scholarly journals Role of Symmetry in Quantum Search via Continuous-Time Quantum Walk

SPIN ◽  
2021 ◽  
pp. 2140002
Author(s):  
Yunkai Wang ◽  
Shengjun Wu
Keyword(s):  
Invariant Subspace ◽  
Continuous Time ◽  
Quantum Walk ◽  
Dimensional Subspace ◽  
Quantum Search ◽  
Trivial Representation ◽  
Time Quantum ◽  
The One ◽  
Dimensionality Reduction Method

For quantum search via the continuous-time quantum walk, the evolution of the whole system is usually limited in a small subspace. In this paper, we discuss how the symmetries of the graphs are related to the existence of such an invariant subspace, which also suggests a dimensionality reduction method based on group representation theory. We observe that in the one-dimensional subspace spanned by each desired basis state which assembles the identically evolving original basis states, we always get a trivial representation of the symmetry group. So, we could find the desired basis by exploiting the projection operator of the trivial representation. Besides being technical guidance in this type of problem, this discussion also suggests that all the symmetries are used up in the invariant subspace and the asymmetric part of the Hamiltonian is very important for the purpose of quantum search.

scholarly journals Noise Characterization: Keeping Reduction Based Per-turbed Quantum Walk Search Optimal

2019 ◽  
Vol 198 ◽  
pp. 00001
Author(s):  
Chen-Fu Chiang ◽  
Chang-Yu Hsieh
Keyword(s):  
Invariant Subspace ◽  
Continuous Time ◽  
Quantum Walk ◽  
Coupling Factor ◽  
Necessary Condition ◽  
Subspace Method ◽  
Noise Distribution ◽  
Noisy Environment ◽  
Time Quantum ◽  
Noise Characterization

In a recent work by Novo et al. (Sci. Rep. 5, 13304, 2015), the invariant subspace method was applied to the study of continuous-time quantum walk (CTQW). In this work, we adopt the aforementioned method to investigate the optimality of a perturbed quantum walk search of a marked element in a noisy environment on various graphs. We formulate the necessary condition of the noise distribution in the system such that the invariant subspace method remains effective and efficient. Based on the noise, we further formulate how to set the appropriate coupling factor to preserve the optimality of the quantum walker.

scholarly journals CONTINUOUS-TIME QUANTUM WALKS ON TREES IN QUANTUM PROBABILITY THEORY

2006 ◽  
Vol 09 (02) ◽  
pp. 287-297 ◽  
Author(s):  
NORIO KONNO
Keyword(s):  
Probability Theory ◽  
Continuous Time ◽  
Quantum Walk ◽  
Quantum Probability ◽  
Quantum Walks ◽  
Homogeneous Tree ◽  
One Dimensional ◽  
Time Quantum ◽  
New Type ◽  
The One

A quantum central limit theorem for a continuous-time quantum walk on a homogeneous tree is derived from quantum probability theory. As a consequence, a new type of limit theorems for another continuous-time walk introduced by the walk is presented. The limit density is similar to that given by a continuous-time quantum walk on the one-dimensional lattice.

scholarly journals Resonant Quantum Kicked Rotor as A Continuous-Time Quantum Walk

2020 ◽  
Vol 5 (1) ◽  
pp. 4 ◽  
Author(s):  
Michele Delvecchio ◽  
Francesco Petiziol ◽  
Sandro Wimberger
Keyword(s):  
Continuous Time ◽  
Probability Distributions ◽  
Quantum Walk ◽  
One Dimension ◽  
Suitable Choice ◽  
Atom Optics ◽  
Quantum Search ◽  
Quantum Resonance ◽  
Kicked Rotor ◽  
Time Quantum

We analytically investigate the analogy between a standard continuous-time quantum walk in one dimension and the evolution of the quantum kicked rotor at quantum resonance conditions. We verify that the obtained probability distributions are equal for a suitable choice of the kick strength of the rotor. We further discuss how to engineer the evolution of the walk for dynamically preparing experimentally relevant states. These states are important for future applications of the atom-optics kicked rotor for the realization of ratchets and quantum search.

scholarly journals Quantum search with a continuous-time quantum walk in momentum space

2020 ◽  
Vol 53 (6) ◽  
pp. 065301 ◽  
Author(s):  
Michele Delvecchio ◽  
Caspar Groiseau ◽  
Francesco Petiziol ◽  
Gil S Summy ◽  
Sandro Wimberger
Keyword(s):  
Continuous Time ◽  
Momentum Space ◽  
Quantum Walk ◽  
Quantum Search ◽  
Time Quantum

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.

