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In this paper we extend general results obtained by V. Kac and J. Liberati,in "Unitary quasifinite representations of $W_\infty$", (Letters Math. Phys.,53 (2000), 11-27), for quasifinite highest weight representations of$\Z$-graded Lie algebras to ${1/2}\Z$-graded Lie superalgebras, and we applythese to classify the irreducible quasifinite highest weight modules of the Liesuperalgebra of quantum pseudo-differential operators.

Quasifinite representations of the Lie superalgebra of quantum pseudo differential operators