# One-dimensional quantum walks

@inproceedings{Ambainis2001OnedimensionalQW, title={One-dimensional quantum walks}, author={Andris Ambainis and Eric Bach and Ashwin Nayak and Ashvin Vishwanath and John Watrous}, booktitle={STOC '01}, year={2001} }

We define and analyze quantum computational variants of random walks on one-dimensional lattices. In particular, we analyze a quantum analog of the symmetric random walk, which we call the <italic>Hadamard walk</italic>. Several striking differences between the quantum and classical cases are observed. For example, when unrestricted in either direction, the Hadamard walk has position that is nearly uniformly distributed in the range <italic>[-t/\sqrt 2, t/\sqrt 2]</italic> after <italic>t…

## 491 Citations

Asymptotic evolution of quantum walks with random coin

- Mathematics
- 2011

We study the asymptotic position distribution of general quantum walks on a lattice, including walks with a random coin, which is chosen from step to step by a general Markov chain. In the unitary…

Coined quantum walks on percolation graphs

- Physics
- 2010

Quantum walks, both discrete (coined) and continuous time, form the basis of several quantum algorithms and have been used to model processes such as transport in spin chains and quantum chemistry.…

Random walk replicating discrete time quantum walk on one dimension

- Mathematics
- 2022

A quantum walk exhibits quite different properties compared to a classical random walk, such as linear spreading and localization. While quantum superposition provides these interesting properties,…

Discrete-time quantum walks on one-dimensional lattices

- Mathematics, Physics
- 2010

Abstract.
In this paper, we study discrete-time quantum walks on
one-dimensional lattices. We find that the coherent dynamics
depends on the initial states and coin parameters. For infinite
size of…

Properties of long quantum walks in one and two dimensions

- PhysicsQuantum Inf. Process.
- 2015

This work performs a detailed numerical study of discrete-time quantum walks, characterized the properties of both walks, demonstrating maximal speed-up and emerging semi-classical behavior in the maximally entangled QW.

One-dimensional Hadamard Quantum Walk on a Cycle with Rotational Implementation

- Mathematics
- 2019

The paper explores two implementations of the quantum walk as a quantum circuit: the first one consists of generalised controlled inversions, as introduced in EffWalk, whereas the second one tries to replace them with rotation operators around the basis states.

Interference-induced localization in quantum random walk on clean cyclic graph

- Physics
- 2018

We quantitatively differentiate between the spreads of discrete-time quantum and classical random walks on a cyclic graph. Due to the closed nature of any cyclic graph, there is additional…

Continuous-Time Quantum Walk on the Line

- Mathematics
- 2004

Quantum walks have recently been introduced and investigated, with the hope that they may be useful in constructing new efficient quantum algorithms. For reviews of quantum walks, see Refs. [4, 16,…

Invariance in Quantum Walks

- Mathematics
- 2016

In this Chapter, we present some interesting properties of quantum walks on the line. We concentrate our attention in the emergence of invariance and provide some insights into the ultimate origin of…

Directivity of quantum walk via its random walk replica

- Physics
- 2022

Quantum walks (QWs) exhibit diﬀerent properties compared with classical random walks (RWs), most notably by linear spreading and localization. In the meantime, random walks that replicate quantum…

## References

SHOWING 1-10 OF 34 REFERENCES

Quantum walks on graphs

- Mathematics, PhysicsSTOC '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.

Quantum simulations of classical random walks and undirected graph connectivity

- Computer ScienceProceedings. Fourteenth Annual IEEE Conference on Computational Complexity (Formerly: Structure in Complexity Theory Conference) (Cat.No.99CB36317)
- 1999

This paper shows that space-bounded quantum Turing machines can efficiently simulate a limited class of random processes-random walks on undirected graphs-without relying on measurements during the computation, and demonstrates that the Undirected graph connectivity problem for regular graphs can be solved by one-sided error quantum Turing Machines that run in logspace and require a single measurement at the end of their computations.

An Example of the Difference Between Quantum and Classical Random Walks

- Physics, MathematicsQuantum Inf. Process.
- 2002

A general definition of quantum random walks on graphs is discussed and with a simple graph the possibility of very different behavior between a classical random walk and its quantum analog is illustrated.

Quantum computation and decision trees

- Computer Science, Mathematics
- 1998

This work devise a quantum-mechanical algorithm that evolves a state, initially localized at the root, through the tree, and proves that if the classical strategy succeeds in reaching level $n$ in time polynomial in $n,$ then so does the quantum algorithm.

From quantum cellular automata to quantum lattice gases

- 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…

A fast quantum mechanical algorithm for database search

- Computer ScienceSTOC '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) .

Faster mixing via average conductance

- MathematicsSTOC '99
- 1999

It is shown that in a convex body in IR, with diameter D, random walk with steps in a ball with radius δ mixes in O(nD/δ) time (if idle steps at the boundary are not counted).

Random walks, universal traversal sequences, and the complexity of maze problems

- Computer Science20th Annual Symposium on Foundations of Computer Science (sfcs 1979)
- 1979

Results are derived suggesting that the undirected reachability problem is structurally different from, and easier than, the directed version of NSPACE(logn), an affirmative answer to a question of S. Cook.

The Markov chain Monte Carlo method: an approach to approximate counting and integration

- Computer Science
- 1996

The introduction of analytical tools with the aim of permitting the analysis of Monte Carlo algorithms for classical problems in statistical physics has spurred the development of new approximation algorithms for a wider class of problems in combinatorial enumeration and optimization.

A probabilistic analysis of multidimensional bin packing problems

- BusinessSTOC '84
- 1984

Probabilistic analyses of two kinds of multidimensional bin packing problems: vector packing and rectangle packing are given and it is shown that the results can be extended to a wide class of distributions of the piece sizes.