scholarly journals Comparing Algorithms for Graph Isomorphism Using Discrete- and Continuous-Time Quantum Random Walks

2013 ◽  
Vol 10 (7) ◽  
pp. 1653-1661 ◽  
Author(s):  
Kenneth Rudinger ◽  
John King Gamble ◽  
Eric Bach ◽  
Mark Friesen ◽  
Robert Joynt ◽  
...  
Keyword(s):  
Random Walks ◽  
Continuous Time ◽  
Graph Isomorphism ◽  
Quantum Random Walks ◽  
Time Quantum

EVOLUTION OF CONTINUOUS-TIME QUANTUM RANDOM WALKS ON CIRCLES

2005 ◽  
Vol 05 (01) ◽  
pp. L73-L83 ◽  
Author(s):  
NORIO INUI ◽  
KOICHIRO KASAHARA ◽  
YOSHINAO KONISHI ◽  
NORIO KONNO
Keyword(s):  
Standard Deviation ◽  
Random Walks ◽  
Discrete Time ◽  
Continuous Time ◽  
Quantum Random Walks ◽  
Time Quantum ◽  
Mixing Property ◽  
Temporal Standard Deviation

This work deals with both instantaneous uniform mixing property and temporal standard deviation for continuous-time quantum random walks on circles in order to study their fluctuations comparing with discrete-time quantum random walks, and continuous- and discrete-time classical random walks.

scholarly journals Marking Vertices to Find Graph Isomorphism Mapping Based on Continuous-Time Quantum Walk

Entropy ◽  
2018 ◽  
Vol 20 (8) ◽  
pp. 586 ◽  
Author(s):  
Xin Wang ◽  
Yi Zhang ◽  
Kai Lu ◽  
Xiaoping Wang ◽  
Kai Liu
Keyword(s):  
Polynomial Time ◽  
Continuous Time ◽  
Graph Isomorphism ◽  
Quantum Walk ◽  
Isomorphism Problem ◽  
Quantum Walks ◽  
Regular Graphs ◽  
Structure Preserving ◽  
Topological Structures ◽  
Time Quantum

The isomorphism problem involves judging whether two graphs are topologically the same and producing structure-preserving isomorphism mapping. It is widely used in various areas. Diverse algorithms have been proposed to solve this problem in polynomial time, with the help of quantum walks. Some of these algorithms, however, fail to find the isomorphism mapping. Moreover, most algorithms have very limited performance on regular graphs which are generally difficult to deal with due to their symmetry. We propose IsoMarking to discover an isomorphism mapping effectively, based on the quantum walk which is sensitive to topological structures. Firstly, IsoMarking marks vertices so that it can reduce the harmful influence of symmetry. Secondly, IsoMarking can ascertain whether the current candidate bijection is consistent with existing bijections and eventually obtains qualified mapping. Thirdly, our experiments on 1585 pairs of graphs demonstrate that our algorithm performs significantly better on both ordinary graphs and regular graphs.

scholarly journals Coherent modification of entanglement: benefits due to extended Hilbert space

2016 ◽  
Vol 16 (11&12) ◽  
pp. 954-968
Author(s):  
Dmitry Solenov
Keyword(s):  
Hilbert Space ◽  
Continuous Time ◽  
Computing System ◽  
Local Network ◽  
Toffoli Gate ◽  
Quantum Random Walks ◽  
Physical Systems ◽  
Time Quantum ◽  
Speed Up ◽  
Non Local

A quantum computing system is typically represented by a set of non-interacting (local) two-state systems—qubits. Many physical systems can naturally have more accessible states, both local and non-local. We show that the resulting non-local network of states connecting qubits can be efficiently addressed via continuous time quantum random walks, leading to substantial speed-up of multiqubit entanglement manipulations. We discuss a three-qubit Toffoli gate and a system of superconducting qubits as an illustration.

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 $1$.

QUANTUM PARRONDO'S GAMES CONSTRUCTED BY QUANTUM RANDOM WALKS

2013 ◽  
Vol 12 (04) ◽  
pp. 1350024 ◽  
Author(s):  
MIN LI ◽  
YONG-SHENG ZHANG ◽  
GUANG-CAN GUO
Keyword(s):  
Random Walks ◽  
Discrete Time ◽  
Quantum Walks ◽  
Quantum Random Walks ◽  
Parrondo’S Paradox ◽  
Time Quantum

We construct a Parrondo's game using discrete-time quantum walks (DTQWs). Two losing games are represented by two different coin operators. By mixing the two coin operators UA(αA, βA, γA) and UB(αB, βB, γB), we may win the game. Here, we mix the two games in position instead of time. With a number of selections of the parameters, we can win the game with sequences ABB, ABBB, etc. If we set βA = 45°, γA = 0, αB = 0, βB = 88°, we find game 1 with [Formula: see text], [Formula: see text] will win and get the most profit. If we set αA = 0, βA = 45°, αB = 0, βB = 88° and game 2 with [Formula: see text], [Formula: see text] will win most. Game 1 is equivalent to game 2 with changes in sequences and steps. But at large enough steps, the game will lose at last. Parrondo's paradox does not exist in classical situation with our model.

scholarly journals Crossovers induced by discrete-time quantum walks

2011 ◽  
Vol 11 (9&10) ◽  
pp. 741-760
Author(s):  
Kota Chisaki ◽  
Norio Konno ◽  
Etsuo Segawa ◽  
Yutaka Shikano
Keyword(s):  
Weak Convergence ◽  
Random Walks ◽  
Discrete Time ◽  
Convergence Theorem ◽  
Continuous Time ◽  
Quantum Walk ◽  
Convergence Theorems ◽  
Final Time ◽  
Time Quantum ◽  
Weak Convergence Theorem

We consider crossovers with respect to the weak convergence theorems from a discrete-time quantum walk (DTQW). We show that a continuous-time quantum walk (CTQW) and discrete- and continuous-time random walks can be expressed as DTQWs in some limits. At first we generalize our previous study [Phys. Rev. A \textbf{81}, 062129 (2010)] on the DTQW with position measurements. We show that the position measurements per each step with probability $p \sim 1/n^\beta$ can be evaluated, where $n$ is the final time and $0<\beta<1$. We also give a corresponding continuous-time case. As a consequence, crossovers from the diffusive spreading (random walk) to the ballistic spreading (quantum walk) can be seen as the parameter $\beta$ shifts from 0 to 1 in both discrete- and continuous-time cases of the weak convergence theorems. Secondly, we introduce a new class of the DTQW, in which the absolute value of the diagonal parts of the quantum coin is proportional to a power of the inverse of the final time $n$. This is called a final-time-dependent DTQW (FTD-DTQW). The CTQW is obtained in a limit of the FTD-DTQW. We also obtain the weak convergence theorem for the FTD-DTQW which shows a variety of spreading properties. Finally, we consider the FTD-DTQW with periodic position measurements. This weak convergence theorem gives a phase diagram which maps sufficiently long-time behaviors of the discrete- and continuous-time quantum and random walks.

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