Submitted manuscript, 435 KB, PDF document
Research output: Working paper
The convergence of unitary quantum random walks. / Belton, Alexander C. R.; Gnacik, Michal; Lindsay, J. Martin.
2014.Research output: Working paper
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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 -