Weak limits for quantum random walks

We formulate and prove a general weak limit theorem for quantum random walks in one and more dimensions. With $X_n$ denoting position at time $n$, we show that $X_n/n$ converges weakly as $n \to \infty$ to a certain distribution which is absolutely continuous and of bounded support. The proof is rigorous and makes use of Fourier transform methods. This approach simplifies and extends certain preceding derivations valid in one dimension that make use of combinatorial and path integral methods

Node Sampling using Random Centrifugal Walks

Sampling a network with a given probability distribution has been identified as a useful operation. In this paper we propose distributed algorithms for sampling networks, so that nodes are selected by a special node, called the \emph{source}, with a given probability distribution. All these algorithms are based on a new class of random walks, that we call Random Centrifugal Walks (RCW). A RCW is a random walk that starts at the source and always moves away from it. Firstly, an algorithm to sample any connected network using RCW is proposed. The algorithm assumes that each node has a weight, so that the sampling process must select a node with a probability proportional to its weight. This algorithm requires a preprocessing phase before the sampling of nodes. In particular, a minimum diameter spanning tree (MDST) is created in the network, and then nodes' weights are efficiently aggregated using the tree. The good news are that the preprocessing is done only once, regardless of the number of sources and the number of samples taken from the network. After that, every sample is done with a RCW whose length is bounded by the network diameter. Secondly, RCW algorithms that do not require preprocessing are proposed for grids and networks with regular concentric connectivity, for the case when the probability of selecting a node is a function of its distance to the source. The key features of the RCW algorithms (unlike previous Markovian approaches) are that (1) they do not need to warm-up (stabilize), (2) the sampling always finishes in a number of hops bounded by the network diameter, and (3) it selects a node with the exact probability distribution

Predicting match outcome in professional Dutch football using tactical performance metrics computed from position tracking data

Quality as well as quantity of tracking data have rapidly increased over the recent years, and multiple leagues have programs for league-wide collection of tracking data. Tracking data enables in-depth performance analysis, especially with regard to tactics. This already resulted in the development of several Key Performance Indicators (KPI’s) related to scoring opportunities, outplaying defenders, numerical balance and territorial advantage. Although some of these KPI’s have gained popularity in the analytics community, little research has been conducted to support the link with performance. Therefore, we aim to study the relationship between match outcome and tactical KPI’s derived from tracking data. Our dataset contains tracking data of all players and the ball, and match outcome, for 118 Dutch premier league matches. Using tracking data, we identified 72.989 passes. For every pass-reception window we computed KPI’s related to numerical superiority, outplayed defenders, territorial gains and scoring opportunities using position data. This individual data was then aggregated over a full match. We then split the dataset in a train and test set, and predicted match outcome using different combinations of features in a logistic regression model. KPI’s related to a combination of off-the-ball features seemed to be the best predictor of match outcome (accuracy of 64.0% and a log loss of 0.67), followed by KPI’s related to the creation of scoring opportunities (accuracy of 58% and a log loss of 0.69). This indicates that although most (commercially) available KPI’s are based on ball-events, the most important information seems to be in off-the-ball activity. We have demonstrated that tactical KPI’s computed from tracking data are relatively good predictors of match outcome. As off-the-ball activity seems to be the main predictor of match outcome, tracking data seems to provide much more insight than notational analysis

Hitting time for the continuous quantum walk

We define the hitting (or absorbing) time for the case of continuous quantum walks by measuring the walk at random times, according to a Poisson process with measurement rate $\lambda$. From this definition we derive an explicit formula for the hitting time, and explore its dependence on the measurement rate. As the measurement rate goes to either 0 or infinity the hitting time diverges; the first divergence reflects the weakness of the measurement, while the second limit results from the Quantum Zeno effect. Continuous-time quantum walks, like discrete-time quantum walks but unlike classical random walks, can have infinite hitting times. We present several conditions for existence of infinite hitting times, and discuss the connection between infinite hitting times and graph symmetry.Comment: 12 pages, 1figur

Exploring effects of response biases in affect induction procedures

This study examined whether self-reports or ratings of experienced affect, often used as manipulation checks on the efficacy of affect induction procedures (AIPs), reflect genuine changes in affective states rather than response biases arising from demand characteristics or social desirability effects. In a between-participants design, participants were exposed to positive, negative and neutral images with valence-congruent music or sound to induce happy, sad and neutral mood. Half of the participants had to actively appraise each image whereas the other half viewed images passively. We hypothesised that if ratings of affective valence are subject to response biases then they should reflect the target mood in the same way for active appraisal and passive exposure as participants encountered the same affective stimuli in both conditions. We also tested whether the AIP resulted in mood-congruent changes in facial expressions analysed by FaceReader to see whether behavioural indicators corroborate the self-reports. The results showed that while participants’ ratings reflected the induced target valence, the difference between positive and negative AIP was significantly attenuated in the active appraisal condition, suggesting that self-reports of mood experienced after the AIP are not entirely a reflection of response biases. However, there were no effects of the AIP on FaceReader valence scores, in line with theories questioning the existence of cross-culturally and inter-individually universal behavioural indicators of affective states. Efficacy of AIPs is therefore best checked using self-reports

