  We study the computational power of unitary Clifford circuits with solely
magic state inputs (CM circuits), supplemented by classical efficient
computation. We show that CM circuits are hard to classically simulate up to
multiplicative error (assuming PH non-collapse), and also up to additive error
under plausible average-case hardness conjectures. Unlike other such known
classes, a broad variety of possible conjectures apply. Along the way we give
an extension of the Gottesman-Knill theorem that applies to universal
computation, showing that for Clifford circuits with joint stabiliser and
non-stabiliser inputs, the stabiliser part can be eliminated in favour of
classical simulation, leaving a Clifford circuit on only the non-stabiliser
part. Finally we discuss implementational advantages of CM circuits.
