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.

翻译：量子全同态加密（QFHE）有望实现安全的委托量子计算，但现有方案对量子资源的巨大需求阻碍了其发展。本文提出一个统一框架，通过协同整合三种理论工具：**模算术程序（MAP）**、**花园软管模型**和**基于测量的量子计算（MBQC）**，实现了效率的**指数级提升**。我们的核心创新在于针对学习带误差（LWE）解密的代数结构设计了一种新型MAP。与产生指数级开销的通用方法不同，该MAP通过追踪模$q$的部分和来计算内积$\langle \boldsymbol{sk}, \boldsymbol{c} \rangle \bmod q$，仅需$O(\log q)$比特的状态宽度。由此产生的分支程序宽度为$O(\log λ)$、长度为$O(λ\log λ)$，从而将核心量子模件的规模从$O(λ^{2.58})$对EPR对降低至$O(λ\log^2 λ)$——对于标准安全参数，具体改进因子可达$2^{15}$至$2^{18}$。关键的是，我们证明了LWE解密并非**对称函数**，这使得我们的专用MAP设计超越了此前对称函数优化的局限性。该框架通过花园软管模型将MAP直接映射为高效量子模件，而MBQC则为同态评估提供了确定性控制流。由此产生的QFHE方案支持**完全经典客户端**，仅依赖**经典LWE假设**（避免循环安全性或量子硬度假设），并保持了紧致性。本工作大幅降低了实用化QFHE的量子资源门槛，为现实中的隐私保护量子云计算铺平了道路。