Mathematics

We study the model of quantum walks on cycles enriched by the addition of 1-step memory. We provide a formula for the probability distribution and the time-averaged limiting probability distribution of the introduced quantum walk. Using the obtained results, we discuss the properties of the introduced model and the difference in comparison to the memoryless model.

We study the model of quantum walks on cycles enriched by the addition of 1-step memory. We provide a formula for the probability distribution and the time-averaged limiting probability distribution of the introduced quantum walk. Using the obtained results, we discuss the properties of the introduced model and the difference in comparison to the memoryless model.

Quantum walks with memory (QWM) are a type of modified quantum walks that record the walker's latest path. As we know, only two kinds of QWM have been presented up to now. It is desired to design more QWM for research, so that we can explore the potential of QWM. In this work, by presenting the one-to-one correspondence between QWM on a regular graph and quantum walks without memory (QWoM) on a line digraph of the regular graph, we construct a generic model of QWM on regular graphs. This construction gives a general scheme for building all possible standard QWM on regular graphs and makes it possible to study properties of different kinds of QWM. Here, by taking the simplest example, which is QWM with one memory on the line, we analyze some properties of QWM, such as variance, occupancy rate, and localization.

It is known that quantum resources can allow us to achieve a family of equilibria that can have sometimes a better social welfare, while guaranteeing privacy. We use graph games to propose a way to build non-cooperative games from graph states, and we show how to achieve an unlimited improvement with quantum advice compared to classical advice.

In this work, we show the importance of considering a city's shape, as much as its population density figures, in urban transport planning. We consider in particular cities that are circular (the most common shape) compared to those that are rectangular: For the latter case we show greater utility for a single line light rail/tram system. We introduce the new concepts of Infeasible Regions and Infeasibility Factors, and show how to calculate them numerically and (in some cases) analytically. A particular case study is presented for Galway City.

In this paper we illustrate how non-stochastic (max,+) techniques can be used to describe partial synchronization in a Discrete Event Dynamical System. Our work uses results from the spectral theory of dioids and analyses (max,+) equations describing various synchronization rules in a simple network. The network in question is a transport network consisting of two routes joined at a single point, and our Discrete Events are the departure times of transport units along these routes. We calculate the maximum frequency of circulation of these units as a function of the synchronization parameter. These functions allow us further to determine the waiting times on various routes, and here we find critical parameters (dependent on the fixed travel times on each route) which dictate the overall behavoiur. We give explicit equations for these parameters and state the rules which enable optimal performance in the network (corresponding to minimum waiting time).

We investigate the quantum versions of a one-dimensional random walk, whose corresponding Markov Chain is of order 2. This corresponds to the walk having a memory of up to two previous steps. We derive the amplitudes and probabilities for these walks, and point out how they differ from both classical random walks, and quantum walks without memory.

In this article we compute analytically the number of Nash Equilibria (NE) for a two-choice game played on a (circular) ladder graph with 2n players. We consider a set of games with generic payoff parameters, with the only requirement that a NE occurs if the players choose opposite strategies (anti-coordination game). The results show that for both, the ladder and circular ladder, the number of NE grows exponentially with (half) the number of players n, as NNE(2n)∼C(φ), where φ = 1.618.. is the golden ratio and Ccirc >Cladder. In addition, the value of the scaling factor Cladder depends on the value of the payoff parameters. However, that is no longer true for the circular ladder (3-degree graph), that is Ccirc is constant, which might suggest that the topology of the graph indeed plays an important role for setting the number of NE.

In this paper, we numerically study quantum walks on two kinds of two-dimensional graphs: cylindrical strip and Mobius strip. The two kinds of graphs are typical two-dimensional topological graph. We study the crossing property of quantum walks on these two models. Also, we study its dependence on the initial state, size of the model. At the same time, we compare the quantum walk and classical walk on these two models to discuss the difference of quantum walk and classical walk.

We investigate the quantum versions of a one-dimensional random walk, whose corresponding Markov Chain is of order 2. This corresponds to the walk having a memory of up to two previous steps. We derive the amplitudes and probabilities for these walks, and point out how they differ from both classical random walks, and quantum walks without memory.

