Quantum fully homomorphic encryption (QFHE) promises secure delegated quantum computation but has been impeded by the prohibitive quantum resource demands of existing constructions. This paper introduces a unified framework that achieves an \textbf{exponential improvement} in efficiency by synergistically integrating three theoretical tools: \textbf{modular arithmetic programs (MAP)}, the \textbf{garden-hose model}, and \textbf{measurement-based quantum computation (MBQC)}. Our central innovation is a novel MAP tailored to the algebraic structure of Learning-with-Errors (LWE) decryption. Unlike generic approaches that incur exponential overhead, our MAP computes the inner product $\langle \boldsymbol{sk}, \boldsymbol{c} \rangle \bmod q$ by tracking a partial sum modulo $q$, requiring only $O(\log q)$ bits of state width. This yields branching programs of width $O(\log λ)$ and length $O(λ\log λ)$, thereby reducing the size of the essential quantum gadget from $O(λ^{2.58})$ to $O(λ\log^2 λ)$ EPR pairs -- a concrete improvement factor of $2^{15}$ to $2^{18}$ for standard security parameters. Critically, we demonstrate that LWE decryption is not a \textbf{symmetric function}, necessitating our specialized MAP design beyond prior symmetric-function optimizations. The framework provides a direct mapping from the MAP to an efficient gadget via the garden-hose model, with MBQC furnishing the deterministic control flow for homomorphic evaluation. The resulting QFHE scheme supports \textbf{fully classical clients}, relies solely on the \textbf{classical LWE assumption} (avoiding circular security or quantum hardness assumptions), and maintains compactness. This work dramatically lowers the quantum resource barrier for practical QFHE, paving the way for realistic privacy-preserving quantum cloud computing.

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