Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:quant-ph/0305182

Continuous-time quantum walks on the symmetric group

Heath Gerhardt, John Watrous

Published May 29, 2003• 0 citations
arXivHTMLPDF

Abstract

In this paper we study continuous-time quantum walks on Cayley graphs of the symmetric group, and prove various facts concerning such walks that demonstrate significant differences from their classical analogues. In particular, we show that for several natural choices for generating sets, these quantum walks do not have uniform limiting distributions, and are effectively blind to large areas of the graphs due to destructive interference.

DOI
10.48550/arxiv.quant-ph/0305182

Related papers

Quantum walks advantage on the dihedral group for uniform sampling problem

Shyam Dhamapurkar, Yu–Hang Dang, Saniya Wagh +1 more

Abstract Random walk algorithms are crucial for sampling and approximation problems in statistical physics and theoretical computer science. The mixing property is necessary for Markov chains to approach stationary distributions and is facilitated by walks. Quantum walks show promise for faster mixing times than classical methods but lack universal proof, especially in finite group settings. Here, we investigate the continuous-time quantum walks on Cayley graphs of the dihedral group D 2n for odd n, generated by the smallest inverse closed symmetric subset. We present a significant finding that, in contrast to the classical mixing time on these Cayley graphs, which typically takes at least order Ω ( n 2 log ( 1 / 2 ϵ ) ) , the continuous-time quantum walk mixing time on D 2n is of order O ( n ( log n ) 5 log ( 1 / ϵ ) ) , achieving a quadratic improvement over the classical case. Our paper advances the general understanding of quantum walk mixing on Cayley graphs, highlighting the improved mixing time achieved by continuous-time quantum walks on D 2n . This work has potential applications in algorithms for a class of sampling problems based on non-abelian groups.

arXiv:2312.15693•Communications in Theoretical Physics•Sep 26, 2024•0 citations

Quantum Walks Advantage on the Dihedral Group for Uniform Sampling Problem

Shyam Dhamapurkar, Yu–Hang Dang, Saniya Wagh +1 more

Random walk algorithms, including sampling and approximations, have played a significant role in statistical physics and theoretical computer science. Mixing through walks is the process for a Markov chain to approximate a stationary distribution for a group. Quantum walks have shown potential advantages in mixing time over the classical case but lack general proof in the finite group case. Here, we investigate the continuous-time quantum walks on Cayley graphs of the dihedral group D2n for odd n, generated by the smallest inverse closed symmetric subset. We present a significant finding that, in contrast to the classical mixing time on these Cayley graphs, which typically takes at least order Ω(n2log(1/2ε)), the continuous-time quantum walk mixing time on D2n is of order Ω($n(log n)$5log(1/ε)), achieving a quadratic improvement over the classical case. Our paper advances the general understanding of quantum walk mixing on Cayley graphs, highlighting the improved mixing time achieved by continuous-time quantum walks on D2n. This work has potential applications in algorithms for a class of sampling problems based on non-abelian groups.

International Conference on Quantum Communications, Networking, and Computing (QCNC)•Jul 1, 2024•1 citation

On mixing in continuous-time quantum walks on some circulant graphs

A. Ahmadi, Ryan Belk, Christino Tamon +1 more

Classical random walks on well-behaved graphs are rapidly mixing towards the uniform distribution. Moore and Russell showed that the continuous-time quantum walk on the hypercube is instantaneously uniform mixing. We show that the continuous-time quantum walks on other well-behaved graphs do not exhibit this uniform mixing. We prove that the only graphs amongst balanced complete multipartite graphs that have the instantaneous exactly uniform mixing property are the complete graphs on two, three and four vertices, and the cycle graph on four vertices. Our proof exploits the circulant structure of these graphs. Furthermore, we conjecture that most complete cycles and Cayley graphs of the symmetric group lack this mixing property as well.

Quantum Information and Computation•Nov 1, 2003•54 citations

Quantum walks on Cayley graphs

Olga Lopez‐Acevedo, Thierry Gobron

We address the problem of the construction of quantum walks on Cayley graphs. Our main motivation is the relationship between quantum algorithms and quantum walks. In particular, we discuss the choice of the dimension of the local Hilbert space and consider various classes of graphs on which the structure of quantum walks may differ. We completely characterize quantum walks on free groups and present partial results on more general cases. Some examples are given including a family of quantum walks on the hypercube involving a Clifford algebra.

arXiv:quant-ph/0503078•Journal of Physics A Mathematical and General•Dec 21, 2005•27 citations

QUANTUM WALKS ON CAYLEY GRAPHS

O. Lopez, Acevedo, Thierry Gobron

Abstract. We address the problem of the construction of quantum walks on Cayley graphs. Our main motivation is the relationship between quantum algorithms and quantum walks. In particular, we discuss the choice of the dimension of the local Hilbert space and consider various classes of graphs on which the structure of quantum walks may differ. We completely characterise quantum walks on free groups and present partial results on more general cases. Some examples are given including a family of quantum walks on the hypercube involving a Clifford Algebra. PACS number: 03.67.LxQuantum walks on Cayley graphs 2 1.

