Quantum Perspectives on the Clifford Group

The Clifford group, a multiplicative group generated by the Hadamard, phase, and CNOT gates, defines a set of unitary operations which normalize the Pauli group. The finite structure of this group promotes a graph-theoretic description, known as a Cayley graph, with vertices indicating each group element and edges representing the particular generators that transform Clifford operations into each other. Acting with the Clifford group on a computational basis state generates the complete set of stabilizer states, the set of all n-qubit quantum states invariant under some 2ⁿ-element subset of the n-qubit Pauli group. Stabilizer states and stabilizer circuits, those quantum circuits exclusively composed of Clifford operations and stabilizer measurements, are famously known to be simulable on a classical computer. The Hilbert space of n-qubit stabilizer states also admits a natural description as a mathematical graph, known as a reachability graph, which can be constructed as a quotient space of the overall Cayley graph after identifying group elements which act trivially on each stabilizer state. When considering the action of some Clifford subgroup this quotient space separates into multiple disconnected subgraphs, which we term restricted graphs. We introduce this definition of reachability graphs as Cayley graph quotient spaces and demonstrate how operator identities on stabilizer states completely describe the identifications that produce restricted graphs. Motivated by an understanding of stabilizer state entropies, we provide a complete description of the Clifford subgroup generated by the Hadamard and CNOT gates acting on any pair of qubits in an n-qubit system, and present some physical insights gained from studying the associated set of restricted graphs.