A complex analogue of Toda's Theorem

Toda \cite{Toda} proved in 1989 that the (discrete) polynomial time hierarchy, $\mathbf{PH}$, is contained in the class \mathbf{P}^{#\mathbf{P}}, namely the class of languages that can be decided by a Turing machine in polynomial time given access to an oracle with the power to compute a function in the counting complexity class #\mathbf{P}. This result, which illustrates the power of counting is considered to be a seminal result in computational complexity theory. An analogous result (with a compactness hypothesis) in the complexity theory over the reals (in the sense of Blum-Shub-Smale real machines \cite{BSS89}) was proved in \cite{BZ09}. Unlike Toda's proof in the discrete case, which relied on sophisticated combinatorial arguments, the proof in \cite{BZ09} is topological in nature in which the properties of the topological join is used in a fundamental way. However, the constructions used in \cite{BZ09} were semi-algebraic -- they used real inequalities in an essential way and as such do not extend to the complex case. In this paper, we extend the techniques developed in \cite{BZ09} to the complex projective case. A key role is played by the complex join of quasi-projective complex varieties. As a consequence we obtain a complex analogue of Toda's theorem. The results contained in this paper, taken together with those contained in \cite{BZ09}, illustrate the central role of the Poincar\'e polynomial in algorithmic algebraic geometry, as well as, in computational complexity theory over the complex and real numbers -- namely, the ability to compute it efficiently enables one to decide in polynomial time all languages in the (compact) polynomial hierarchy over the appropriate field.Comment: 31 pages. Final version to appear in Foundations of Computational Mathematic

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

Multidimensional decision analysis in public investment analysis: theory and practice

Most important investment decisions involve several criteria or dimensions, e.g. a weapon system could be judged by cost, portability, reliability and firepower. Moreover, the values of these dimensions that alternative courses of action will produce is rarely known for certain, because they will occur in the future or because analysis to obtain the information would be too expensive or too time- consuming. These comments apply to public sector investment particularly, because the market does not provide a price that incorporates several dimensions for public goods, and because much public investment is one-off, having rarely been done before, e.g., Medibank

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