Homomorphic encryption (HE) delivers cloud-scale compute power with rigorous end-to-end data privacy. While classical HE schemes support encrypted search, analytics, and limited machine learning workloads, two critical gaps remain:

- Post-quantum security challenge: Most mature HE systems rest on RSA, elliptic-curve, or pairing assumptions–all vulnerable to Shor-style attacks once large, fault-tolerant quantum computers arrive.

- Quantum computing limitation: None of the widely deployed schemes allow a user to outsource quantum computations (e.g., variational kernels, error-correction gadgets) while keeping both the algorithm and the quantum data hidden.

Our research addresses both gaps simultaneously, building on the categorical HE framework for intuitionistic proofs and total functional programs introduced in Dr. @bengoertzel's paper on "Homomorphic encryption of intuitionistic logic proofs and functional programs: A categorical approach inspired by composite-order bilinear groups" and extending it to the quantum domain: