Quantum Certificate Complexity

Given a Boolean function f, we study two natural generalizations of the certificate complexity C(f): the randomized certificate complexity RC(f) and the quantum certificate complexity QC(f). Using Ambainis' adversary method, we exactly characterize QC(f) as the square root of RC(f). We then use this result to prove the new relation R0(f) = O(Q2(f)^2 Q0(f) log n) for total f, where R0, Q2, and Q0 are zero-error randomized, bounded-error quantum, and zero-error quantum query complexities respectively. Finally we give asymptotic gaps between the measures, including a total f for which C(f) is superquadratic in QC(f), and a symmetric partial f for which QC(f) = O(1) yet Q2(f) = Omega(n/log n).Comment: 9 page

A Metaheuristic Adaptive Cubature Based Algorithm to Find Bayesian Optimal Designs for Nonlinear Models

Finding Bayesian optimal designs for nonlinear models is a difficult task because the optimality criteriontypically requires us to evaluate complex integrals before we perform a constrained optimization. Wepropose a hybridized method where we combine an adaptive multidimensional integration algorithm anda metaheuristic algorithm called imperialist competitive algorithm to find Bayesian optimal designs. Weapply our numerical method to a few challenging design problems to demonstrate its efficiency. Theyinclude finding D-optimal designs for an item response model commonly used in education, Bayesianoptimal designs for survivalmodels, and Bayesian optimal designs for a four-parameter sigmoid Emax doseresponse model. Supplementary materials for this article are available online and they contain an R packagefor implementing the proposed algorithm and codes for reproducing all the results in this paper

Improved Quantum Query Upper Bounds Based on Classical Decision Trees

We consider the following question in query complexity: Given a classical query algorithm in the form of a decision tree, when does there exist a quantum query algorithm with a speed-up (i.e., that makes fewer queries) over the classical one? We provide a general construction based on the structure of the underlying decision tree, and prove that this can give us an up-to-quadratic quantum speed-up in the number of queries. In particular, our results give a bounded-error quantum query algorithm of cost O(?s) to compute a Boolean function (more generally, a relation) that can be computed by a classical (even randomized) decision tree of size s. This recovers an O(?n) algorithm for the Search problem, for example. Lin and Lin [Theory of Computing\u2716] and Beigi and Taghavi [Quantum\u2720] showed results of a similar flavor. Their upper bounds are in terms of a quantity which we call the "guessing complexity" of a decision tree. We identify that the guessing complexity of a decision tree equals its rank, a notion introduced by Ehrenfeucht and Haussler [Information and Computation\u2789] in the context of learning theory. This answers a question posed by Lin and Lin, who asked whether the guessing complexity of a decision tree is related to any measure studied in classical complexity theory. We also show a polynomial separation between rank and its natural randomized analog for the complete binary AND-OR tree. Beigi and Taghavi constructed span programs and dual adversary solutions for Boolean functions given classical decision trees computing them and an assignment of non-negative weights to edges of the tree. We explore the effect of changing these weights on the resulting span program complexity and objective value of the dual adversary bound, and capture the best possible weighting scheme by an optimization program. We exhibit a solution to this program and argue its optimality from first principles. We also exhibit decision trees for which our bounds are strictly stronger than those of Lin and Lin, and Beigi and Taghavi. This answers a question of Beigi and Taghavi, who asked whether different weighting schemes in their construction could yield better upper bounds

Seeing into Darkness: Scotopic Visual Recognition

Images are formed by counting how many photons traveling from a given set of directions hit an image sensor during a given time interval. When photons are few and far in between, the concept of `image' breaks down and it is best to consider directly the flow of photons. Computer vision in this regime, which we call `scotopic', is radically different from the classical image-based paradigm in that visual computations (classification, control, search) have to take place while the stream of photons is captured and decisions may be taken as soon as enough information is available. The scotopic regime is important for biomedical imaging, security, astronomy and many other fields. Here we develop a framework that allows a machine to classify objects with as few photons as possible, while maintaining the error rate below an acceptable threshold. A dynamic and asymptotically optimal speed-accuracy tradeoff is a key feature of this framework. We propose and study an algorithm to optimize the tradeoff of a convolutional network directly from lowlight images and evaluate on simulated images from standard datasets. Surprisingly, scotopic systems can achieve comparable classification performance as traditional vision systems while using less than 0.1% of the photons in a conventional image. In addition, we demonstrate that our algorithms work even when the illuminance of the environment is unknown and varying. Last, we outline a spiking neural network coupled with photon-counting sensors as a power-efficient hardware realization of scotopic algorithms.Comment: 23 pages, 6 figure

Fault-ignorant Quantum Search

We investigate the problem of quantum searching on a noisy quantum computer. Taking a 'fault-ignorant' approach, we analyze quantum algorithms that solve the task for various different noise strengths, which are possibly unknown beforehand. We prove lower bounds on the runtime of such algorithms and thereby find that the quadratic speedup is necessarily lost (in our noise models). However, for low but constant noise levels the algorithms we provide (based on Grover's algorithm) still outperform the best noiseless classical search algorithm.Comment: v1: 15+8 pages, 4 figures; v2: 19+8 pages, 4 figures, published version (Introduction section significantly expanded, presentation clarified, results and order unchanged

A Nearly Optimal Lower Bound on the Approximate Degree of AC0^0

The approximate degree of a Boolean function $f \colon \{-1, 1\}^n \rightarrow \{-1, 1\}$ is the least degree of a real polynomial that approximates $f$ pointwise to error at most $1/3$. We introduce a generic method for increasing the approximate degree of a given function, while preserving its computability by constant-depth circuits. Specifically, we show how to transform any Boolean function $f$ with approximate degree $d$ into a function $F$ on $O(n \cdot \operatorname{polylog}(n))$ variables with approximate degree at least $D = \Omega(n^{1/3} \cdot d^{2/3})$. In particular, if $d= n^{1-\Omega(1)}$, then $D$ is polynomially larger than $d$. Moreover, if $f$ is computed by a polynomial-size Boolean circuit of constant depth, then so is $F$. By recursively applying our transformation, for any constant $\delta > 0$ we exhibit an AC$^0$ function of approximate degree $\Omega(n^{1-\delta})$. This improves over the best previous lower bound of $\Omega(n^{2/3})$ due to Aaronson and Shi (J. ACM 2004), and nearly matches the trivial upper bound of $n$ that holds for any function. Our lower bounds also apply to (quasipolynomial-size) DNFs of polylogarithmic width. We describe several applications of these results. We give: * For any constant $\delta > 0$, an $\Omega(n^{1-\delta})$ lower bound on the quantum communication complexity of a function in AC$^0$. * A Boolean function $f$ with approximate degree at least $C(f)^{2-o(1)}$, where $C(f)$ is the certificate complexity of $f$. This separation is optimal up to the $o(1)$ term in the exponent. * Improved secret sharing schemes with reconstruction procedures in AC$^0$.Comment: 40 pages, 1 figur