One of the proposals for the exploitation of two-dimensional quantum walks has been the efficient generation of entanglement. Unfortunately, the technological effort required for the experimental realization of standard two-dimensional quantum walks is significantly demanding. In this respect, an alternative scheme with less challenging conditions has been recently studied, particularly in terms of spatial-entanglement generation [C. Di Franco, M. Mc Gettrick, and Th. Busch, Phys. Rev. Lett. 106, 080502 (2011)]. Here, we extend the investigation to a scenario where a measurement is performed on the coin degree of freedom after the evolution, allowing a further comparison with the standard two-dimensional Grover walk.

The non-localized case of the spatial density probability of the two-dimensional Grover walk can be obtained using only a two-dimensional coin space and a quantum walk in alternate directions. This significantly reduces the resources necessary for its feasible experimental realization. We present a formal proof of this correspondence and analyze the behavior of the coin-position entanglement as well as the x-y spatial entanglement in our scheme with respect to the Grover one. Our scheme allows us to entangle the two orthogonal directions of the walk more efficiently.

We have recently proposed a two-dimensional quantum walk where the requirement of a higher dimensionality of the coin space is substituted with the alternance of the directions in which the walker can move [C. Di Franco, M. Mc Gettrick, and Th. Busch, Phys. Rev. Lett. 106, 080502 (2011)]. For a particular initial state of the coin, this walk is able to perfectly reproduce the spatial probability distribution of the non-localized case of the Grover walk. Here, we present a more detailed proof of this equivalence. We also extend the analysis to other initial states, in order to provide a more complete picture of our walk. We show that this scheme outperforms the Grover walk in the generation of x-y spatial entanglement for any initial condition, with the maximum entanglement obtained in the case of the particular aforementioned state. Finally, the equivalence is generalized to wider classes of quantum walks and a limit theorem for the alternate walk in this context is presented.

Lazy quantum walks were presented by Andrew M. Childs to prove that the continuous-time quantum walk is a limit of the discrete-time quantum walk [Commun.Math.Phys.294,581-603(2010)]. In this paper, we discuss properties of lazy quantum walks. Our analysis shows that lazy quantum walks have O(t n) order of the n-th moment of the corresponding probability distribution, which is the same as that for normal quantum walks. Also, the lazy quantum walk with DFT (Discrete Fourier Transform) coin operator has a similar probability distribution concentrated interval to that of the normal Hadamard quantum walk. Most importantly, we introduce the concepts of occupancy number and occupancy rate to measure the extent to which the walk has a (relatively) high probability at every position in its range. We conclude that lazy quantum walks have a higher occupancy rate than other walks such as normal quantum walks, classical walks and lazy classical walks.

Quantum walks are a kind of basic quantum computation model. Quantum walks with memory(QWM) are types of modified quantum walks that record the walker's latest path. In this paper we present QWM-P, a kind of QWM whose evolution depends on the parity of memory. QWM-P has an identity coin shift function which helps in analyzing and designing algorithms. Then we discuss some properties of QWM-P. We find that the parity of memory length affects the appearance of QWM-P.

There is a well-known equivalence between the homotopy types of connected CW-spaces X with π n X=0 for n = 1, 2 and the quasi-isomorphism classes of crossed modules ∂ : M → P [11]. When the homotopy groups π 1 X and π 2 X are finite one can represent the homotopy type of X by a crossed module in which M and P are finite groups. We define the order of such a crossed module to be |∂| = |M | × |P |, and the order of a quasi-isomorphism class of crossed modules to be the least order of any crossed module in the class. We then define the order of a homotopy 2-type X to be the order of the corresponding quasi-isomorphism class of crossed modules. In this paper we describe an implemented computer function that inputs a finite crossed module of reasonably small order and returns a quasi-isomorphic crossed module of least order. The function is used to enumerate all homotopy 2-types of order m ≤ 127, m = 32, 64, 81, 96. Underlying the function is a catalogue of all isomorphism classes of crossed modules of order m ≤ 255.