Extending matchgates into universal quantum computation

Matchgates are a family of two-qubit gates associated with noninteracting fermions. They are classically simulatable if acting only on nearest neighbors, but become universal for quantum computation if we relax this restriction or use SWAP gates [Jozsa and Miyake, Proc. R. Soc. A 464, 3089 (2008)]. We generalize this result by proving that any nonmatchgate parity-preserving unitary is capable of extending the computational power of matchgates into universal quantum computation. We identify the single local invariant of parity-preserving unitaries responsible for this, and discuss related results in the context of fermionic systems.Comment: 9 pages, 2 figure

Analysis of a Simple, Multi-Receiver GPS Spoof Detector

GPS spoofing is a hot topic of late; technical discussions vary widely based upon the assumed capabilities and a priori knowledge of the spoofer. For a single GPS receiver, various methods to detect a spoofing event have been proposed in the literature. These range from simple ideas (e.g. monitoring the power levels of the GPS signals) to more complex concepts (e.g. looking for vestigial peaks in the correlator outputs) to the comparison to non- GPS signals (e.g. an IMU). Much of this prior work has been on the conceptual level with limited experimentation; little appears to have been done to analyze the resulting detection performance. The detector of interest here monitors the GPS signals using not one, but two or more receivers with their antennas at known relative positions. The assumption is that during a spoofing event these multiple receivers will receive the same spoofer RF signal in that the satellites’ characteristics (i.e. relative times of arrival) are identical at all of the antennas. With no spoofer present, each antenna would receive a unique RF signal, consistent with its position in space. The concept of the detector, then, is that the presence of spoofing is discernible from the near equivalence of the receivers’ receptions. While one could compare these multiple receptions at the RF level, we compare the position solutions across receivers, declaring a spoofing event if the resulting position solutions are too close to each other as compared to the (known) relative locations of the antennas. The primary advantage of such an approach is that the hypothesis test does not require receiver hardware modification or even access to software GPS methods; a separate processor could easily monitor the positions output from the receivers. In this paper we analyze such a detector from a Neyman-Pearson perspective assuming Gaussian statistics on the position solution data. We consider four cases: (1) two receivers with fixed (known) locations, (2) two receivers with fixed separation and known orientation (but unknown absolute position), (3) two receivers with fixed separation and unknown orientation, and (4) a three receiver example

The power of quantum systems on a line

We study the computational strength of quantum particles (each of finite dimensionality) arranged on a line. First, we prove that it is possible to perform universal adiabatic quantum computation using a one-dimensional quantum system (with 9 states per particle). This might have practical implications for experimentalists interested in constructing an adiabatic quantum computer. Building on the same construction, but with some additional technical effort and 12 states per particle, we show that the problem of approximating the ground state energy of a system composed of a line of quantum particles is QMA-complete; QMA is a quantum analogue of NP. This is in striking contrast to the fact that the analogous classical problem, namely, one-dimensional MAX-2-SAT with nearest neighbor constraints, is in P. The proof of the QMA-completeness result requires an additional idea beyond the usual techniques in the area: Not all illegal configurations can be ruled out by local checks, so instead we rule out such illegal configurations because they would, in the future, evolve into a state which can be seen locally to be illegal. Our construction implies (assuming the quantum Church-Turing thesis and that quantum computers cannot efficiently solve QMA-complete problems) that there are one-dimensional systems which take an exponential time to relax to their ground states at any temperature, making them candidates for being one-dimensional spin glasses.Comment: 21 pages. v2 has numerous corrections and clarifications, and most importantly a new author, merged from arXiv:0705.4067. v3 is the published version, with additional clarifications, publisher's version available at http://www.springerlink.co

Modelling team performance in soccer using tactical features derived from position tracking data

Decision-makers in soccer routinely assess the tactical behaviour of a team and its opponents both during and after the game to optimize performance. Currently, this assessment is typically driven by notational analysis and observation. Therefore, potential high-impact decisions are often made based on limited or even biased information. With the current study, we aimed to quantitatively assess tactical performance by abstracting a set of spatiotemporal features from the general offensive principles of play in soccer using position tracking data, and to train a machine learning classifier to predict match outcome based on these features computed over the full game as well as only parts of the game. Based on the results of these analyses, we describe a proof of concept of a decision support system for coaches and managers. In an analysis of 302 professional Dutch Eredivisie matches, we were able to train a Linear Discriminant Analysis model to predict match outcome with fair to good (74.1%) accuracy with features computed over the full match, and 67.9% accuracy with features computed over only 1/4th of the match. We therefore conclude that using only position tracking data, we can provide valuable feedback to coaches about how their team is executing the various principles of play, and how these principles are contributing to overall performance

Lambda's, V's and optimal cloning with stimulated emission

We show that optimal universal cloning of the polarization state of photons can be achieved via stimulated emission in three-level systems, both of the Lambda and the V type. We establish the equivalence of our systems with coupled harmonic oscillators, which permits us to analyze the structure of the cloning transformations realized. These transformations are shown to be equivalent to the optimal cloning transformations for qubits discovered by Buzek and Hillery, and Gisin and Massar. The down-conversion cloner discovered previously by some of the authors is obtained as a limiting case. We demonstrate an interesting equivalence between systems of Lambda atoms and systems of pairwise entangled V atoms. Finally we discuss the physical differences between our photon cloners and the qubit cloners considered previously and prove that the bounds on the fidelity of the clones derived for qubits also apply in our situation.Comment: 10 page