scholarly journals Quantum fast-forwarding: Markov chains and graph property testing

2019 ◽  
Vol 19 (3&4) ◽  
pp. 181-213 ◽  
Author(s):  
Simon Apers ◽  
Alain Scarlet
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Markov Chains ◽  
Quantum State ◽  
Quantum Algorithms ◽  
Quantum Algorithm ◽  
Quantum Walk ◽  
Transient Behavior ◽  
Quantum Walks ◽  
Learning Context

We introduce a new tool for quantum algorithms called quantum fast-forwarding (QFF). The tool uses quantum walks as a means to quadratically fast-forward a reversible Markov chain. More specifically, with P the Markov chain transition matrix and D = \sqrt{P\circ P^T} its discriminant matrix (D=P if P is symmetric), we construct a quantum walk algorithm that for any quantum state |v> and integer t returns a quantum state \epsilon-close to the state D^t|v>/\|D^t|v>. The algorithm uses O(|D^t|v>|^{-1}\sqrt{t\log(\epsilon\|D^t|v>})^{-1}}) expected quantum walk steps and O(\|D^t|v>|^{-1}) expected reflections around |v>. This shows that quantum walks can accelerate the transient dynamics of Markov chains, complementing the line of results that proves the acceleration of their limit behavior. We show that this tool leads to speedups on random walk algorithms in a very natural way. Specifically we consider random walk algorithms for testing the graph expansion and clusterability, and show that we can quadratically improve the dependency of the classical property testers on the random walk runtime. Moreover, our quantum algorithm exponentially improves the space complexity of the classical tester to logarithmic. As a subroutine of independent interest, we use QFF for determining whether a given pair of nodes lies in the same cluster or in separate clusters. This solves a robust version of s-t connectivity, relevant in a learning context for classifying objects among a set of examples. The different algorithms crucially rely on the quantum speedup of the transient behavior of random walks.

scholarly journals Efficient circuits for quantum walks

2010 ◽  
Vol 10 (5&6) ◽  
pp. 420-434
Author(s):  
C.-F. Chiang ◽  
D. Nagaj ◽  
P. Wocjan
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Random Walks ◽  
Probability Distributions ◽  
Quantum Walk ◽  
Quantum Walks ◽  
Problem Size ◽  
Integrable Probability ◽  
The Difference ◽  
General Method

We present an efficient general method for realizing a quantum walk operator corresponding to an arbitrary sparse classical random walk. Our approach is based on Grover and Rudolph's method for preparing coherent versions of efficiently integrable probability distributions \cite{GroverRudolph}. This method is intended for use in quantum walk algorithms with polynomial speedups, whose complexity is usually measured in terms of how many times we have to apply a step of a quantum walk \cite{Szegedy}, compared to the number of necessary classical Markov chain steps. We consider a finer notion of complexity including the number of elementary gates it takes to implement each step of the quantum walk with some desired accuracy. The difference in complexity for various implementation approaches is that our method scales linearly in the sparsity parameter and poly-logarithmically with the inverse of the desired precision. The best previously known general methods either scale quadratically in the sparsity parameter, or polynomially in the inverse precision. Our approach is especially relevant for implementing quantum walks corresponding to classical random walks like those used in the classical algorithms for approximating permanents \cite{Vigoda, Vazirani} and sampling from binary contingency tables \cite{Stefankovi}. In those algorithms, the sparsity parameter grows with the problem size, while maintaining high precision is required.

scholarly journals QUANTUM HITTING TIME ON THE COMPLETE GRAPH

2010 ◽  
Vol 08 (05) ◽  
pp. 881-894 ◽  
Author(s):  
RAQUELINE AZEVEDO MEDEIROS SANTOS ◽  
RENATO PORTUGAL
Keyword(s):  
Markov Chain ◽  
Complete Graph ◽  
Quantum Algorithms ◽  
Success Probability ◽  
Natural Extension ◽  
Quantum Walk ◽  
Quantum Walks ◽  
Hitting Time ◽  
Quantum Markov Chain ◽  
Definition Of

Quantum walks play an important role in the area of quantum algorithms. Many interesting problems can be reduced to searching marked states in a quantum Markov chain. In this context, the notion of quantum hitting time is very important, because it quantifies the running time of the algorithms. Markov chain-based algorithms are probabilistic, therefore the calculation of the success probability is also required in the analysis of the computational complexity. Using Szegedy's definition of quantum hitting time, which is a natural extension of the definition of the classical hitting time, we present analytical expressions for the hitting time and success probability of the quantum walk on the complete graph.

