Efficient Quantum Fully Homomorphic Encryption

TL;DR

This paper presents a highly efficient quantum fully homomorphic encryption scheme that significantly reduces quantum resource requirements by integrating modular arithmetic programs, the garden-hose model, and measurement-based quantum computation.

Contribution

It introduces a novel modular arithmetic program tailored for LWE decryption, achieving exponential efficiency improvements in quantum resource usage for QFHE.

Findings

Reduces quantum gadget size from O(λ^{2.58}) to O(λ log^2 λ) EPR pairs.

Supports fully classical clients using only classical LWE assumptions.

Provides a practical pathway for privacy-preserving quantum cloud computing.

Abstract

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 $O(\log \lambda)$ and length $O(\lambda \log \lambda)$, thereby reducing the size of the essential quantum gadget from $O(\lambda^{2.58})$ to $O(\lambda \log^2 \lambda)$ 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.