# Quantum random walk polynomial and quantum random walk measure

Quantum random walk polynomial and quantum random walk measure In the paper, we introduce a quantum random walk polynomial (QRWP) that can be defined as a polynomial $$\{P_{n}(x)\}$$ { P n ( x ) } , which is orthogonal with respect to a quantum random walk measure (QRWM) on $$[-1, 1]$$ [ - 1 , 1 ] , such that the parameters $$\alpha _{n},\omega _{n}$$ α n , ω n are in the recurrence relations \begin{aligned} P_{n+1}(x)= (x - \alpha _{n})P_{n}(x) - \omega _{n}P_{n-1}(x) \end{aligned} P n + 1 ( x ) = ( x - α n ) P n ( x ) - ω n P n - 1 ( x ) and satisfy $$\alpha _{n}\in \mathfrak {R},\omega _{n}> 0$$ α n ∈ R , ω n > 0 . We firstly obtain some results of QRWP and QRWM, in which case the correspondence between measures and orthogonal polynomial sequences is one-to-one. It shows that any measure with respect to which a quantum random walk polynomial sequence is orthogonal is a quantum random walk measure. We next collect some properties of QRWM; moreover, we extend Karlin and McGregor’s representation formula for the transition probabilities of a quantum random walk (QRW) in the interacting Fock space, which is a parallel result with the CGMV method. Using these findings, we finally obtain some applications for QRWM, which are of interest in the study of quantum random walk, highlighting the role played by QRWP and QRWM. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Quantum Information Processing Springer Journals

# Quantum random walk polynomial and quantum random walk measure

Quantum Information Processing, Volume 13 (5) – Jan 5, 2014
19 pages

/lp/springer-journals/quantum-random-walk-polynomial-and-quantum-random-walk-measure-QSTE1uQiIQ
Publisher
Springer Journals
Subject
Physics; Quantum Information Technology, Spintronics; Quantum Computing; Data Structures, Cryptology and Information Theory; Quantum Physics; Mathematical Physics
ISSN
1570-0755
eISSN
1573-1332
D.O.I.
10.1007/s11128-013-0724-4
Publisher site
See Article on Publisher Site

### Abstract

In the paper, we introduce a quantum random walk polynomial (QRWP) that can be defined as a polynomial $$\{P_{n}(x)\}$$ { P n ( x ) } , which is orthogonal with respect to a quantum random walk measure (QRWM) on $$[-1, 1]$$ [ - 1 , 1 ] , such that the parameters $$\alpha _{n},\omega _{n}$$ α n , ω n are in the recurrence relations \begin{aligned} P_{n+1}(x)= (x - \alpha _{n})P_{n}(x) - \omega _{n}P_{n-1}(x) \end{aligned} P n + 1 ( x ) = ( x - α n ) P n ( x ) - ω n P n - 1 ( x ) and satisfy $$\alpha _{n}\in \mathfrak {R},\omega _{n}> 0$$ α n ∈ R , ω n > 0 . We firstly obtain some results of QRWP and QRWM, in which case the correspondence between measures and orthogonal polynomial sequences is one-to-one. It shows that any measure with respect to which a quantum random walk polynomial sequence is orthogonal is a quantum random walk measure. We next collect some properties of QRWM; moreover, we extend Karlin and McGregor’s representation formula for the transition probabilities of a quantum random walk (QRW) in the interacting Fock space, which is a parallel result with the CGMV method. Using these findings, we finally obtain some applications for QRWM, which are of interest in the study of quantum random walk, highlighting the role played by QRWP and QRWM.

### Journal

Quantum Information ProcessingSpringer Journals

Published: Jan 5, 2014

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