondeterministic thrifty branching programs (NTBP) would separate $NL$ from $P$ for thrifty models solving the tree evaluation problem. First, we show that {\em Read-Once NTBPs} are equivalent to whole black-white pebbling algorithms thus showing a tight lower bound (ignoring polynomial factors) for this model. We then introduce a weaker restriction of NTBPs called {\em Bitwise Independence}. The best known NTBPs (of size $O(k^{h/2+1})$) for the tree evaluation problem given by Cook et. al. (2012) are Bitwise Independent. As our main result, we show that any Bitwise Independent NTBP solving $TEP_{2}^{h}(k)$ must have at least $\frac{1}{2}k^{h/2}$ states. Prior to this work, lower bounds were known for NTBPs only for fixed heights $h=2,3,4$ (See Cook et. al. (2012)). We prove our results by associating a fractional black-white pebbling strategy with any bitwise independent NTBP solving the Tree Evaluation Problem. Such a connection was not known previously even for fixed heights. Our main technique is the entropy method introduced by Jukna and Z{\'a}k (2001) originally in the context of proving lower bounds for read-once branching programs. We also show that the previous lower bounds given by Cook et. al. (2012) for deterministic branching programs for Tree Evaluation Problem can be obtained using this approach. Using this method, we also show tight lower bounds for any $k$-way deterministic branching program solving Tree Evaluation Problem when the instances are restricted to have the same group operation in all internal nodes.Comment: 25 Pages, Manuscript submitted to Journal in June 2013 This version includes a proof for tight size bounds for (syntactic) read-once NTBPs. The proof is in the same spirit as the proof for size bounds for bitwise independent NTBPs present in the earlier version of the paper and is included in the journal version of the paper submitted in June 201

Tunneling transport in NSN junctions made of Majorana nanowires across the topological quantum phase transition

We theoretically consider transport properties of a normal metal (N)- superconducting semiconductor nanowire (S)-normal metal (N) structure (NSN) in the context of the possible existence of Majorana bound states in disordered semiconductor-superconductor hybrid systems in the presence of spin-orbit coupling and Zeeman splitting induced by an external magnetic field. We study in details the transport signatures of the topological quantum phase transition as well as the existence of the Majorana bound states in the electrical transport properties of the NSN structure. Our theory includes the realistic nonperturbative effects of disorder, which is detrimental to the topological phase (eventually suppressing the superconducting gap completely), and the effects of the tunneling barriers (or the transparency at the tunneling NS contacts), which affect (and suppress) the zero bias conductance peak associated with the zero energy Majorana bound states. We show that in the presence of generic disorder and barrier transparency the interpretation of the zero bias peak as being associated with the Majorana bound state is problematic since the nonlocal correlations between the two NS contacts at two ends may not manifest themselves in the tunneling conductance through the whole NSN structure. We establish that a simple modification of the standard transport measurements using conductance differences (rather than the conductance itself as in a single NS junction) as the measured quantity can allow direct observation of the nonlocal correlations inherent in the Majorana bound states and enables the mapping out of the topological phase diagram (even in the presence of considerable disorder) by precisely detecting the topological quantum phase transition point.Comment: 34 pages, 7 figures, 1 table. New version with minor modifications and more physical discussion

Studies on gluon evolution and geometrical scaling in kinematic constrained unitarized BFKL equation: application to high-precision HERA DIS data

We suggest a modified form of a unitarized BFKL equation imposing the so-called kinematic constraint on the gluon evolution in multi-Regge kinematics. The underlying nonlinear effects on the gluon evolution are investigated by solving the unitarized BFKL equation analytically. We obtain an equation of the critical boundary between dilute and dense partonic system, following a new differential geometric approach and sketch a phenomenological insight on geometrical scaling. Later we illustrate the phenomenological implication of our solution for unintegrated gluon distribution $f(x,k_T^2)$ towards exploring high precision HERA DIS data by theoretical prediction of proton structure functions ($F_2$ and $F_L$) as well as double differential reduced cross section $(\sigma_r)$. The validity of our theory in the low $Q^2$ transition region is established by studying virtual photon-proton cross section in light of HERA data

Inverse Design of Perfectly Transmitting Eigenchannels in Scattering Media

Light-matter interactions inside turbid medium can be controlled by tailoring the spatial distribution of energy density throughout the system. Wavefront shaping allows selective coupling of incident light to different transmission eigenchannels, producing dramatically different spatial intensity profiles. In contrast to the density of transmission eigenvalues that is dictated by the universal bimodal distribution, the spatial structures of the eigenchannels are not universal and depend on the confinement geometry of the system. Here, we develop and verify a model for the transmission eigenchannel with the corresponding eigenvalue close to unity. By projecting the original problem of two-dimensional diffusion in a homogeneous scattering medium onto a one-dimensional inhomogeneous diffusion, we obtain an analytical expression relating the intensity profile to the shape of the confining waveguide. Inverting this relationship enables the inverse design of the waveguide shape to achieve the desired energy distribution for the perfectly transmitting eigenchannel. Our approach also allows to predict the intensity profile of such channel in a disordered slab with open boundaries, pointing to the possibility of controllable delivery of light to different depths with local illumination.Comment: 9 pages, 6 figure