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.

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 Spatial search on grids with minimum memory

2015 ◽  
Vol 15 (13&14) ◽  
pp. 1233-1247
Author(s):  
Andris Ambainis ◽  
Renato Portugal ◽  
Nikolay Nahimov
Keyword(s):  
Numerical Simulations ◽  
Quantum Algorithms ◽  
Success Probability ◽  
Quantum Walk ◽  
Quantum Walks ◽  
Optimal Number ◽  
Rigorous Analysis ◽  
Spatial Search ◽  
Finite Dimensional

We study quantum algorithms for spatial search on finite dimensional grids. Patel \textit{et al.}~and Falk have proposed algorithms based on a quantum walk without a coin, with different operators applied at even and odd steps. Until now, such algorithms have been studied only using numerical simulations. In this paper, we present the first rigorous analysis for an algorithm of this type, showing that the optimal number of steps is $O(\sqrt{N\log N})$ and the success probability is $O(1/\log N)$, where $N$ is the number of vertices. This matches the performance achieved by algorithms that use other forms of quantum 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 Floquet-engineered quantum walks

2020 ◽  
Vol 10 (1) ◽  
Author(s):  
Haruna Katayama ◽  
Noriyuki Hatakenaka ◽  
Toshiyuki Fujii
Keyword(s):  
Quantum Computation ◽  
Computational Models ◽  
Quantum Algorithms ◽  
Quantum Walk ◽  
Quantum Walks ◽  
Research Areas ◽  
Wide Range ◽  
Mechanical Analogue ◽  
Universal Quantum ◽  
Walking Mechanism

Abstract The quantum walk is the quantum-mechanical analogue of the classical random walk, which offers an advanced tool for both simulating highly complex quantum systems and building quantum algorithms in a wide range of research areas. One prominent application is in computational models capable of performing any quantum computation, in which precisely controlled state transfer is required. It is, however, generally difficult to control the behavior of quantum walks due to stochastic processes. Here we unveil the walking mechanism based on its particle-wave duality and then present tailoring quantum walks using the walking mechanism (Floquet oscillations) under designed time-dependent coins, to manipulate the desired state on demand, as in universal quantum computation primitives. Our results open the path towards control of quantum walks.

scholarly journals Unstructured search by random and quantum walk

2022 ◽  
Vol 22 (1&2) ◽  
pp. 53-85
Author(s):  
Thomas G. Wong
Keyword(s):  
Random Walks ◽  
Continuous Time ◽  
Quantum Computer ◽  
Success Probability ◽  
Quantum Walk ◽  
Quantum Walks ◽  
Search Problem ◽  
Large N ◽  
Time Quantum ◽  
Classical Computer

The task of finding an entry in an unsorted list of $N$ elements famously takes $O(N)$ queries to an oracle for a classical computer and $O(\sqrt{N})$ queries for a quantum computer using Grover's algorithm. Reformulated as a spatial search problem, this corresponds to searching the complete graph, or all-to-all network, for a marked vertex by querying an oracle. In this tutorial, we derive how discrete- and continuous-time (classical) random walks and quantum walks solve this problem in a thorough and pedagogical manner, providing an accessible introduction to how random and quantum walks can be used to search spatial regions. Some of the results are already known, but many are new. For large $N$, the random walks converge to the same evolution, both taking $N \ln(1/\epsilon)$ time to reach a success probability of $1-\epsilon$. In contrast, the discrete-time quantum walk asymptotically takes $\pi\sqrt{N}/2\sqrt{2}$ timesteps to reach a success probability of $1/2$, while the continuous-time quantum walk takes $\pi\sqrt{N}/2$ time to reach a success probability of $1$.

scholarly journals Implementing graph-theoretic quantum algorithms on a silicon photonic quantum walk processor

2021 ◽  
Vol 7 (9) ◽  
pp. eabb8375
Author(s):  
Xiaogang Qiang ◽  
Yizhi Wang ◽  
Shichuan Xue ◽  
Renyou Ge ◽  
Lifeng Chen ◽  
...  
Keyword(s):  
Large Scale ◽  
Quantum Algorithms ◽  
Quantum Walk ◽  
Quantum Walks ◽  
Graph Theoretic ◽  
Two Photon ◽  
Full Control ◽  
Silicon Photonic ◽  
Vertex Graph ◽  
Particle Exchange

Applications of quantum walks can depend on the number, exchange symmetry and indistinguishability of the particles involved, and the underlying graph structures where they move. Here, we show that silicon photonics, by exploiting an entanglement-driven scheme, can realize quantum walks with full control over all these properties in one device. The device we realize implements entangled two-photon quantum walks on any five-vertex graph, with continuously tunable particle exchange symmetry and indistinguishability. We show how this simulates single-particle walks on larger graphs, with size and geometry controlled by tuning the properties of the composite quantum walkers. We apply the device to quantum walk algorithms for searching vertices in graphs and testing for graph isomorphisms. In doing so, we implement up to 100 sampled time steps of quantum walk evolution on each of 292 different graphs. This opens the way to large-scale, programmable quantum walk processors for classically intractable applications.

