Quantum walks on Cayley graphs

We address the problem of the construction of quantum walks on Cayley graphs. Our main motivation is the relationship between quantum algorithms and quantum walks. In particular, we discuss the choice of the dimension of the local Hilbert space and consider various classes of graphs on which the structure of quantum walks may differ. We completely characterise quantum walks on free groups and present partial results on more general cases. Some examples are given, including a family of quantum walks on the hypercube involving a Clifford Algebra.Comment: J. Phys. A (accepted for publication

Linking adult second language learning and diachronic change:a cautionary note

It has been suggested that the morphological complexity of a language is negatively correlated with the size of its population of speakers. This relationship may be driven by the proportion of non-native speakers, among other things, and reflects adaptations to learning constraints imposed by adult language learners. Here we sound a note of caution with respect to these claims by arguing that (a) morphological complexity is defined in somewhat contradictory ways and hence not straightforward to measure, and (b) there is insufficient evidence to suggest that children’s cognitive limitations support mechanisms beneficial for learning of complex morphology relative to adults. We suggest that considering the informational value of morphological cues may be a better way to capture learnability of morphology. To settle the issue of how age related constraints on learning might impact language change, more cross-linguistic studies comparing learning trajectories of different second languages and laboratory experiments examining language transmission in children and adults are needed

More is more in language learning:reconsidering the less-is-more hypothesis

The Less-is-More hypothesis was proposed to explain age-of-acquisition effects in first language (L1) acquisition and second language (L2) attainment. We scrutinize different renditions of the hypothesis by examining how learning outcomes are affected by (1) limited cognitive capacity, (2) reduced interference resulting from less prior knowledge, and (3) simplified language input. While there is little-to-no evidence of benefits of limited cognitive capacity, there is ample support for a More-is-More account linking enhanced capacity with better L1- and L2-learning outcomes, and reduced capacity with childhood language disorders. Instead, reduced prior knowledge (relative to adults) may afford children with greater flexibility in inductive inference; this contradicts the idea that children benefit from a more constrained hypothesis space. Finally, studies of childdirected speech (CDS) confirm benefits from less complex input at early stages, but also emphasize how greater lexical and syntactic complexity of the input confers benefits in L1-attainment

Separable states are more disordered globally than locally

A remarkable feature of quantum entanglement is that an entangled state of two parties, Alice (A) and Bob (B), may be more disordered locally than globally. That is, S(A) > S(A,B), where S(.) is the von Neumann entropy. It is known that satisfaction of this inequality implies that a state is non-separable. In this paper we prove the stronger result that for separable states the vector of eigenvalues of the density matrix of system AB is majorized by the vector of eigenvalues of the density matrix of system A alone. This gives a strong sense in which a separable state is more disordered globally than locally and a new necessary condition for separability of bipartite states in arbitrary dimensions. We also investigate the extent to which these conditions are sufficient to characterize separability, exhibiting examples that show separability cannot be characterized solely in terms of the local and global spectra of a state. We apply our conditions to give a simple proof that non-separable states exist sufficiently close to the completely mixed state of $n$ qudits.Comment: 4 page

How are exemplar representations transformed by encoding, retrieval, and explicit knowledge? A commentary on Ambridge (2020)

The radical exemplar model resonates with work on perceptual classification and categorization highlighting the role of exemplars in memory representations. Further development of the model requires acknowledgment of both the fleeting and fragile nature of perceptual representations and the gist-based, good-enough quality of long-term memory representations. Retrieval operations potentially serve as a mechanism for abstraction as representations of exemplars are distorted through reconstructive processes. As a framework applicable to both first and second language acquisition, the model needs to account for how explicit knowledge arises and its role in filtering input via selective attention

Fault-Tolerant Quantum Computation via Exchange interactions

Quantum computation can be performed by encoding logical qubits into the states of two or more physical qubits, and controlling a single effective exchange interaction and possibly a global magnetic field. This "encoded universality" paradigm offers potential simplifications in quantum computer design since it does away with the need to perform single-qubit rotations. Here we show that encoded universality schemes can be combined with quantum error correction. In particular, we show explicitly how to perform fault-tolerant leakage correction, thus overcoming the main obstacle to fault-tolerant encoded universality.Comment: 5 pages, including 1 figur

