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arXiv:2604.23490

Efficient Quantum Fully Homomorphic Encryption

Fengxia Liu, Zixian Gong, Kun Tian, Yi Zhang, Zhiming Zheng, Maozhi Xu

quant-phcs.CRSubmitted Apr 26, 2026• Announced Apr 28, 2026• 0 citations• Revised Jun 21, 2026 (v2)
v1 · Apr 26, 2026v2 · Jun 21, 2026v3 · Oct 13, 2026
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Abstract

Quantum fully homomorphic encryption (QFHE) enables arbitrary quantum computations on encrypted data, but prior constructions require prohibitive quantum resources--specifically, O(lambda^2) EPR pairs per T-gate evaluation using the Barrington-based approach (DSS16). This paper introduces a unified framework achieving exponential improvement over the generic Barrington-based approach in program length. The central innovation is a novel modular arithmetic program (MA-Program) tailored to learning with errors (LWE) decryption. We show that LWE decryption computes the inner product <sk,ct> mod q, a modular inner product that is NOT a symmetric function. Thus, prior symmetric-function optimizations (Sinha's O(n)-state branching programs) do not apply. Our MA-Program tracks partial sums modulus q with state space Z_q requiring O(log q) bits, yielding programs of state count O(lambda) with binary encoding O(log lambda) and length O(lambda log lambda). This reduces the quantum gadget size from O(lambda^2) to O(lambda log^2 lambda) EPR pairs. To achieve a fully classical client, we transfer all quantum resources (EPR preparation, Bell measurements, adaptive error correction) to the server via the MA-Program gadget framework. Clients only perform classical LWE key generation, Pauli key encryption under classical FHE, and no quantum operations; a layered key structure further eliminates circular security assumptions. For parallel computation, we adopt the MBQC framework with flow functions, supporting up to O(log lambda) parallel measurements per layer. This separates offline resource preparation from online adaptive measurement, enabling parallel processing while maintaining deterministic evaluation.

