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 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 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 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 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 Lackadaisical quantum walk for spatial search

2019 ◽  
Vol 35 (08) ◽  
pp. 2050043 ◽  
Author(s):  
Pulak Ranjan Giri ◽  
Vladimir Korepin
Keyword(s):  
Search Algorithm ◽  
Success Probability ◽  
Quantum Walk ◽  
Two Dimensions ◽  
Quantum Search Algorithm ◽  
Dimensional Lattice ◽  
One Dimensional ◽  
Target State ◽  
Spatial Search ◽  
Target States

Lackadaisical quantum walk (LQW) has been an efficient technique in searching for a target state in a database which is distributed in a two-dimensional lattice. We numerically study the quantum search algorithm based on the lackadaisical quantum walk in one and two dimensions. It is observed that specific values of the self-loop weight at each vertex of the graph is responsible for such a speedup of the algorithm. Searching for a target state in one-dimensional lattice with periodic boundary conditions is possible using lackadaisical quantum walk, which can find a target state with [Formula: see text] success probability after [Formula: see text] time steps. In two dimensions, our numerical simulation up to [Formula: see text] for specific sets of target states suggests that the lackadaisical quantum walk can search one of the [Formula: see text] target states in [Formula: see text] time steps.

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.

Global Optimization With Quantum Walk Enhanced Grover Search

2014 ◽  
Author(s):  
Yan Wang
Keyword(s):  
Global Optimization ◽  
Optimization Problems ◽  
Early Stage ◽  
Hybrid Approach ◽  
Quantum Walk ◽  
Quantum Walks ◽  
Tunneling Effect ◽  
Grover’S Algorithm ◽  
Grover's Algorithm ◽  
Grover Search

One of the significant breakthroughs in quantum computation is Grover’s algorithm for unsorted database search. Recently, the applications of Grover’s algorithm to solve global optimization problems have been demonstrated, where unknown optimum solutions are found by iteratively improving the threshold value for the selective phase shift operator in Grover rotation. In this paper, a hybrid approach that combines continuous-time quantum walks with Grover search is proposed. By taking advantage of quantum tunneling effect, local barriers are overcome and better threshold values can be found at the early stage of search process. The new algorithm based on the formalism is demonstrated with benchmark examples of global optimization. The results between the new algorithm and the Grover search method are also compared.

Higher-dimensional open quantum walk in environment of quantum Bernoulli noises

2021 ◽  
pp. 2250001
Author(s):  
Ce Wang
Keyword(s):  
Quantum Walk ◽  
Quantum Walks ◽  
Initial State ◽  
Quantum Random Walks ◽  
Open Quantum ◽  
Higher Dimensional ◽  
Gauss Type ◽  
Open Quantum Random Walks ◽  
Limit Probability ◽  
Creation Operators

Open quantum walks (OQWs) (also known as open quantum random walks) are quantum analogs of classical Markov chains in probability theory, and have potential application in quantum information and quantum computation. Quantum Bernoulli noises (QBNs) are annihilation and creation operators acting on Bernoulli functionals, and can be used as the environment of an open quantum system. In this paper, by using QBNs as the environment, we introduce an OQW on a general higher-dimensional integer lattice. We obtain a quantum channel representation of the walk, which shows that the walk is indeed an OQW. We prove that all the states of the walk are separable provided its initial state is separable. We also prove that, for some initial states, the walk has a limit probability distribution of higher-dimensional Gauss type. Finally, we show links between the walk and a unitary quantum walk recently introduced in terms of QBNs.

Sign in / Sign up

Export Citation Format

Share Document