arXiv subjects

Adam Glos

Publications and source records attributed to Adam Glos.

At least 37 records · Page 2Linked to original sources

arXiv2021

Asymptotic entropy of the Gibbs state of complex networks

In this work we study the entropy of the Gibbs state corresponding to a graph. The Gibbs state is obtained from the Laplacian, normalized Laplacian or adjacency matrices associated with a graph. We calculated the entropy of the Gibbs state for a few classes of graphs and studied their behavior with changing graph order and temperature. We illustrate our analytical results with numerical simulations for Erdős-Rényi, Watts-Strogatz, Barabási-Albert and Chung-Lu graph models and a few real-world graphs. Our results show that the behavior of Gibbs entropy as a function of the temperature differs for a choice of real networks when compared to the random Erdős-Rényi graphs.

math.PR↗

arXiv2020

LP WAN Gateway Location Selection Using Modified K-Dominating Set Algorithm

The LP WAN networks use gateways or base stations to communicate with devices distributed on large distances, up to tens of kilometres. The selection of optimal gateway locations in wireless networks should allow providing the complete coverage for a given set of nodes, taking into account the limitations, such as the number of nodes served per access point or required redundancy. In this paper, we describe the problem of selecting the base stations in a network using the concept of $k$-dominating set. In our model, we include information about the required redundancy and spectral efficiency. We consider the additional requirements on the resulting connections and provide the greedy algorithm for solving the problem. The algorithm is evaluated in randomly generated network topologies and using the coordinates of sample real smart metering networks.

cs.NI↗

arXiv2020

Comment to Spatial Search by Quantum Walk is Optimal for Almost all Graphs

This comment is to correct the proof of optimality of quantum spatial search for Erdős-Rényi graphs presented in `Spatial Search by Quantum Walk is Optimal for Almost all Graphs' (https://doi.org/10.1103/PhysRevLett.116.100501). The authors claim that if $p\geq \frac{\log^{3/2}(n)}{n}$, then the CTQW-based search is optimal for almost all graphs. Below we point the issues found in the main paper, and propose corrections, which in fact improve the result to $p=ω(\log(n)/n)$ in case of transition rate $γ= 1/λ_1$. In the case of the proof for simplified transition rate $1/(np)$ we pointed a possible issue with applying perturbation theory.

quant-ph↗

arXiv2020

Upperbounds on the probability of finding marked connected components using quantum walks

Quantum walk search may exhibit phenomena beyond the intuition from a conventional random walk theory. One of such examples is exceptional configuration phenomenon -- it appears that it may be much harder to find any of two or more marked vertices, that if only one of them is marked. In this paper, we analyze the probability of finding any of marked vertices in such scenarios and prove upper bounds for various sets of marked vertices. We apply the upper bounds to large collection of graphs and show that the quantum search may be slow even when taking real-world networks.

quant-ph↗

arXiv2020

Space-efficient binary optimization for variational computing

In the era of Noisy Intermediate-Scale Quantum (NISQ) computers it is crucial to design quantum algorithms which do not require many qubits or deep circuits. Unfortunately, the most well-known quantum algorithms are too demanding to be run on currently available quantum devices. Moreover, even the state-of-the-art algorithms developed for the NISQ era often suffer from high space complexity requirements for particular problem classes. In this paper, we show that it is possible to greatly reduce the number of qubits needed for the Traveling Salesman Problem (TSP), a paradigmatic optimization task, at the cost of having deeper variational circuits. While the focus is on this particular problem, we claim that the approach can be generalized for other problems where the standard bit-encoding is highly inefficient. Finally, we also propose encoding schemes which smoothly interpolate between the qubit-efficient and the circuit depth-efficient models. All the proposed encodings remain efficient to implement within the Quantum Approximate Optimization Algorithm framework.

quant-ph↗

arXiv2020

Quantum Optimization for the Graph Coloring Problem with Space-Efficient Embedding

Current quantum computing devices have different strengths and weaknesses depending on their architectures. This means that flexible approaches to circuit design are necessary. We address this task by introducing a novel space-efficient quantum optimization algorithm for the graph coloring problem. Our circuits are deeper than the ones of the standard approach. However, the number of required qubits is exponentially reduced in the number of colors. We present extensive numerical simulations demonstrating the performance of our approach. Furthermore, to explore currently available alternatives, we perform a study of random graph coloring on a quantum annealer to test the limiting factors of that approach, too.