On the absorption probabilities and mean time for absorption for discrete Markov chains

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Nikolaos Halidias
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Markov Chains ◽  
Generating Function ◽  
Discrete Time ◽  
Probability Generating Function ◽  
The Mean ◽  
Mean Time ◽  
Absorption Probabilities

Abstract In this note we study the probability and the mean time for absorption for discrete time Markov chains. In particular, we are interested in estimating the mean time for absorption when absorption is not certain and connect it with some other known results. Computing a suitable probability generating function, we are able to estimate the mean time for absorption when absorption is not certain giving some applications concerning the random walk. Furthermore, we investigate the probability for a Markov chain to reach a set A before reach B generalizing this result for a sequence of sets A 1 , A 2 , … , A k {A_{1},A_{2},\dots,A_{k}} .

scholarly journals Quantum walks and elliptic integrals

2010 ◽  
Vol 20 (6) ◽  
pp. 1091-1098 ◽  
Author(s):  
NORIO KONNO
Keyword(s):  
Random Walk ◽  
Generating Function ◽  
Elliptic Integral ◽  
Quantum Walk ◽  
Quantum Walks ◽  
Elliptic Integrals ◽  
Two Dimensional ◽  
One Dimensional ◽  
Similar Expression ◽  
Return Probability

Pólya showed in his 1921 paper that the generating function of the return probability for a two-dimensional random walk can be written in terms of an elliptic integral. In this paper we present a similar expression for a one-dimensional quantum walk.

scholarly journals One dimensional quantum walks with memory

2010 ◽  
Vol 10 (5&6) ◽  
pp. 509-524
Author(s):  
M. Mc Gettrick
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Random Walks ◽  
Quantum Walks ◽  
One Dimensional ◽  
With Memory ◽  
Previous Step

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 one previous step. 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.

scholarly journals Quantum Generative Adversarial Networks for learning and loading random distributions

2019 ◽  
Vol 5 (1) ◽  
Author(s):  
Christa Zoufal ◽  
Aurélien Lucchi ◽  
Stefan Woerner
Keyword(s):  
Quantum State ◽  
Quantum Channel ◽  
Quantum Algorithms ◽  
Probability Distributions ◽  
Quantum Algorithm ◽  
Quantum Circuit ◽  
Quantum Simulation ◽  
Quantum States ◽  
Generative Adversarial Networks ◽  
Adversarial Networks

AbstractQuantum algorithms have the potential to outperform their classical counterparts in a variety of tasks. The realization of the advantage often requires the ability to load classical data efficiently into quantum states. However, the best known methods require $${\mathcal{O}}\left({2}^{n}\right)$$O2n gates to load an exact representation of a generic data structure into an $$n$$n-qubit state. This scaling can easily predominate the complexity of a quantum algorithm and, thereby, impair potential quantum advantage. Our work presents a hybrid quantum-classical algorithm for efficient, approximate quantum state loading. More precisely, we use quantum Generative Adversarial Networks (qGANs) to facilitate efficient learning and loading of generic probability distributions - implicitly given by data samples - into quantum states. Through the interplay of a quantum channel, such as a variational quantum circuit, and a classical neural network, the qGAN can learn a representation of the probability distribution underlying the data samples and load it into a quantum state. The loading requires $${\mathcal{O}}\left(poly\left(n\right)\right)$$Opolyn gates and can thus enable the use of potentially advantageous quantum algorithms, such as Quantum Amplitude Estimation. We implement the qGAN distribution learning and loading method with Qiskit and test it using a quantum simulation as well as actual quantum processors provided by the IBM Q Experience. Furthermore, we employ quantum simulation to demonstrate the use of the trained quantum channel in a quantum finance application.

