# Random Routing and Concentration in Quantum Switching Networks

##### Abstract

Flexible distribution of data in the form of quantum bits or qubits
among spatially separated entities is an essential component of
envisioned scalable quantum computing architectures. Accordingly, we
consider the problem of dynamically permuting groups of quantum bits,
i.e., qubit packets, using networks of reconfigurable quantum
switches.
We demonstrate and then explore the equivalence between the quantum
process of creation of packet superpositions and the process of
randomly routing packets in the corresponding classical network. In
particular, we consider an <italic>n × n</italic> Baseline network for which we
explicitly relate the pairwise input-output routing probabilities in
the classical random routing scenario to the probability amplitudes of
the individual packet patterns superposed in the quantum output state.
We then analyze the effect of using quantum random routing on a
classically non-blocking configuration like the Benes network. We
prove that for an <italic>n × n</italic> quantum Benes network, any input
packet assignment with no output contention is probabilistically
self-routable. In particular, we prove that with random routing on the
first (log <italic>n</italic>-1) stages and bit controlled self-routing on the last
log <italic>n</italic> stages of a quantum Benes network, the output packet
pattern corresponding to routing with no blocking is always present in
the output quantum state with a non-zero probability. We give a lower
bound on the probability of observing such patterns on measurement at
the output and identify a class of 2<super><italic>n</italic>-1</super> permutation patterns for
which this bound is equal to 1, i.e., for all the permutation
patterns in this class the following is true: in <italic>every</italic> pattern
in the quantum output assignment all the valid input packets are
present at their correct output addresses.
In the second part of this thesis we give the complete design of
quantum sparse crossbar concentrators. Sparse crossbar concentrators
are rectangular grids of simple 2 × 2 switches or crosspoints,
with the switches arranged such that any <italic>k</italic> inputs can be connected
to some <italic>k</italic> outputs. We give the design of the quantum crosspoints for
such concentrators and devise a self-routing method to concentrate
quantum packets. Our main result is a rigorous proof that certain
crossbar structures, namely, the fat-slim and banded quantum crossbars
allow, without blocking, the realization of all concentration patterns
with self-routing.
In the last part we consider the scenario in which quantum packets are
queued at the inputs to an <italic>n × n </italic> quantum non-blocking
switch. We assume that each packet is a superposition of <italic>m</italic> classical
packets. Under the assumption of uniform traffic, i.e., any output is
equally likely to be accessed by a packet at an input we find the
minimum value of <italic>m</italic> such that the output quantum state contains at
least one packet pattern in which no two packets contend for the same
output. Our calculations show that for <italic>m</italic>=9 the probability of a
non-contending output pattern occurring in the quantum output is
greater than 0.99 for all <italic>n</italic> up to 64.

University of Maryland, College Park, MD 20742-7011 (301)314-1328.

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