Quantum margulis expanders

Received (received date) Revised (revised date) We present a simple way to quantize the well-known Margulis expander map. The result is a quantum expander which acts on discrete Wigner functions in the same way the classical Margulis expander acts on probability distributions. The quantum version shares all essential properties of the classical counterpart, e.g., it has the same degree and spectrum. Unlike previous constructions of quantum expanders, our method does not rely on non-Abelian harmonic analysis. Analogues for continuous variable systems are mentioned. Indeed, the construction seems one of the few instances where applications based on discrete and continuous phase space methods can be developed in complete analogy. Motivated by the prominent role expander graphs play in theoretical computer science [1], quantum expanders have recently received a great deal of attention [2, 3, 4, 5, 6, 7]. In this work, we report an observation which allows for the simple explicit construction of such quantum expanders. The method relies heavily on quantum phase space techniques: Once familiar with this techniques, the result is an almost trivial corollary of the analogous classical statement. We further discuss continuous analogues of quantum expanders, where again, phase space methods render this an obvious generalization. Hence, the present work can equally be regarded as the presentation of a simple quantum expander, as as a short exposition of the strengths of the phase space formalism as such. 1.1 Expanders Expander graphs turn up in various areas of combinatorics and computer science (for all claims made in this section, the reader is referred to the excellent survey article Ref. [1]). They often come into play when one is concerned with a property which " typically " holds, but defies systematic understanding. A simple example is given by classical error correction codes. One can show that a randomly chosen code is extremely likely to have favorable properties, but it seems very difficult to come up with a deterministic construction of codes