The existence of moments for stationary Markov chains

1983 ◽  
Vol 20 (01) ◽  
pp. 191-196 ◽  
Author(s):  
R. L. Tweedie
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Markov Chains ◽  
Stationary Distribution ◽  
Time Dependent ◽  
General Function ◽  
Convergence Of Moments ◽  
Special Cases

We give conditions under which the stationary distribution π of a Markov chain admits moments of the general form ∫ f(x)π(dx), where f is a general function; specific examples include f(x) = xr and f(x) = esx . In general the time-dependent moments of the chain then converge to the stationary moments. We show that in special cases this convergence of moments occurs at a geometric rate. The results are applied to random walk on [0, ∞).

scholarly journals A random walk approach to quantum algorithms

2006 ◽  
Vol 364 (1849) ◽  
pp. 3407-3422 ◽  
Author(s):  
Vivien M Kendon
Keyword(s):  
Random Walk ◽  
Quantum Dynamics ◽  
Quantum Computer ◽  
Quantum Algorithms ◽  
Quantum Walk ◽  
Quantum Random Walk ◽  
Small Quantum ◽  
Starting Point ◽  
Quantum Version ◽  
Speed Up

The development of quantum algorithms based on quantum versions of random walks is placed in the context of the emerging field of quantum computing. Constructing a suitable quantum version of a random walk is not trivial; pure quantum dynamics is deterministic, so randomness only enters during the measurement phase, i.e. when converting the quantum information into classical information. The outcome of a quantum random walk is very different from the corresponding classical random walk owing to the interference between the different possible paths. The upshot is that quantum walkers find themselves further from their starting point than a classical walker on average, and this forms the basis of a quantum speed up, which can be exploited to solve problems faster. Surprisingly, the effect of making the walk slightly less than perfectly quantum can optimize the properties of the quantum walk for algorithmic applications. Looking to the future, even with a small quantum computer available, the development of quantum walk algorithms might proceed more rapidly than it has, especially for solving real problems.

scholarly journals Quantum algorithms for search with wildcards and combinatorial group testing

2014 ◽  
Vol 14 (5&6) ◽  
pp. 439-453
Author(s):  
Andris Ambainis ◽  
Ashley Montanaro
Keyword(s):  
Lower Bound ◽  
Quantum Algorithms ◽  
Group Testing ◽  
Quantum Algorithm ◽  
Quantum Walk ◽  
Combinatorial Problems ◽  
Special Items ◽  
State Discrimination ◽  
Combinatorial Group Testing ◽  
Combinatorial Group

We consider two combinatorial problems. The first we call ``search with wildcards'': given an unknown $n$-bit string $x$, and the ability to check whether any subset of the bits of $x$ is equal to a provided query string, the goal is to output $x$. We give a nearly optimal $O(\sqrt{n} \log n)$ quantum query algorithm for search with wildcards, beating the classical lower bound of $\Omega(n)$ queries. Rather than using amplitude amplification or a quantum walk, our algorithm is ultimately based on the solution to a state discrimination problem. The second problem we consider is combinatorial group testing, which is the task of identifying a subset of at most $k$ special items out of a set of $n$ items, given the ability to make queries of the form ``does the set $S$ contain any special items?''\ for any subset $S$ of the $n$ items. We give a simple quantum algorithm which uses $O(k)$ queries to solve this problem, as compared with the classical lower bound of $\Omega(k \log(n/k))$ queries.

On the existence of quasi-stationary distributions in denumerable R-transient Markov chains

1992 ◽  
Vol 29 (01) ◽  
pp. 21-36 ◽  
Author(s):  
Masaaki Kijima
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Markov Chains ◽  
State Space ◽  
Stationary Distribution ◽  
Practical Importance ◽  
Prime Concern ◽  
Continuous Random Walk ◽  
Transient Markov Chain ◽  
Quasi Stationary Distribution

Let {Xn, n= 0, 1, 2, ···} be a transient Markov chain which, when restricted to the state space 𝒩+= {1, 2, ···}, is governed by an irreducible, aperiodic and strictly substochastic matrix𝐏= (pij), and letpij(n) =P∈Xn=j, Xk∈ 𝒩+fork= 0, 1, ···,n|X0=i],i, j𝒩+. The prime concern of this paper is conditions for the existence of the limits,qijsay, ofasn →∞. Ifthe distribution (qij) is called the quasi-stationary distribution of {Xn} and has considerable practical importance. It will be shown that, under some conditions, if a non-negative non-trivial vectorx= (xi) satisfyingrxT=xT𝐏andexists, whereris the convergence norm of𝐏, i.e.r=R–1andand T denotes transpose, then it is unique, positive elementwise, andqij(n) necessarily converge toxjasn →∞.Unlike existing results in the literature, our results can be applied even to theR-null andR-transient cases. Finally, an application to a left-continuous random walk whose governing substochastic matrix isR-transient is discussed to demonstrate the usefulness of our results.

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