Elliptic curves in continuous-variable quantum systems

Elliptic curves are planar curves which can be used to define an abelian group. The computation of this group's addition operation, as well as related operations such as the discrete logarithm, could be performed on quantum computing devices, potentially leading to better performance. Currently, however, thousands of logical qubits are required for elliptic curve group addition, putting this application out of reach for near-term quantum hardware. Here we give an algorithm for computing elliptic curve group addition (over the real numbers) using a single continuous-variable mode, based on weak measurements of a system with a cubic potential energy. This result could lead to improvements in the efficiency of elliptic curve operations using a quantum device.