Higher-genus GKP codes via geometric quantization of algebraic curves

Planar GKP codes can be viewed as bosonic encodings that emulate a system with a compact, toroidal phase space. Using the perspective of geometric quantization, we extend this phase-space viewpoint to compact Riemann surfaces of higher genus. As a concrete example, we instantiate our construction on the genus-two Bolza curve and characterize the resulting code space and logical operators. A key consequence of the higher-genus geometry is that the stabilizers and logical operators are generated by Gaussian squeezing operators, in contrast to the displacement operators of conventional planar GKP codes. We outline syndrome-extraction protocols for circuit QED and trapped-ion platforms using controlled-squeezing gates. Our framework provides a systematic route to constructing new continuous-variable codes grounded in algebraic geometry, with natural extensions to higher-dimensional algebraic varieties.