Efficient Quantum Fully Homomorphic Encryption

2026-04-25

AI Summary

Existing quantum fully homomorphic encryption (QFHE) schemes remain impractical due to their prohibitive quantum resource overhead. This work proposes an efficient QFHE framework that integrates modular arithmetic programs (MAP), the garden-hose model, and measurement-based quantum computation (MBQC), relying solely on the classical learning-with-errors (LWE) assumption and enabling a purely classical client. The key innovation lies in a novel MAP tailored to the algebraic structure of LWE decryption, which overcomes limitations imposed by symmetric functions and reduces the size of the critical quantum gadget from O(λ²·⁵⁸) to O(λ log² λ) EPR pairs. Under standard security parameters, this advancement achieves a resource compression factor of 2¹⁵–2¹⁸ compared to prior constructions.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Learning to Search
• Knowledge Representation and Reasoning: Qualitative Reasoning

Application Category

• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs

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 $O(\log λ)$ 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.

Problem

• Quantum Fully Homomorphic Encryption
• quantum resource efficiency
• secure delegated quantum computation
• LWE decryption
• practical QFHE

Innovation

• Quantum Fully Homomorphic Encryption
• Modular Arithmetic Programs
• Learning-with-Errors
• Garden-Hose Model
• Measurement-Based Quantum Computation