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32pages, 2figures

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Quantum fully homomorphic encryption (QFHE) enables arbitrary quantum computations on encrypted data, but prior constructions require prohibitive quantum resources--specifically, O(lambda^2) EPR pairs per T-gate evaluation using the Barrington-based approach (DSS16). This paper introduces a unified framework achieving exponential improvement over the generic Barrington-based approach in program length.
The central innovation is a novel modular arithmetic program (MA-Program) tailored to learning with errors (LWE) decryption. We show that LWE decryption computes the inner product <sk,ct> mod q, a modular inner product that is NOT a symmetric function. Thus, prior symmetric-function optimizations (Sinha's O(n)-state branching programs) do not apply. Our MA-Program tracks partial sums modulus q with state space Z_q requiring O(log q) bits, yielding programs of state count O(lambda) with binary encoding O(log lambda) and length O(lambda log lambda). This reduces the quantum gadget size from O(lambda^2) to O(lambda log^2 lambda) EPR pairs.
To achieve a fully classical client, we transfer all quantum resources (EPR preparation, Bell measurements, adaptive error correction) to the server via the MA-Program gadget framework. Clients only perform classical LWE key generation, Pauli key encryption under classical FHE, and no quantum operations; a layered key structure further eliminates circular security assumptions. For parallel computation, we adopt the MBQC framework with flow functions, supporting up to O(log lambda) parallel measurements per layer. This separates offline resource preparation from online adaptive measurement, enabling parallel processing while maintaining deterministic evaluation.
We present a new scheme for quantum homomorphic encryption which is compact and allows for efficient evaluation of arbitrary quantum circuits, as long as those circuits are of size, polynomial in the security parameter. Building on the framework of Broadbent and Jeffery (CRYPTO'15) and recent results in the area of instantaneous non-local quantum computation by Speelman (TQC'16), we show how to construct quantum gadgets that allow perfect correction of the errors which occur during the homomorphic evaluation of T\mathsf{T}T gates on encrypted quantum data. Our scheme can be based on any classical (leveled) fully homomorphic encryption (FHE) scheme and requires no computational assumptions besides those already used by the classical scheme. The size of our quantum gadget depends on the space complexity of the classical decryption function -- which aligns well with the current efforts to minimize the complexity of the decryption function. Our scheme (or slight variants of it) offers a number of additional advantages such as ideal compactness, the ability to supply gadgets “on demand,” circuit privacy for the evaluator against passive adversaries, and a three-round scheme for blind delegated quantum computation which puts only very limited demands on the quantum abilities of the client. A conference version of this paper appeared in Advances in Cryptology: Proceedings of the 36th International Cryptology Conference (CRYPTO 2016).
Quantum fully homomorphic encryption (QFHE) allows to evaluate quantum circuits on encrypted data. We present a novel QFHE scheme, which extends Pauli one-time pad encryption by relying on the quaternion representation of SU(2). With the scheme, evaluating 1-qubit gates is more efficient, and evaluating general quantum circuits is polynomially improved in asymptotic complexity. Technically, a new encrypted multi-bit control technique is proposed, which allows to perform any 1-qubit gate whose parameters are given in the encrypted form. With this technique, we establish a conversion between the new encryption and previous Pauli one-time pad encryption, bridging our QFHE scheme with previous ones. Also, this technique is useful for private quantum circuit evaluation. The security of the scheme relies on the hardness of the underlying quantum capable FHE scheme, and the latter sets its security on the learning with errors problem and the circular security assumption.
Cloud computing allows clients with limited computational resources to offload computations to more powerful remote servers. In this paradigm, homomorphic encryption (HE) schemes enable a server to run any computation on a client's encrypted data. These schemes are widely used in cloud computing protocols such as delegated computing, two-party secure computation, and zero-knowledge proofs. Quantum homomorphic encryption (QHE) aims to achieve the objectives of HE with quantum data and quantum circuits, enabling cloud quantum servers to compute on encrypted quantum data uploaded by clients. In this work, we consider a scenario where a client has access to a quantum “encryption/decryption device”, which allows the encryption, transmission, reception, and decryption of quantum states, but not universal quantum computation. In this setting, we provide a proof-of-concept software simulation of quantum homomorphic encryption. Our code implements the “EPR scheme” of Broadbent and Jeffery, which allows for the execution of universal quantum circuits by the server at the cost of requiring shared EPR pairs between the client and server. Our implementation explores the near-term viability of the EPR scheme. Perhaps unsurprisingly, our experiments indicate that the additional cost of homomorphic circuit evaluation is minor in comparison to the simulation cost of the quantum operations. Our simulation toolkit is implemented in Python and is open-source.
As quantum computing moves toward remote and cloud-based access, protecting quantum data during outsourced execution becomes important. Quantum one-time pad (QOTP) encryption provides information-theoretic hiding of quantum state, but evaluating gates on Pauli-encrypted data is nontrivial: Clifford operations allow classical Pauli-frame updates, whereas non-Clifford operations require key-dependent corrections. We introduce QOTPH, a client-side key-tracked compilation framework for encrypted quantum computation. The client retains the QOTP keys and transforms each operation into a corrected gate or decomposition. The resulting circuit is executed non-adaptively by the quantum server, which neither receives the keys nor performs key updates. We develop a unified Pauli frame compilation layer spanning Clifford operations, one- and two-qubit Pauli-generated rotations, explicit key-dependent parametrized controlled rotations, and recursively composed gates, and establish a global correctness result for circuits composed of the formally verified primitive rules. We also characterize the evaluator's view under a hidden logical circuit model, in which the QOTP frame, the uncompiled logical circuit, and the logical-to-physical instruction mapping remain client-side; the security of this model against various algorithms is also analyzed. The implementation is evaluated in Qiskit using noiseless simulation and IBM Quantum hardware, including in-circuit and local-decryption workflows. This work bridges the gap between algebraic QOTP key tracking and practical encrypted quantum computation by providing a unified, hardware-validated compilation framework for current gate-based quantum processors.
Homomorphic encryption is a cryptographic primitive that enables information processing on encrypted data. Such primitives are useful in a delegated computing setting in which a client delegates processing to a server, but does not trust the server fully with her information. Here, I discuss a way of performing quantum homomorphic encryption. A homomorphic encryption protocol comprises of four algorithms: random key generation, encryption, evaluation, and decryption. Our technique requires that a pair of commuting operators be used to implement the encryption and evaluation. The client randomly selects the former, while the server implements the latter. Then post-evaluation, if the client decrypts using the inverse of the encryption, he retrieves the input state with the desired computation implemented. The code space has to be chosen carefully so that non-trivial evaluations can be performed on the logical qubits. Since the encryption key is not known to the server, the randomness introduced by key generation hides some information about the encoded data despite it being put into the hands of the server. This framework is powerful because it gives a general description for implementing quantum homomorphic encryption. For example, families of commuting groups of operators are known to exist via the Schur-Weyl duality. However, it is limited by a no-go theorem that states that an exponential overhead is needed if arbitrary computation and perfect security are desired. Nonetheless, our protocols are still interesting as an application for near-term quantum computers, especially when the computational tasks require only low-depth circuits.
Suppose some data have been encrypted, can you compute with the data without decrypting them? This problem has been studied as homomorphic encryption and blind computing. We consider this problem in the context of quantum information processing, and present the definitions of quantum homomorphic encryption (QHE) and quantum fully homomorphic encryption (QFHE). Then, based on quantum one-time pad (QOTP), we construct a symmetric QFHE scheme, where the evaluate algorithm depends on the secret key. This scheme permits any unitary transformation on any nnn-qubit state that has been encrypted. Compared with classical homomorphic encryption, the QFHE scheme has perfect security. Finally, we also construct a QOTP-based symmetric QHE scheme, where the evaluate algorithm is independent of the secret key.