quant-ph↗

arXiv2019

Impact of the malicious input data modification on the efficiency of quantum spatial search

In this paper we demonstrate that the efficiency of quantum algorithms can be significantly altered by malicious manipulation of the input data. We exemplify the possibility of attacks on quantum spatial search based on Szegedy walk. We achieve this by proposing a framework suitable for analysing efficiency of attacks on quantum search algorithms. We provide the analysis of proposed attacks for different models of random graphs.

quant-ph↗

arXiv2019

Impact of global and local interaction on quantum spatial search on chimera graph

In the paper, we investigated the influence of local and global interaction on the efficiency of continuous-time quantum spatial search. To do so, we analyzed numerically chimera graph, which is defined as 2D grid with each node replaced by complete bipartite graph. Our investigation provides a numerical evidence that with a large number of local interactions the quantum spatial search is optimal, contrary to the case with limited number of such interactions. The result suggests that relatively large number of local interactions with the marked vertex is necessary for optimal search, which in turn would imply that poorly connected vertices are hard to be found.

quant-ph↗

arXiv2019

Comment on Nahimovs et al. `On the probability of finding marked connected components using quantum walks'

In this comment paper we present two misconceptions found in paper of Nahimovs et al. \emph{On the probability of finding marked connected components using quantum walks}. First, we show that the Theorem 2 (sufficient and necessary condition for a state to be stationary) is incomplete -- it works only if unmarked vertices form a single connected component. Second, we correct derivation of \emph{a} coefficient in the Theorem 3 (lower bound on the probability) and show how to upper bound value of \emph{a}.

quant-ph↗

arXiv2018

Optimal Quantum Walk Search on Kronecker Graphs with Dominant or Fixed Regular Initiators

In network science, graphs obtained by taking the Kronecker or tensor power of the adjacency matrix of an initiator graph are used to construct complex networks. In this paper, we analytically prove sufficient conditions under which such Kronecker graphs can be searched by a continuous-time quantum walk in optimal $Θ(\sqrt{N})$ time. First, if the initiator is regular and its adjacency matrix has a dominant principal eigenvalue, meaning its unique largest eigenvalue asymptotically dominates the other eigenvalues in magnitude, then the Kronecker graphs generated by this initiator can be quantum searched with probability 1 in $π\sqrt{N}/2$ time, asymptotically, and we give the critical jumping rate of the walk that enables this. Second, for any fixed initiator that is regular, non-bipartite, and connected, the Kronecker graphs generated by it are quantum searched in $Θ(\sqrt{N})$ time. This greatly extends the number of Kronecker graphs on which quantum walks are known to optimally search. If the fixed, regular, connected initiator is bipartite, however, then search on its Kronecker powers is not optimal, but is still better than classical computer's $O(N)$ runtime if the initiator has more than two vertices.

quant-ph↗

arXiv2018

QSWalk.jl: Julia package for quantum stochastic walks analysis

The presented paper describes QSWalk.jl package for Julia programming language, developed for the purpose of simulating the evolution of open quantum systems. The package enables the study of quantum procedures developed using stochastic quantum walks on arbitrary directed graphs. We provide a detailed description of the implemented functions, along with a number of usage examples. The package is compared with the existing software offering a similar functionality.

quant-ph↗

arXiv2018

Spectral similarity for Barabási-Albert and Chung-Lu models

In the paper we have analyzed spectral similarity between Barabási-Albert and Chung-lu models. We have shown the similarity of spectral distribution for sufficiently large Barabási-Albert parameter value. Contrary, extreme eigenvalues and principal eigenvector are not similar for those model. We provide applications of obtained results related to the spectral graph theory and efficiency of quantum spatial search

math.SP↗

arXiv2018

Introducing higher order correlations to marginals' subset of multivariate data by means of Archimedean copulas

In this paper, we present the algorithm that alters the subset of marginals of multivariate standard distributed data into such modelled by an Archimedean copula. Proposed algorithm leaves a correlation matrix almost unchanged, but introduces a higher order correlation into a subset of marginals. Our data transformation algorithm can be used to analyse whether particular machine learning algorithm, especially a dimensionality reduction one, utilises higher order correlations or not. We present an exemplary application on two features selection algorithms, mention that features selection is one of the approaches to dimensionality reduction. To measure higher order correlation, we use multivariate higher order cumulants, hence to utilises higher order correlations be to use the Joint Skewness Band Selection (JSBS) algorithm that uses third-order multivariate cumulant. We show the robust performance of the JSBS in contrary to the poor performance of the Maximum Ellipsoid Volume (MEV) algorithm that does not utilise such higher order correlations. With this result, we confirm the potential application of our data generation algorithm to analyse a performance of various dimensionality reduction algorithms.