Correlation between the continuous-time quantum walk and cliques in graphs and its application

2021 ◽  
pp. 2150009
Author(s):  
Mingyou Wu ◽  
Xi Li ◽  
Zhihao Liu ◽  
Hanwu Chen
Keyword(s):  
Continuous Time ◽  
Maximum Clique ◽  
Quantum Walk ◽  
Approximate Algorithm ◽  
Maximum Clique Problem ◽  
High Approximation ◽  
Transmission Characteristics ◽  
Original Algorithm ◽  
Time Quantum ◽  
General Graphs

The continuous-time quantum walk (CTQW) provides a new approach to problems in graph theory. In this paper, the correlation between the CTQW and cliques in graphs is studied, and an approximate algorithm for the maximum clique problem (MCP) based on the CTQW is given. Via both numerical and theoretical analyses, it is found that the maximum clique is related to the transmission characteristics of the CTQW on some special graphs. For general graphs, the correlation is difficult to describe analytically. Therefore, the transmission characteristics of the CTQW are applied as a vertex selection criterion to a classical MCP algorithm and it is compared with the original algorithm. Numerous simulation on general graphs shows that the new algorithm is more efficient. Furthermore, an approximate MCP algorithm based on the CTQW is introduced, which only requires a very small number of searches with a high approximation ratio.

The fog on: Generalized teleportation by means of discrete-time quantum walks on N-lines and N-cycles

2019 ◽  
Vol 33 (23) ◽  
pp. 1950270 ◽  
Author(s):  
Duc Manh Nguyen ◽  
Sunghwan Kim
Keyword(s):  
Discrete Time ◽  
Quantum Teleportation ◽  
Quantum Walk ◽  
Quantum Walks ◽  
One Dimensional ◽  
Time Quantum ◽  
Infinite Line ◽  
The One

The recent paper entitled “Generalized teleportation by means of discrete-time quantum walks on [Formula: see text]-lines and [Formula: see text]-cycles” by Yang et al. [Mod. Phys. Lett. B 33(6) (2019) 1950069] proposed the quantum teleportation by means of discrete-time quantum walks on [Formula: see text]-lines and [Formula: see text]-cycles. However, further investigation shows that the quantum walk over the one-dimensional infinite line can be based over the [Formula: see text]-cycles and cannot be based on [Formula: see text]-lines. The proofs of our claims on quantum walks based on finite lines are also provided in detail.

scholarly journals NON-UNIFORM MIXING OF QUANTUM WALK ON CYCLES

2007 ◽  
Vol 05 (06) ◽  
pp. 781-793 ◽  
Author(s):  
WILLIAM ADAMCZAK ◽  
KEVIN ANDREW ◽  
LEON BERGEN ◽  
DILLON ETHIER ◽  
PETER HERNBERG ◽  
...  
Keyword(s):  
Random Walk ◽  
Uniform Distribution ◽  
Continuous Time ◽  
Quantum Walk ◽  
Complete Graphs ◽  
Circulant Graph ◽  
Mixing Properties ◽  
Time Quantum ◽  
Average Distribution ◽  
Instantaneous Distribution

A classical lazy random walk on cycles is known to mix with the uniform distribution. In contrast, we show that a continuous-time quantum walk on cycles exhibits strong non-uniform mixing properties. First, we prove that the instantaneous distribution of a quantum walk on most even-length cycles is never uniform. More specifically, we prove that a quantum walk on a cycle Cnis not instantaneous uniform mixing, whenever n satisfies either: (a) n = 2u, for u ≥ 3; or (b) n = 2uq, for u ≥ 1 and q ≡ 3 (mod 4). Second, we prove that the average distribution of a quantum walk on any Abelian circulant graph is never uniform. As a corollary, the average distribution of a quantum walk on any standard circulant graph, such as the cycles, complete graphs, and even hypercubes, is never uniform. Nevertheless, we show that the average distribution of a quantum walk on the cycle Cnis O(1/n)-uniform.

Sign in / Sign up

Export Citation Format

Share Document