Quantized Markov Chain Couplings that Prepare Qsamples

Positive and doubly stochastic maps, and majorization in Euclidean Jordan algebras

We present in this paper a mathematical method based on stochastic comparisons of Markov chains in order to compute in a mobile network performance measures that are important Q parameters for users: the dropping handover of voice and the blocking probability of a new voice call. The key idea of this methodology is that given a complex system represented by a multi-dimensional Markov chain which is too large to be solved, we propose to reduce the state space, and so to define a new Markov chain which is a simplified version of the original one. This reduced Markov chain is defined as an aggregated one, which represents a stochastic bound for performance measures written as increasing reward function on the stationary distribution. The main steps of the construction of the aggregated Markov chain applied into mobile networks are presented in this paper. As the number of mobile users with different kinds of applications increases, the associated model is more complex and it can be represented by multi-dimensional continuous time Markov chain with a very large size. Thus, we define an aggregated Markov chain represented by a multi-dimensional birth and death process which is very easy to solve. We have proved that there is a weak ordering between the original Markov chain and the aggregated one using increasing set formalism. We have computed upper bounds of dropping handover and blocking probability for different values of input parameters. Numerical results prove that upper bounds give good results and so the stochastic methodology is an interesting mathematical tool for the performance evaluation of complex systems. Keywords: Mobile Networks, Quality of Service, Stochastic ordering, stochastic comparisons, Continuous Time Markov Chains.

In this paper, we extend the fluctuation theorems used for quantum channels to multitime processes. The fluctuation theorems for quantum channels are less restrictive. We show that the given entropy production can be equal to the result of a closed system environment. The assumption that the system evolves under a completely positive and trace preserving map is quite general, but it is more specific for cases in which the system is initially correlated with the environment. System-environment correlations arise naturally in multitime processes, with which we can give clear and physical interpretations regarding the effects of correlations. Multitime processes can provide many-body channels. The Choi state of such a many-body channel is called a process tensor. One can derive channels by executing the process tensor on a set of operations. We establish a general quantum fluctuation theorem framework for a many-body channel and its derived channels. In this framework, the effects of correlations are reflected in a Markovian property. For Markovian processes, we can extend the two-point measurement to a three-point measurement and obtain that the fluctuation theorems contain complete information about the intermediate state. For non-Markovian processes, the complete measurement of the intermediate state leads to conflicts. Therefore, we use a general measurement, which only provides partial information, for the intermediate state. The corresponding fluctuation theorems show that memory effects can reduce these fluctuations. This is consistent with the fact that system states can be recovered under non-Markovian processes.

Completely positive maps are useful in modeling the discrete evolution of quantum systems. Spectral properties of operators associated with such maps are relevant for determining the asymptotic dynamics of quantum systems subjected to multiple interactions described by the same quantum channel. We discuss a connection between the properties of the peripheral spectrum of completely positive and trace preserving map and the algebra generated by its Kraus operators A(A1,…,AK). By applying the Shemesh and Amitsur-Levitzki theorems to analyse the structure of the algebra A(A1,…,AK), one can predict the asymptotic dynamics for a class of operations.

Article Free Access Share on Exact sampling and approximate counting techniques Author: Mark Huber School of Operations Research and Industrial Engineering, Cornell University, Ithaca, NY School of Operations Research and Industrial Engineering, Cornell University, Ithaca, NYView Profile Authors Info & Claims STOC '98: Proceedings of the thirtieth annual ACM symposium on Theory of computingMay 1998 Pages 31–40https://doi.org/10.1145/276698.276709Published:23 May 1998Publication History 25citation666DownloadsMetricsTotal Citations25Total Downloads666Last 12 Months66Last 6 weeks15 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteeReaderPDF

This work analyzes distributed linear averaging within a connected network of sensors that each track the stationary distribution of an ergodic Markov chain with a slowly switching regime. Our approach is based on a two-time-scale stochastic approximation. A hyperparameter modeled as a Markov chain on a slower time-scale modulates the regime of each observed Markov chain. The average of all currently observed stationary distributions constitutes the average-consensus estimate to be reached by all sensors. Assuming the Markov chains do not share a common stationary distribution conditioned on their regime, then under the proposed linear averaging algorithm, the exchange graph conditions required for the sequence of sensor state values to converge weakly to the average-consensus are obtained. Estimation of a weighted average of all observed stationary distributions, not only the current ones, is proved feasible over a long-run time horizon, provided an additional communication condition holds. The sensor st...