cs.DS↗

arXiv2018

Vertices cannot be hidden from quantum spatial search for almost all random graphs

In this paper we show that all nodes can be found optimally for almost all random Erdős-Rényi ${\mathcal G}(n,p)$ graphs using continuous-time quantum spatial search procedure. This works for both adjacency and Laplacian matrices, though under different conditions. The first one requires $p=ω(\log^8(n)/n)$, while the seconds requires $p\geq(1+\varepsilon)\log (n)/n$, where $\varepsilon>0$. The proof was made by analyzing the convergence of eigenvectors corresponding to outlying eigenvalues in the $\|\cdot\|_\infty $ norm. At the same time for $p<(1-\varepsilon)\log(n)/n$, the property does not hold for any matrix, due to the connectivity issues. Hence, our derivation concerning Laplacian matrix is tight.

quant-ph↗

arXiv2017

Limit properties of global interaction stochastic quantum walks on directed graphs

The main results of our work is determining the differences between limiting properties in various models of quantum stochastic walks. In particular, we prove that in the case of strongly connected and a class of weakly connected directed graphs, local environment interaction evolution is relaxing, and in the case of undirected graphs, global environment interaction evolution is convergent. For other classes of directed graphs we show, that the character of connectivity large influence on the limiting properties. We also study the limiting properties for the non-moralizing global interaction case. We demonstrate that the digraph observance is recovered in this case.

quant-ph↗

arXiv2017

The role of quantum correlations in Cop and Robber game

We introduce and study quantized versions of Cop and Robber game. We achieve this by using graph-preserving quantum operations, which are the quantum analogues of stochastic operations preserving the graph. We provide the tight bound for the number of operations required to reach the given state. By extending them to the controlled operations, we define a quantum-controlled Cop and Robber game, which expands the classical Cop and Robber game, as well as the classically controlled quantum Cop and Robber game. In contrast to the typical scheme for introducing quantum games, we assume that both parties can utilise full information about the opponent's strategy. We show that the utilisation of the full knowledge about the opponent's state does not provide the advantage. Moreover, the chances of catching the Robber decrease for classical cop-win graphs. This result does not depend on the chosen model of evolution. On the other hand, the possibility to execute controlled quantum operations allows catching the Robber on almost all classical cop-win graphs. By this, we demonstrate that it is necessary to enrich the structure of correlations between the players' systems to provide a non-trivial quantized Cop and Robber game. Thus the quantum controlled operations offer a significant advantage over the classically controlled quantum operations.

quant-ph↗

arXiv2017

Properties of quantum stochastic walks from the asymptotic scaling exponent

This work focuses on the study of quantum stochastic walks, which are a generalization of coherent, i. e. unitary quantum walks. Our main goal is to present a measure of a coherence of the walk. To this end, we utilize the asymptotic scaling exponent of the second moment of the walk i. e. of the mean squared distance covered by a walk. As the quantum stochastic walk model encompasses both classical random walks and quantum walks, we are interested how the continuous change from one regime to the other influences the asymptotic scaling exponent. Moreover this model allows for behavior which is not found in any of the previously mentioned model -- the model with global dissipation. We derive the probability distribution for the walker, and determine the asymptotic scaling exponent analytically, showing that ballistic regime of the walk is maintained even at large dissipation strength.

quant-ph↗

arXiv2016

Constructive quantum scaling of unitary matrices

In this work we present a method of decomposition of arbitrary unitary matrix $U\in\mathbf U(2^k)$ into a product of single-qubit negator and controlled-$\sqrt{\mbox{NOT}}$ gates. Since the product results with negator matrix, which can be treated as complex analogue if bistochastic matrix, our method can be seen as complex analogue of Sinkhorn-Knopp algorithm, where diagonal matrices are replaced by adding and removing an one-qubit ancilla. The decomposition can be found constructively and resulting circuit consists of $O(4^k)$ entangling gates, which is proved to be optimal. An example of such transformation is presented.

quant-ph↗