  The representation theory of the Clifford group is playing an increasingly
prominent role in quantum information theory, including in such diverse use
cases as the construction of protocols for quantum system certification,
quantum simulation, and quantum cryptography. In these applications, the tensor
powers of the defining representation seem particularly important. The
representation theory of these tensor powers is understood in two regimes. 1.
For odd qudits in the case where the power t is not larger than the number of
systems n: Here, a duality theory between the Clifford group and certain
discrete orthogonal groups can be used to make fairly explicit statements about
the occurring irreps (this theory is related to Howe duality and the
eta-correspondence). 2. For qubits: Tensor powers up to t=4 have been analyzed
on a case-by-case basis. In this paper, we provide a unified framework for the
duality approach that also covers qubit systems. To this end, we translate the
notion of rank of symplectic representations to representations of the qubit
Clifford group, and generalize the eta correspondence between symplectic and
orthogonal groups to a correspondence between the Clifford and certain
orthogonal-stochastic groups. As a sample application, we provide a protocol to
efficiently implement the complex conjugate of a black-box Clifford unitary
evolution.