The problem of sampling from the stationary distribution of a Markov chain finds widespread applications in a variety of fields. The time required for a Markov chain to converge to its stationary distribution is known as the classical mixing time. In this article, we deal with analog quantum algorithms for mixing. First, we provide an analog quantum algorithm that, given a Markov chain, allows us to sample from its stationary distribution in a time that scales as the sum of the square root of the classical mixing time and the square root of the classical hitting time. Our algorithm makes use of the framework of interpolated quantum walks and relies on Hamiltonian evolution in conjunction with von Neumann measurements. There also exists a different notion for quantum mixing: the problem of sampling from the limiting distribution of quantum walks, defined in a time-averaged sense. In this scenario, the quantum mixing time is defined as the time required to sample from a distribution that is close to this limiting distribution. Recently, we provided an upper bound on the quantum mixing time for Erd\ifmmode \mbox{\H{o}}\else \H{o}\fi{}s-R\'enyi random graphs [Phys. Rev. Lett. 124, 050501 (2020)]. Here, we also extend and expand upon our findings therein. Namely, we provide an intuitive understanding of the state-of-the-art random matrix theory tools used to derive our results. In particular, for our analysis we require information about macroscopic, mesoscopic, and microscopic statistics of eigenvalues of random matrices which we highlight here. Furthermore, we provide numerical simulations that corroborate our analytical findings and extend this notion of mixing from simple graphs to any ergodic, reversible, Markov chain.

In this paper, we develop a new "robust mixing" framework for reasoning about adversarially modified Markov Chains (AMMC). Let $\mathbb{P}$ be the transition matrix of an irreducible Markov Chain with stationary distribution $\pi$. An adversary announces a sequence of stochastic matrices $\{\mathbb{A}_t\}_{t > 0}$ satisfying $\pi\mathbb{A}_t = \pi$. An AMMC process involves an application of $\mathbb{P}$ followed by $\mathbb{A}_t$ at time $t$. The robust mixing time of an ergodic Markov Chain $\mathbb{P}$ is the supremum over all adversarial strategies of the mixing time of the corresponding AMMC process. Applications include estimating the mixing times for certain non-Markovian processes and for reversible liftings of Markov Chains. Non-Markovian card shuffling processes: The random-to-cyclic transposition process is a non-Markovian card shuffling process, which at time $t$, exchanges the card at position $L_t := t {\pmod n}$ with a random card. Mossel, Peres and Sinclair (2004) showed a lower bound of $(0.0345+o(1))n\log n$ for the mixing time of the random-to-cyclic transposition process. They also considered a generalization of this process where the choice of $L_t$ is adversarial, and proved an upper bound of $C n\log n + O(n)$ (with $C \approx 4\times 10^5$) on the mixing time. We reduce the constant to $1$ by showing that the random-to-top transposition chain (a Markov Chain) has robust mixing time $\leq n\log n + O(n)$ when the adversarial strategies are limited to holomorphic strategies, i.e. those strategies which preserve the symmetry of the underlying Markov Chain. We also show a $O(n\log^2 n)$ bound on the robust mixing time of the lazy random-to-top transposition chain when the adversary is not limited to holomorphic strategies. Reversible liftings: Chen, Lovasz and Pak showed that for a reversible ergodic Markov Chain $\mathbb{P}$, any reversible lifting $\mathbb{Q}$ of $\mathbb{P}$ must satisfy $\mathcal{T}(\mathbb{P}) \leq \mathcal{T}(\mathbb{Q})\log (1/\pi_*)$ where $\pi_*$ is the minimum stationary probability. Looking at a specific adversarial strategy allows us to show that $\mathcal{T}(\mathbb{Q}) \geq r(\mathbb{P})$ where $r(\mathbb{P})$ is the relaxation time of $\mathbb{P}$. This gives an alternate proof of the reversible lifting result and helps identify cases where reversible liftings cannot improve the mixing time by more than a constant factor.