Hardness of approximation for quantum problems

The polynomial hierarchy plays a central role in classical complexity theory. Here, we define a quantum generalization of the polynomial hierarchy, and initiate its study. We show that not only are there natural complete problems for the second level of this quantum hierarchy, but that these problems are in fact hard to approximate. Using these techniques, we also obtain hardness of approximation for the class QCMA. Our approach is based on the use of dispersers, and is inspired by the classical results of Umans regarding hardness of approximation for the second level of the classical polynomial hierarchy [Umans, FOCS 1999]. The problems for which we prove hardness of approximation for include, among others, a quantum version of the Succinct Set Cover problem, and a variant of the local Hamiltonian problem with hybrid classical-quantum ground states.Comment: 21 pages, 1 figure, extended abstract appeared in Proceedings of the 39th International Colloquium on Automata, Languages and Programming (ICALP), pages 387-398, Springer, 201

Distributional effects and individual differences in L2 morphology learning

Second language (L2) learning outcomes may depend on the structure of the input and learners’ cognitive abilities. This study tested whether less predictable input might facilitate learning and generalization of L2 morphology while evaluating contributions of statistical learning ability, nonverbal intelligence, phonological short-term memory, and verbal working memory. Over three sessions, 54 adults were exposed to a Russian case-marking paradigm with a balanced or skewed item distribution in the input. Whereas statistical learning ability and nonverbal intelligence predicted learning of trained items, only nonverbal intelligence also predicted generalization of case-marking inflections to new vocabulary. Neither measure of temporary storage capacity predicted learning. Balanced, less predictable input was associated with higher accuracy in generalization but only in the initial test session. These results suggest that individual differences in pattern extraction play a more sustained role in L2 acquisition than instructional manipulations that vary the predictability of lexical items in the input

Quasi-regular sequences and optimal schedules for security games

We study security games in which a defender commits to a mixed strategy for protecting a finite set of targets of different values. An attacker, knowing the defender's strategy, chooses which target to attack and for how long. If the attacker spends time $t$ at a target $i$ of value $\alpha_i$, and if he leaves before the defender visits the target, his utility is $t \cdot \alpha_i$; if the defender visits before he leaves, his utility is 0. The defender's goal is to minimize the attacker's utility. The defender's strategy consists of a schedule for visiting the targets; it takes her unit time to switch between targets. Such games are a simplified model of a number of real-world scenarios such as protecting computer networks from intruders, crops from thieves, etc. We show that optimal defender play for this continuous time security games reduces to the solution of a combinatorial question regarding the existence of infinite sequences over a finite alphabet, with the following properties for each symbol $i$: (1) $i$ constitutes a prescribed fraction $p_i$ of the sequence. (2) The occurrences of $i$ are spread apart close to evenly, in that the ratio of the longest to shortest interval between consecutive occurrences is bounded by a parameter $K$. We call such sequences $K$-quasi-regular. We show that, surprisingly, $2$-quasi-regular sequences suffice for optimal defender play. What is more, even randomized $2$-quasi-regular sequences suffice for optimality. We show that such sequences always exist, and can be calculated efficiently. The question of the least $K$ for which deterministic $K$-quasi-regular sequences exist is fascinating. Using an ergodic theoretical approach, we show that deterministic $3$-quasi-regular sequences always exist. For $2 \leq K < 3$ we do not know whether deterministic $K$-quasi-regular sequences always exist.Comment: to appear in Proc. of SODA 201

Acquisition of gender agreement in Lithuanian:exploring the effect of diminutive usage in an elicited production task

This study examines Lithuanian children's acquisition of gender agreement using an elicited production task. Lithuanian is a richly inflected Baltic language, with two genders and seven cases. Younger (N=24, mean 3;1, 2;5–3;8) and older (N=24, mean 6;3, 5;6–6;9) children were shown pictures of animals and asked to describe them after hearing the animal's name. Animal names differed with respect to familiarity (novel vs. familiar), derivational status (diminutive vs. simplex) and gender (masculine vs. feminine). Analyses of gender-agreement errors based on adjective and pronoun usage indicated that younger children made more errors than older children, with errors more prevalent for novel animal names. For novel animals, and for feminine nouns, children produced fewer errors with nouns introduced in diminutive form. These results complement findings from several Slavic languages (Russian, Serbian and Polish) that diminutives constitute a salient cluster of word forms that may provide an entry point for the child's acquisition of noun morphology