•Jan 1, 2005•21 citations

Uniform mixing in continuous-time quantum walks on oriented, nonabelian Cayley graphs

Peter Sin

A family of oriented, normal, nonabelian Cayley graphs is presented, whose continuous-time quantum walks exhibit uniform mixing.

arXiv:2510.08376•Oct 9, 2025•0 citations
In this paper we study continuous-time quantum walks on Cayley graphs of the symmetric group, and prove various facts concerning such walks that demonstrate significant differences from their classical analogues. In particular, we show that for several natural choices for generating sets, these quantum walks do not have uniform limiting distributions, and are effectively blind to large areas of the graphs due to destructive interference.
Abstract Random walk algorithms are crucial for sampling and approximation problems in statistical physics and theoretical computer science. The mixing property is necessary for Markov chains to approach stationary distributions and is facilitated by walks. Quantum walks show promise for faster mixing times than classical methods but lack universal proof, especially in finite group settings. Here, we investigate the continuous-time quantum walks on Cayley graphs of the dihedral group D 2n for odd n, generated by the smallest inverse closed symmetric subset. We present a significant finding that, in contrast to the classical mixing time on these Cayley graphs, which typically takes at least order Ω ( n 2 log ( 1 / 2 ϵ ) ) , the continuous-time quantum walk mixing time on D 2n is of order O ( n ( log n ) 5 log ( 1 / ϵ ) ) , achieving a quadratic improvement over the classical case. Our paper advances the general understanding of quantum walk mixing on Cayley graphs, highlighting the improved mixing time achieved by continuous-time quantum walks on D 2n . This work has potential applications in algorithms for a class of sampling problems based on non-abelian groups.
Random walk algorithms, including sampling and approximations, have played a significant role in statistical physics and theoretical computer science. Mixing through walks is the process for a Markov chain to approximate a stationary distribution for a group. Quantum walks have shown potential advantages in mixing time over the classical case but lack general proof in the finite group case. Here, we investigate the continuous-time quantum walks on Cayley graphs of the dihedral group D2n for odd n, generated by the smallest inverse closed symmetric subset. We present a significant finding that, in contrast to the classical mixing time on these Cayley graphs, which typically takes at least order Ω(n2log(1/2ε)), the continuous-time quantum walk mixing time on D2n is of order Ω($n(log n)$5log(1/ε)), achieving a quadratic improvement over the classical case. Our paper advances the general understanding of quantum walk mixing on Cayley graphs, highlighting the improved mixing time achieved by continuous-time quantum walks on D2n. This work has potential applications in algorithms for a class of sampling problems based on non-abelian groups.
Classical random walks on well-behaved graphs are rapidly mixing towards the uniform distribution. Moore and Russell showed that the continuous-time quantum walk on the hypercube is instantaneously uniform mixing. We show that the continuous-time quantum walks on other well-behaved graphs do not exhibit this uniform mixing. We prove that the only graphs amongst balanced complete multipartite graphs that have the instantaneous exactly uniform mixing property are the complete graphs on two, three and four vertices, and the cycle graph on four vertices. Our proof exploits the circulant structure of these graphs. Furthermore, we conjecture that most complete cycles and Cayley graphs of the symmetric group lack this mixing property as well.
We address the problem of the construction of quantum walks on Cayley graphs. Our main motivation is the relationship between quantum algorithms and quantum walks. In particular, we discuss the choice of the dimension of the local Hilbert space and consider various classes of graphs on which the structure of quantum walks may differ. We completely characterize quantum walks on free groups and present partial results on more general cases. Some examples are given including a family of quantum walks on the hypercube involving a Clifford algebra.
Abstract. We address the problem of the construction of quantum walks on Cayley graphs. Our main motivation is the relationship between quantum algorithms and quantum walks. In particular, we discuss the choice of the dimension of the local Hilbert space and consider various classes of graphs on which the structure of quantum walks may differ. We completely characterise quantum walks on free groups and present partial results on more general cases. Some examples are given including a family of quantum walks on the hypercube involving a Clifford Algebra. PACS number: 03.67.LxQuantum walks on Cayley graphs 2 1.