In this paper, we develop a new “robust mixing” framework for reasoning about adversarially modified Markov Chains (AMMC). Let ℙ be the transition matrix of an irreducible Markov Chain with stationary distribution π. An adversary announces a sequence of stochastic matrices $\{{\mathbb{A}}_t\}_{t > 0}$ satisfying $\pi{\mathbb{A}}_t = \pi$. An AMMC process involves an application of ℙ followed by ${\mathbb{A}}_t$ at time t. The robust mixing time of an irreducible Markov Chain ℙ is the supremum over all adversarial strategies of the mixing time of the corresponding AMMC process. Applications include estimating the mixing times for certain non-Markovian processes and for reversible liftings of Markov Chains. Non-Markovian card shuffling processes: The random-to-cyclic transposition process is a non-Markovian card shuffling process, which at time t, exchanges the card at position $t {\pmod n}$ with a random card. Mossel, Peres and Sinclair (2004) showed that the mixing time of this process lies between (0.0345+o(1))nlogn and Cnlogn + O(n) (with C ≈4 ×105). We reduce the constant C to 1 by showing that the random-to-top transposition chain (a Markov Chain) has robust mixing time ≤nlogn + O(n) when the adversarial strategies are limited to those which preserve the symmetry of the underlying Markov Chain. Reversible liftings: Chen, Lovasz and Pak showed that for a reversible ergodic Markov Chain ℙ, any reversible lifting ℚ of ℙ must satisfy ${\mathcal{T}}({\mathbb{P}}) \leq {\mathcal{T}}(\mathbb{Q})\log (1/\pi_*)$ where π* is the minimum stationary probability. Looking at a specific adversarial strategy allows us to show that ${\mathcal{T}}(\mathbb{Q}) \geq r({\mathbb{P}})$ where r(ℙ) is the relaxation time of ℙ. This helps identify cases where reversible liftings cannot improve the mixing time by more than a constant factor.

A general model for the description of a discrete-time Markov chain in random environment is introduced. We study the stationary distribution of this model and its relationship with the Palm (embedded) distributions at certain environmental transition epochs. We also study the Events See Time Averages (ESTA) property for this model and we show that it is related to certain partial balance equations and a product-form stationary distribution. Next, we study certain forms of interaction between the environmental process and the process of interest that lead to tractable stationary distributions. More specifically, the stationary distributions of a large class of Markov chains in random environment can be computed efficiently using α-potentials of smaller Markov chains. This decomposability result is also the basis for developing approximations for the stationary distributions of intractable models that evolve in a slowly changing random environment. Moreover, explicit error bounds for these approximations are given. The model can be used for the performance evaluation of discrete time queueing systems in random environment.

We describe how randomized benchmarking can be used to reconstruct the unital part of any trace-preserving quantum map, which in turn is sufficient for the full characterization of any unitary evolution, or more generally, any unital trace-preserving evolution. This approach inherits randomized benchmarking's robustness to preparation, measurement, and gate imperfections, therefore avoiding systematic errors caused by these imperfections. We also extend these techniques to efficiently estimate the average fidelity of a quantum map to unitary maps outside of the Clifford group. The unitaries we consider correspond to large circuits commonly used as building blocks to achieve scalable, universal, and fault-tolerant quantum computation. Hence, we can efficiently verify all such subcomponents of a circuit-based universal quantum computer. In addition, we rigorously bound the time and sampling complexities of randomized benchmarking procedures, proving that the required non-linear estimation problem can be solved efficiently.

