- BELTON.GNACIK.LINDSAYcvgce.unitaryQRWs-
Submitted manuscript, 435 KB, PDF document

Research output: Working paper

Published

**The convergence of unitary quantum random walks.** / Belton, Alexander C. R.; Gnacik, Michal; Lindsay, J. Martin.

Research output: Working paper

Belton, A. C. R., Gnacik, M., & Lindsay, J. M. (2014). *The convergence of unitary quantum random walks*.

@techreport{c4badcaec1014a3f894833bee772bb37,

title = "The convergence of unitary quantum random walks",

abstract = "We give a simple and direct treatment of the convergence of quantum random walks to quantum stochastic operator cocycles, using the semigroup method. The pointwise product of two such quantum random walks is shown to converge to the quantum stochastic Trotter product of the respective limit cocycles. Since such Trotter products themselves reduce to pointwise products when the cocycles inhabit commuting subspaces of the system algebra, this yields an elementary approach to the quantum random walk approximation of the 'tensorisation' of cocycles with common noise dimension space. The repeated quantum interactions model is shown to fit nicely into the convergence scheme described.",

keywords = "quantum random walk, repeated interactions, noncommutative Markov chain, toy Fock space, quantum stochastic cocycle, series product, quantum stochastic Trotter product",

author = "Belton, {Alexander C. R.} and Michal Gnacik and Lindsay, {J. Martin}",

year = "2014",

month = apr,

day = "25",

language = "English",

type = "WorkingPaper",

}

TY - UNPB

T1 - The convergence of unitary quantum random walks

AU - Belton, Alexander C. R.

AU - Gnacik, Michal

AU - Lindsay, J. Martin

PY - 2014/4/25

Y1 - 2014/4/25

N2 - We give a simple and direct treatment of the convergence of quantum random walks to quantum stochastic operator cocycles, using the semigroup method. The pointwise product of two such quantum random walks is shown to converge to the quantum stochastic Trotter product of the respective limit cocycles. Since such Trotter products themselves reduce to pointwise products when the cocycles inhabit commuting subspaces of the system algebra, this yields an elementary approach to the quantum random walk approximation of the 'tensorisation' of cocycles with common noise dimension space. The repeated quantum interactions model is shown to fit nicely into the convergence scheme described.

AB - We give a simple and direct treatment of the convergence of quantum random walks to quantum stochastic operator cocycles, using the semigroup method. The pointwise product of two such quantum random walks is shown to converge to the quantum stochastic Trotter product of the respective limit cocycles. Since such Trotter products themselves reduce to pointwise products when the cocycles inhabit commuting subspaces of the system algebra, this yields an elementary approach to the quantum random walk approximation of the 'tensorisation' of cocycles with common noise dimension space. The repeated quantum interactions model is shown to fit nicely into the convergence scheme described.

KW - quantum random walk

KW - repeated interactions

KW - noncommutative Markov chain

KW - toy Fock space

KW - quantum stochastic cocycle

KW - series product

KW - quantum stochastic Trotter product

M3 - Working paper

BT - The convergence of unitary quantum random walks

ER -