Geometric Approach to Digital Quantum Information

We present geometric methods for uniformly discretizing the N -qubit Hilbert space, producing the equivalent of the Platonic solids in 2 dimensional Hilbert space. The group of transformations leaving these uniform Hilbertian polytopes invariant known as the Clifford group provides a general language for describing discrete transformations on N qubits. Our geometric approach to discretization sheds light on the relationships between the discrete quantum states and the generalized π/2 rotations belonging to the Clifford group. We describe in detail the uniform Hilbertian polytope HN for one and two qubits.