Random walks are a fundamental model in applied mathematics and are a common example of a Markov chain. The limiting stationary distribution of the Markov chain represents the fraction of the time spent in each state during the stochastic process. A standard way to compute this distribution for a random walk on a finite set of states is to compute the Perron vector of the associated transition matrix. There are algebraic analogues of this Perron vector in terms of transition probability tensors of higher-order Markov chains. These vectors are nonnegative, have dimension equal to the dimension of the state space, and sum to one and are derived by making an algebraic substitution in the equation for the joint-stationary distribution of a higher-order Markov chains. Here, we present the spacey random walk, a non-Markovian stochastic process whose stationary distribution is given by the tensor eigenvector. The process itself is a vertex-reinforced random walk, and its discrete dynamics are related to a continuous dynamical system. We analyze the convergence properties of these dynamics and discuss numerical methods for computing the stationary distribution. Finally, we provide several applications of the spacey random walk model in population genetics, ranking, and clustering data, and we use the process to analyze taxi trajectory data in New York. This example shows definite non-Markovian structure.

short-paper Share on Tail asymptotics of the stationary distribution for a two-node generalized Jackson network Author: Masakiyo Miyazawa Tokyo University of Science, Noda, Chiba, Japan Tokyo University of Science, Noda, Chiba, JapanView Profile Authors Info & Claims ACM SIGMETRICS Performance Evaluation ReviewVolume 42Issue 2September 2014pp 70–72https://doi.org/10.1145/2667522.2667545Published:04 September 2014Publication History 1citation26DownloadsMetricsTotal Citations1Total Downloads26Last 12 Months0Last 6 weeks0 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteGet Access

We assume that a Poisson flow of vehicles arrives at isolated signalized intersection, and each vehicle, independently of others, represents a random number X of passenger car units (PCU’s). We analyze numerically the stationary distribution of the queue process {Zn}, where Zn is the number of PCU’s in a queue at the beginning of the n-th red phase, n → ∞. We approximate the number Yn of PCU’s arriving during one red-green cycle by a two-parameter Negative Binomial Distribution (NBD). The well-known fact is that {Zn} follow an infinite-state Markov chain. We approximate its stationary distribution using a finite-state Markov chain. We show numerically that there is a strong dependence of the mean queue length E[Zn] in equilibrium on the input distribution of Yn and, in particular, on the ”over dispersion” parameter γ= Var[Yn ]/E[Yn ]. For Poisson input, γ = 1. γ > 1 indicates presence of heavy-tailed input. In reality it means that a relatively large ”portion” of PCU’s, considerably exceeding the average, may arrive with high probability during one red-green cycle. Empirical formulas are presented for an accurate estimation of mean queue length as a function of load and g of the input flow. Using the Markov chain technique, we analyze the mean ”virtual” delay time for a car which always arrives at the beginning of the red phase.

In this paper, we analyze a multi-server queueing system with a marked Markov arrival process of two types of customers and a phase-type distribution of service time depending on the type of customer. Customers of both types are assumed to be impatient and renege from the buffers after an exponentially distributed number of times. The strategy of flexible provisioning of priorities is analyzed. It assumes a randomized choice of the customers from the buffers, with probabilities dependent on the relation between the number of customers in a priority finite buffer and the fixed threshold value. To simplify the construction of the underlying Markov chain and the derivation of the explicit form of its generator, we use the so-called generalized phase-type distribution. It is shown that the created Markov chain fits the category of asymptotically quasi-Toeplitz Markov chains. Using this fact, we show that the considered Markov chain is ergodic for any value of the system parameters and compute its stationary distribution. Expressions for key performance measures are presented. Numerical results that show how the parameters of the control strategy affect the system’s performance measurements are given. It is shown that the results can be used for managerial purposes and that it is crucial to take correlation in the arrival process into account.