# Quantum walks on quotient graphs

@article{Krovi2007QuantumWO, title={Quantum walks on quotient graphs}, author={Hari Krovi and Todd A. Brun}, journal={Physical Review A}, year={2007}, volume={75}, pages={062332} }

A discrete-time quantum walk on a graph {gamma} is the repeated application of a unitary evolution operator to a Hilbert space corresponding to the graph. If this unitary evolution operator has an associated group of symmetries, then for certain initial states the walk will be confined to a subspace of the original Hilbert space. Symmetries of the original graph, given by its automorphism group, can be inherited by the evolution operator. We show that a quantum walk confined to the subspace…

## 50 Citations

Symmetry in quantum walks

- Mathematics, Physics
- 2007

A discrete-time quantum walk on a graph is the repeated application of a unitary evolution operator to a Hilbert space corresponding to the graph. Hitting times for discrete quantum walks on graphs…

Influence of coin symmetry on infinite hitting times in quantum walks

- Physics
- 2021

Classical random walks on finite graphs have an underrated property: a walk from any vertex can reach every other vertex in finite time, provided they are connected. Discrete-time quantum walks on…

STUDY OF CONTINUOUS-TIME QUANTUM WALKS ON QUOTIENT GRAPHS VIA QUANTUM PROBABILITY THEORY

- Mathematics, Physics
- 2007

In the present paper, we study the continuous-time quantum walk on quotient graphs. On such graphs, there is a straightforward reduction of the problem to a subspace that can be considerably smaller…

Limit theorems for the discrete-time quantum walk on a graph with joined half lines

- Physics, MathematicsQuantum Inf. Comput.
- 2012

It is shown that Wt,κ can be reduced to the walk on a half line even if the initial state is asymmetric, which plays an important role to analyze the walks on some class of graphs with symmetric initial states.

Systematic Dimensionality Reduction for Quantum Walks: Optimal Spatial Search and Transport on Non-Regular Graphs

- Computer Science, MedicineScientific reports
- 2015

This work uses invariant subspace methods, that can be computed systematically using the Lanczos algorithm, to obtain the reduced set of states that encompass the dynamics of the problem at hand without the specific knowledge of underlying symmetries.

Chirality from quantum walks without a quantum coin

- PhysicsPhysical Review A
- 2019

Quantum walks (QWs) describe the evolution of quantum systems on graphs. An intrinsic degree of freedom---called the coin and represented by a finite-dimensional Hilbert space---is associated to each…

Finding structural anomalies in star graphs using quantum walks.

- Mathematics, MedicinePhysical review letters
- 2014

The theory tells us how the initial state of the walk should be chosen, and how many steps the walk must make in order to find G, and the eigenvalues associated with these two parts in the N→∞ limit must be the same.

Continuous-time quantum walks on semi-regular spidernet graphs via quantum probability theory

- Physics, Computer ScienceQuantum Inf. Process.
- 2010

The characteristic time tc, which is the time when the first maximum of the probabilities occur on an infinite graph, for the quantum walk is shorter than that of the classical walk, which can interpret that the quantum transport speed on spidernet is faster than the classical one.

Perfect State Transfer in Quantum Walks on Graphs

- Mathematics
- 2011

We provide a brief survey of perfect state transfer in quantum walks on finite graphs. The ability to transfer a quantum state from one part of a quantum computer to another is a key ingredient of…

Search algorithm on strongly regular graphs based on scattering quantum walk

- Physics
- 2017

Janmark, Meyer, and Wong showed that continuous-time quantum walk search on known families of strongly regular graphs (SRGs) with parameters achieves full quantum speedup. The problem is reconsidered…

## References

SHOWING 1-8 OF 8 REFERENCES

Quantum walks on directed graphs

- Computer Science, PhysicsQuantum Inf. Comput.
- 2007

It is shown that reversibility is a necessary and sufficient condition for a directed graph to allow the notion of a discrete-time quantum walk, and some implications of this condition are discussed.

Quantum Walk on the Line

- Mathematics
- 2000

Motivated by the immense success of random walk and Markov chain methods in the design of classical algorithms, we consider {\em quantum\/} walks on graphs. We analyse in detail the behaviour of…

Controlling discrete quantum walks: coins and initial states

- Computer Science, Physics
- 2003

This paper explores some of the possibilities on regular graphs, and also reports periodic behaviour on small cyclic graphs.

THE HIDDEN SUBGROUP PROBLEM - REVIEW AND OPEN PROBLEMS

- Mathematics, Physics
- 2004

An overview of quantum computing and in particular the Hidden Subgroup Problem are presented from a mathematical viewpoint. Detailed proofs are supplied for many important results from the…

A fast quantum mechanical algorithm for database search

- Computer Science, PhysicsSTOC '96
- 1996

In early 1994, it was demonstrated that a quantum mechanical computer could efficiently solve a well-known problem for which there was no known efficient algorithm using classical computers, i.e. testing whether or not a given integer, N, is prime, in a time which is a finite power of o (logN) .

Randomized algorithms

- Computer ScienceCSUR
- 1996

A randomized algorithm is an algorithm that uses random numbers to influence the choices it makes in the course of its computation, so its behavior varies from one execution to another even with a fixed input.

Topological Graph Theory

- Mathematics, Computer ScienceHandbook of Graph Theory
- 2003

Introduction Voltage Graphs and Covering Spaces Surfaces and Graph Imbeddings Imbedded Voltage Graphs and Current Graphs Map Colorings The Genus of A Group References.