On the SU(1,1)-based stabilizer formalism

Abstract

This work is motivated by the geometry and symmetry of continuous-variable (CV) and open quantum systems. We describe a stabilizer formalism based on the noncompact group SU(1,1). In contrast to the Pauli stabilizer codes, which are finite and discrete, and the GKP code, which uses displacement stabilizers on a flat phase-space lattice, the SU(1,1) approach is naturally connected to hyperbolic geometry. Errors can be organized into elliptic, parabolic, and hyperbolic types according to the subgroup structure of SU(1,1). This provides new classes of stabilizer operations that go beyond the Pauli-Clifford setting and at the same time, can capture the encoding structure of the GKP code. The construction is preliminary, but it suggests a more general framework for building fault-tolerant codes tailored to continuous-variable systems. It is natural to conjecture that SU(1,1)-based stabilizers admit a coset-like decomposition, with elliptic, parabolic, and hyperbolic subgroups playing the role of error classes, in analogy with the Pauli case. This perspective offers a pathway toward defining logical operators and error classification in hyperbolic phase space.