Structure of computations in parallel complexity classes

Issued as Annual report, and Final project report, Project no. G-36-67

Pseudorandomness and Average-Case Complexity via Uniform Reductions

Impagliazzo and Wigderson (36th FOCS, 1998) gave the first construction of pseudorandom generators from a uniform complexity assumption on EXP (namely EXP [not equal to] BPP). Unlike results in the nonuniform setting, their result does not provide a continuous trade-off between worst-case hardness and pseudorandomness, nor does it explicitly establish an average-case hardness result. In this paper: 1. We obtain an optimal worst-case to average-case connection for EXP: if EXP is not a subset of BPTIME(t(n)), EXP has problems that cannot be solved on a fraction 1/2 +1/t'(n) of the inputs by BPTIME(t'(n)) algorithms, for t'=t^{\Omega(1)}. 2. We exhibit a PSPACE-complete self-correctible and downward self-reducible problem. This slightly simplifies and strengthens the proof of Impaglaizzo and Wigderson, which used a a #P-complete problem with these properties. 3. We argue that the results of Impagliazzo and Wigderson, and the ones in this paper, cannot be proved via "black-box" uniform reductions.Engineering and Applied Science

The Computational Power of Non-interacting Particles

Shortened abstract: In this thesis, I study two restricted models of quantum computing related to free identical particles. Free fermions correspond to a set of two-qubit gates known as matchgates. Matchgates are classically simulable when acting on nearest neighbors on a path, but universal for quantum computing when acting on distant qubits or when SWAP gates are available. I generalize these results in two ways. First, I show that SWAP is only one in a large family of gates that uplift matchgates to quantum universality. In fact, I show that the set of all matchgates plus any nonmatchgate parity-preserving two-qubit gate is universal, and interpret this fact in terms of local invariants of two-qubit gates. Second, I investigate the power of matchgates in arbitrary connectivity graphs, showing they are universal on any connected graph other than a path or a cycle, and classically simulable on a cycle. I also prove the same dichotomy for the XY interaction. Free bosons give rise to a model known as BosonSampling. BosonSampling consists of (i) preparing a Fock state of n photons, (ii) interfering these photons in an m-mode linear interferometer, and (iii) measuring the output in the Fock basis. Sampling approximately from the resulting distribution should be classically hard, under reasonable complexity assumptions. Here I show that exact BosonSampling remains hard even if the linear-optical circuit has constant depth. I also report several experiments where three-photon interference was observed in integrated interferometers of various sizes, providing some of the first implementations of BosonSampling in this regime. The experiments also focus on the bosonic bunching behavior and on validation of BosonSampling devices. This thesis contains descriptions of the numerical analyses done on the experimental data, omitted from the corresponding publications.Comment: PhD Thesis, defended at Universidade Federal Fluminense on March 2014. Final version, 208 pages. New results in Chapter 5 correspond to arXiv:1106.1863, arXiv:1207.2126, and arXiv:1308.1463. New results in Chapter 6 correspond to arXiv:1212.2783, arXiv:1305.3188, arXiv:1311.1622 and arXiv:1412.678

Symmetry and complexity in propositional reasoning

We establish computational complexity results for a number of simple problem formulations connecting group action and prepositional formulas. The results are discussed in the context of complexity results arising from established work in the area of automated reasoning techniques which exploit symmetry

Solitons from sine waves: Analytical and numerical methods for non-integrable solitary and cnoidal waves

The "FKDV" equation, ut+uux-uxxxxx=0, is used as a testbed for a variety of analytical and numerical methods that can be applied to solitary waves and cnoidal waves of "non-integrable" differential equations, that is to say, to equations which cannot be solved by the inverse scattering transform. The basic tools are (i) Pade approximants formed from power series in the amplitude; (ii) a Newton-Kantorovich/pseudospectral Fourier/continuation numerical method; (iii) singular perturbation theory for two interacting solitons of almost identical phase speed; (iv) bifurcation and branch-switching methods; (v) the imbricate-soliton series. A number of new results for the FKDV equation are obtained including extensive numerical calculations of the spatially periodic solutions with one peak ("cnoidal wave") and two peaks ("bicnoidal wave") per period, an analytical expression for the double-peaked soliton ("bion"), calculation of both the limit and bifurcation points for the bicnoidal wave, and finally the computation of accurate analytical approximations to the cnoidal wave for all amplitudes. More important, all of these analytical and numerical tools are highly effective for this equation in spite of the fact that it cannot be solved by the inverse scattering transform. Work now in progress will apply these methods to non-integrable equations in two space dimensions.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/26057/1/0000131.pd

Conflicting Objectives in Decisions

This book deals with quantitative approaches in making decisions when conflicting objectives are present. This problem is central to many applications of decision analysis, policy analysis, operational research, etc. in a wide range of fields, for example, business, economics, engineering, psychology, and planning. The book surveys different approaches to the same problem area and each approach is discussed in considerable detail so that the coverage of the book is both broad and deep. The problem of conflicting objectives is of paramount importance, both in planned and market economies, and this book represents a cross-cultural mixture of approaches from many countries to the same class of problem

4.Uluslararası Öğrenciler Fen Bilimleri Kongresi Bildiriler Kitabı

Çevrimiçi ( XIII, 495 Sayfa ; 26 cm.)

A circuit-based proof of Toda's theorem

TIB: RN 4052(90651-OR) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekSIGLEDEGerman