Dequantizing read-once quantum formulas

Quantum formulas, defined by Yao [FOCS '93], are the quantum analogs of classical formulas, i.e., classical circuits in which all gates have fanout one. We show that any read-once quantum formula over a gate set that contains all single-qubit gates is equivalent to a read-once classical formula of the same size and depth over an analogous classical gate set. For example, any read-once quantum formula over Toffoli and single-qubit gates is equivalent to a read-once classical formula over Toffoli and NOT gates. We then show that the equivalence does not hold if the read-once restriction is removed. To show the power of quantum formulas without the read-once restriction, we define a new model of computation called the one-qubit model and show that it can compute all boolean functions. This model may also be of independent interest.Comment: 14 pages, 8 figures, to appear in proceedings of TQC 201

On some combinatorial properties of graph states

I graph state sono particolari stati quantistici, rappresentabili tramite grafi indiretti semplici, che giocano un ruolo fondamentale in informatica quantistica, in particolare nell'ambito dei codici a correzione di errore e nel modello di computazione one-way. Lo scopo di questa tesi è studiare alcune proprietà dei graph state attraverso un approccio combinatorio. Innanzitutto si sono analizzate le proprietà di un'invariante dei grafi: il numero di sottografi indotti con numero dispari di archi. Questo numero è stato valutato per famiglie importanti di grafi ed è stato trovato un algoritmo efficiente per calcolarlo. Abbiamo inoltre caratterizzato questo numero rispetto all'azione di local complementation. Alcune proprietà delle funzioni booleane associate ai graph state sono state studiate analizzando la struttura dei grafi di Cayley delle suddette funzioni. Infine sono stati definiti, e ne è stata analizzata la struttura, grafi che permettono di tracciare le trasformazioni di local complementation e switching fra graph state. Un congettura è stata fatta sulla struttura di questi grafi. A margine del lavoro, sono state introdotte alcune estensioni alla definizione classica di graph state. In particolare sono stati definiti gli "edge graph state" e i "3-hypergraph state"

PPT-indistinguishable states via semidefinite programming

We show a simple semidefinite program whose optimal value is equal to the maximum probability of perfectly distinguishing orthogonal maximally entangled states using any PPT measurement (a measurement whose operators are positive under partial transpose). When the states to be distinguished are given by the tensor product of Bell states, the semidefinite program simplifies to a linear program. In [Phys. Rev. Lett. 109, 020506 (2012) -- arXiv:1107.3224v1], Yu, Duan and Ying exhibit a set of 4 maximally entangled states in $C^4 \otimes C^4$, which is distinguishable by any PPT measurement only with probability strictly less than 1. Using semidefinite programming, we show a tight bound of 7/8 on this probability (3/4 for the case of unambiguous PPT measurements). We generalize this result by demonstrating a simple construction of a set of k states in $C^k \otimes C^k$ with the same property, for any k that is a power of 2. Finally, by running numerical experiments, we obtain some non-trivial results about the PPT-distinguishability of certain interesting sets of generalized Bell states in $C^5 \otimes C^5$ and $C^6 \otimes C^6$.Comment: 12 pages. Added section 4. Updated reference

Universal Features in the Genome-level Evolution of Protein Domains

Protein domains are found on genomes with notable statistical distributions, which bear a high degree of similarity. Previous work has shown how these distributions can be accounted for by simple models, where the main ingredients are probabilities of duplication, innovation, and loss of domains. However, no one so far has addressed the issue that these distributions follow definite trends depending on protein-coding genome size only. We present a stochastic duplication/innovation model, falling in the class of so-called Chinese Restaurant Processes, able to explain this feature of the data. Using only two universal parameters, related to a minimal number of domains and to the relative weight of innovation to duplication, the model reproduces two important aspects: (a) the populations of domain classes (the sets, related to homology classes, containing realizations of the same domain in different proteins) follow common power-laws whose cutoff is dictated by genome size, and (b) the number of domain families is universal and markedly sublinear in genome size. An important ingredient of the model is that the innovation probability decreases with genome size. We propose the possibility to interpret this as a global constraint given by the cost of expanding an increasingly complex interactome. Finally, we introduce a variant of the model where the choice of a new domain relates to its occurrence in genomic data, and thus accounts for fold specificity. Both models have general quantitative agreement with data from hundreds of genomes, which indicates the coexistence of the well-known specificity of proteomes with robust self-organizing phenomena related to the basic evolutionary ``moves'' of duplication and innovation

Markov chain approximations to stochastic differential equations by recombination on lattice trees

We revisit the classical problem of approximating a stochastic differential equation by a discrete-time and discrete-space Markov chain. Our construction iterates Caratheodory's theorem over time to match the moments of the increments locally. This allows to construct a Markov chain with a sparse transition matrix where the number of attainable states grows at most polynomially as time increases. Moreover, the MC evolves on a tree whose nodes lie on a "universal lattice" in the sense that an arbitrary number of different SDEs can be approximated on the same tree. The construction is not tailored to specific models, we discuss both the case of uni-variate and multi-variate case SDEs, and provide an implementation and numerical experiments

Grid-free computation of probabilistic safety with Malliavin Calculus

This work concerns continuous-time, continuous-space stochastic dynamical systems described by stochastic differential equations (SDE). It presents a new approach to compute probabilistic safety regions, namely sets of initial conditions of the SDE associated to trajectories that are safe with a probability larger than a given threshold. The approach introduces a functional that is minimised at the border of the probabilistic safety region, then solves an optimisation problem using techniques from Malliavin Calculus, which computes such region. Unlike existing results in the literature, the new approach allows one to compute probabilistic safety regions without gridding the state space of the SDE

Carathéodory sampling for stochastic gradient descent

Many problems require to optimize empirical risk functions over large data sets. Gradient descent methods that calculate the full gradient in every descent step do not scale to such datasets. Various flavours of Stochastic Gradient Descent (SGD) replace the expensive summation that computes the full gradient by approximating it with a small sum over a randomly selected subsample of the data set that in turn suffers from a high variance. We present a different approach that is inspired by classical results of Tchakaloff and Carathéodory about measure reduction. These results allow to replace an empirical measure with another, carefully constructed probability measure that has a much smaller support, but can preserve certain statistics such as the expected gradient. To turn this into scalable algorithms we firstly, adaptively select the descent steps where the measure reduction is carried out; secondly, we combine this with Block Coordinate Descent so that measure reduction can be done very cheaply. This makes the resulting methods scalable to high-dimensional spaces. Finally, we provide an experimental validation and comparison

Current quantization and fractal hierarchy in a driven repulsive lattice gas

Driven lattice gases are widely regarded as the paradigm of collective phenomena out of equilibrium. While such models are usually studied with nearest-neighbor interactions, many empirical driven systems are dominated by slowly decaying interactions such as dipole-dipole and Van der Waals forces. Motivated by this gap, we study the non-equilibrium stationary state of a driven lattice gas with slow-decayed repulsive interactions at zero temperature. By numerical and analytical calculations of the particle current as a function of the density and of the driving field, we identify (i) an abrupt breakdown transition between insulating and conducting states, (ii) current quantization into discrete phases where a finite current flows with infinite differential resistivity, and (iii) a fractal hierarchy of excitations, related to the Farey sequences of number theory. We argue that the origin of these effects is the competition between scales, which also causes the counterintuitive phenomenon that crystalline states can melt by increasing the density

Quantum State Local Distinguishability via Convex Optimization

Entanglement and nonlocality play a fundamental role in quantum computing. To understand the interplay between these phenomena, researchers have considered the model of local operations and classical communication, or LOCC for short, which is a restricted subset of all possible operations that can be performed on a multipartite quantum system. The task of distinguishing states from a set that is known a priori to all parties is one of the most basic problems among those used to test the power of LOCC protocols, and it has direct applications to quantum data hiding, secret sharing and quantum channel capacity. The focus of this thesis is on state distinguishability problems for classes of quantum operations that are more powerful than LOCC, yet more restricted than global operations, namely the classes of separable and positive-partial-transpose (PPT) measurements. We build a framework based on convex optimization (on cone programming, in particular) to study such problems. Compared to previous approaches to the problem, the method described in this thesis provides precise numerical bounds and quantitative analytic results. By combining the duality theory of cone programming with the channel-state duality, we also establish a novel connection between the state distinguishability problem and the study of positive linear maps, which is a topic of independent interest in quantum information theory. We apply our framework to several questions that were left open in previous works regarding the distinguishability of maximally entangled states and unextendable product sets. First, we exhibit small sets of orthogonal maximally entangled states in that are not perfectly distinguishable by LOCC. As a consequence of this, we show a gap between the power of PPT and separable measurements for the task of distinguishing sets consisting only of maximally entangled states. Furthermore, we prove tight bounds on the entanglement cost that is necessary to distinguish any sets of Bell states, thus showing that quantum teleportation is optimal for this task. Finally, we provide an easily checkable characterization of when an unextendable product set is perfectly discriminated by separable measurements, along with the first known example of an unextendable product set that cannot be perfectly discriminated by separable measurements