# An Example of the Difference Between Quantum and Classical Random Walks

@article{Childs2002AnEO, title={An Example of the Difference Between Quantum and Classical Random Walks}, author={Andrew M. Childs and E. Farhi and S. Gutmann}, journal={Quantum Information Processing}, year={2002}, volume={1}, pages={35-43} }

AbstractIn this note, we discuss a general definition of quantum random walks on graphs and illustrate with a simple graph the possibility of very different behavior between a classical random walk and its quantum analog. In this graph, propagation between a particular pair of nodes is exponentially faster in the quantum case.
PACS: 03.67.Hk

#### Topics from this paper

#### 331 Citations

Quantum Random Walks

- 2012

The notion of a quantum random walk has received notable attention recently from the field of quantum information, revealing many surprising and subtle properties that range from significant speedups… Expand

Continuous-Time Quantum Walks on the Symmetric Group

- Mathematics, Physics
- RANDOM-APPROX
- 2003

It is shown that for several natural choices for generating sets, quantum walks on Cayley graphs of the symmetric group do not have uniform limiting distributions, and are effectively blind to large areas of the graphs due to destructive interference. Expand

An Analysis of Absorbing Times of Quantum Walks

- Computer Science
- UMC
- 2002

The absorbing probability and time of quantum walks is introduced and both numerical simulation results and theoretical analyses on Hadamard walks on the line and symmetric walked on the hypercube from the viewpoint of absorbing probabilities and time are given. Expand

Quantum Walks

- Physics
- 2008

This tutorial article showcases the many varieties and uses of quantum walks. Discrete time quantum walks are introduced as counterparts of classical random walks. The emphasis is on the connections… Expand

Quantum walks on Cayley graphs

- Physics, Mathematics
- 2006

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… Expand

Limit theorems and absorption problems for quantum random walks in one dimension

- Physics, Computer Science
- Quantum Inf. Comput.
- 2002

In this paper we consider limit theorems, symmetry of distribution, and absorption problems for two types of one-dimensional quantum random walks determined by 2 × 2 unitary matrices using our PQRS… Expand

Quantum random walks in one dimension via generating functions

- Mathematics
- 2007

We analyze nearest neighbor one-dimensional quantum random walks with arbitrary unitary coin-flip matrices. Using a multivariate generating function analysis we give a simplified proof of a known… Expand

Quantum Random Walks Hit Exponentially Faster

- Computer Science, Mathematics
- ArXiv
- 2002

We show that the hitting time of the discrete time quantum random walk on the n-bit hypercube from one corner to its opposite is polynomial in n. This gives the first exponential quantum-classical… Expand

Two Photon Quantum Walks

- Physics
- 2013

Classical random walks are a stochastic model of a particle or walker, moving according to classical physics about a discrete space represented by a combinatoric graph—a set of vertices… Expand

Finding paths in tree graphs with a quantum walk

- Physics
- 2018

In this paper, we analyze the potential for new types of searches using the formalism of scattering random walks on Quantum Computers. Given a particular type of graph consisting of nodes and… Expand

#### References

SHOWING 1-8 OF 8 REFERENCES

Quantum random walks.

- Physics, Medicine
- Physical review. A, Atomic, molecular, and optical physics
- 1993

We introduce the concept of quantum random walk, and show that due to quantum interference effects the average path length can be much larger than the maximum allowed path in the corresponding… Expand

Quantum walks on graphs

- Mathematics, Computer Science
- STOC '01
- 2001

A lower bound on the possible speed up by quantum walks for general graphs is given, showing that quantum walks can be at most polynomially faster than their classical counterparts. Expand

One-dimensional quantum walks

- Mathematics, Computer Science
- STOC '01
- 2001

A quantum analog of the symmetric random walk, which the authors call the Hadamard walk, is analyzed, which has position that is nearly uniformly distributed in the range after steps, in sharp contrast to the classical random walk. Expand

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… Expand

On the absence of homogeneous scalar unitary cellular automata

- Physics
- 1996

Abstract Failure to find homogeneous scalar unitary cellular automata (CA) in one dimension led to consideration of only “approximately unitary” CA - which motivated our recent proof of a No-go Lemma… Expand

From quantum cellular automata to quantum lattice gases

- Mathematics, Physics
- 1996

A natural architecture for nanoscale quantum computation is that of a quantum cellular automaton. Motivated by this observation, we begin an investigation of exactly unitary cellular automata. After… Expand

The functional integral constructed directly from the hamiltonian

- Physics
- 1992

Abstract Starting with any Hamiltonian and a countable basis we construct a functional integral for the matrix elements of the time evolution operator of the quantum system. Our functional integral… Expand

Quantum computation and decision trees

- Physics
- 1998

Many interesting computational problems can be reformulated in terms of decision trees. A natural classical algorithm is to then run a random walk on the tree, starting at the root, to see if the… Expand