scholarly journals Perfect state transfer on signed graphs

2013 ◽  
Vol 13 (5&6) ◽  
pp. 511-530
Author(s):  
John Brown ◽  
Chris Godsil ◽  
Devlin Mallory ◽  
Abigail Raz ◽  
Christino Tamon
Keyword(s):  
Regular Graph ◽  
Complete Graph ◽  
Quantum Walk ◽  
Double Cover ◽  
Quantum Walks ◽  
Signed Graph ◽  
Perfect State ◽  
Signed Graphs ◽  
State Transfer ◽  
Perfect State Transfer

We study perfect state transfer of quantum walks on signed graphs. Our aim is to show that negative edges are useful for perfect state transfer. First, we show that the signed join of a negative $2$-clique with any positive $(n,3)$-regular graph has perfect state transfer even if the unsigned join does not. Curiously, the perfect state transfer time improves as $n$ increases. Next, we prove that a signed complete graph has perfect state transfer if its positive subgraph is a regular graph with perfect state transfer and its negative subgraph is periodic. This shows that signing is useful for creating perfect state transfer since no complete graph (except for the $2$-clique) has perfect state transfer. Also, we show that the double-cover of a signed graph has perfect state transfer if the positive subgraph has perfect state transfer and the negative subgraph is periodic.Here, signing is useful for constructing unsigned graphs with perfect state transfer. Finally, we study perfect state transfer on a family of signed graphs called the exterior powers which is derived from a many-fermion quantum walk on graphs.

Quantum walks on directed graphs

2007 ◽  
Vol 7 (1&2) ◽  
pp. 93-102
Author(s):  
A. Montanaro
Keyword(s):  
Directed Graph ◽  
Discrete Time ◽  
Directed Graphs ◽  
Quantum Walk ◽  
Quantum Walks ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Time Quantum ◽  
Definition Of ◽  
Necessary And Sufficient

We consider the definition of quantum walks on directed graphs. Call a directed graph reversible if, for each pair of vertices $(v_i, v_j)$, if $v_i$ is connected to $v_j$ then there is a path from $v_j$ to $v_i$. We show that reversibility is a necessary and sufficient condition for a directed graph to allow the notion of a discrete-time quantum walk, and discuss some implications of this condition. We present a method for defining a "partially quantum'' walk on directed graphs that are not reversible.

scholarly journals On Limiting Distributions of Quantum Markov Chains

2011 ◽  
Vol 2011 ◽  
pp. 1-12 ◽  
Author(s):  
Chaobin Liu ◽  
Nelson Petulante
Keyword(s):  
Markov Chain ◽  
Quantum System ◽  
Orthogonal Projection ◽  
Quantum Walks ◽  
Quantum Operation ◽  
Limiting Distributions ◽  
Initial State ◽  
Quantum Markov Chain ◽  
Classic Result ◽  
Temporal Succession

In a quantum Markov chain, the temporal succession of states is modeled by the repeated action of a “bistochastic quantum operation” on the density matrix of a quantum system. Based on this conceptual framework, we derive some new results concerning the evolution of a quantum system, including its long-term behavior. Among our findings is the fact that the Cesàro limit of any quantum Markov chain always exists and equals the orthogonal projection of the initial state upon the eigenspace of the unit eigenvalue of the bistochastic quantum operation. Moreover, if the unit eigenvalue is the only eigenvalue on the unit circle, then the quantum Markov chain converges in the conventional sense to the said orthogonal projection. As a corollary, we offer a new derivation of the classic result describing limiting distributions of unitary quantum walks on finite graphs (Aharonov et al., 2001).

scholarly journals Analysis of lackadaisical quantum walks

2020 ◽  
Vol 20 (13&14) ◽  
pp. 1138-1153
Author(s):  
Peter Hoyer ◽  
Zhan Yu
Keyword(s):  
Random Walk ◽  
Success Probability ◽  
Quantum Walk ◽  
Quantum Walks ◽  
Transitive Graph ◽  
Marked Vertex ◽  
Johnson Graphs ◽  
Vertex Transitive ◽  
Vertex Transitive Graphs ◽  
Transitive Graphs

The lackadaisical quantum walk is a quantum analogue of the lazy random walk obtained by adding a self-loop to each vertex in the graph. We analytically prove that lackadaisical quantum walks can find a unique marked vertex on any regular locally arc-transitive graph with constant success probability quadratically faster than the hitting time. This result proves several speculations and numerical findings in previous work, including the conjectures that the lackadaisical quantum walk finds a unique marked vertex with constant success probability on the torus, cycle, Johnson graphs, and other classes of vertex-transitive graphs. Our proof establishes and uses a relationship between lackadaisical quantum walks and quantum interpolated walks for any regular locally arc-transitive graph.

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