David Feder

We present a framework where the inherent physical imperfections of Gottesman-Kitaev-Preskill (GKP) states—specifically the finite-energy envelopes required for normalizability—are harnessed as a resource for universal quantum computation. Contrary to the standard view that treats the finite squeezing of GKP states solely as a source of error to be corrected, we demonstrate that ``Fock damping'' enables the implementation of non-Clifford gates using only Gaussian operations and homodyne detection. We prove a dichotomy: while ideal (infinite energy) stabilizer GKP states are restricted to Clifford operations under Gaussian operation and measurement, realistic (finite energy) states allow for projection onto non-Pauli eigenstates, including magic states. At experimentally realistic squeezing levels, our protocol yields magic states with fidelities sufficient for distillation with success probabilities exceeding 40\%. This resolves an open question regarding the universality of realistic GKP circuits without requiring auxiliary vacuum modes or resource greedy GKP error correction.

Graph states admit a clean graph-theoretic classification of local-Clifford equivalence: two graph states are equivalent under tensor products of single-qubit Cliffords if and only if their graphs are related by a sequence of local complementations. Graph states are themselves a special case of graph codes, in which a vertex subset selects a logical qubit encoded across the vertices of the underlying graph. We provide the analog of the local-Clifford story for the single-qubit-encoded case: an explicit graph-theoretic recipe for the action of every logical single-qubit Clifford on the graph-code descriptor and the encoded-state coefficients. Each logical Clifford is implemented at the physical level by a circuit of single-qubit Cliffords alone, with no two-qubit entangling gates. We discuss the calculus, its closure, its physical implementation, and several open directions.