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arXiv:2604.19256

Quantum Homomorphic Encryption: Towards Practical and Private Computation on Untrusted Quantum Hardware

Jon Hernández-Bueno, Oscar Lage, Marivi Higuero, Jasone Astorga

quant-phSubmitted Apr 21, 2026• Announced Apr 22, 2026• 0 citations
v1 · Apr 21, 2026v2 · Sep 15, 2026
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Abstract

As quantum computing matures into a practical paradigm, the need for secure and private quantum computation on untrusted hardware becomes increasingly urgent. While classical fully homomorphic encryption has enabled computation over encrypted data in untrusted environments, a fully homomorphic and practically implementable quantum counterpart remains elusive. In this work, we propose a universal quantum homomorphic encryption (QHE) framework developed from the Quantum One-Time Pad (QOTP) scheme. Our approach (QOTPH) maintains information-theoretic security and supports a broad class of quantum operations on encrypted quantum states through a systematic set of homomorphic gate decompositions and key update rules. By leveraging the symmetric structure of QOTP and exploiting the transformation properties of quantum gates under Pauli encryption, we enable non-interactive homomorphic evaluation of arbitrary circuits expressible in the Clifford+T gate set, as well as controlled and parameterized operations relevant to variational quantum algorithms and delegated computation. We provide a formal specification of the proposed encryption model, detail its implementation procedure, and report the results obtained from both simulated environments and real quantum processors. Experimental validation demonstrates the correctness of the homomorphic operations and the preservation of key secrecy under circuit-level noise and real-device constraints. This work takes a step toward bridging the gap between theoretical quantum homomorphic encryption and practical realization on near-term quantum hardware, offering a scalable and symmetric cryptographic primitive for privacy-preserving quantum computation.

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14 pages, 9 figures, 4 tables

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As quantum computing matures into a practical paradigm, the need for secure and private quantum computation on untrusted hardware becomes increasingly urgent. While classical fully homomorphic encryption has enabled computation over encrypted data in untrusted environments, a fully homomorphic and practically implementable quantum counterpart remains elusive. In this work, we propose a universal quantum homomorphic encryption (QHE) framework developed from the Quantum One-Time Pad (QOTP) scheme. Our approach (QOTPH) maintains information-theoretic security and supports a broad class of quantum operations on encrypted quantum states through a systematic set of homomorphic gate decompositions and key update rules.
By leveraging the symmetric structure of QOTP and exploiting the transformation properties of quantum gates under Pauli encryption, we enable non-interactive homomorphic evaluation of arbitrary circuits expressible in the Clifford+T gate set, as well as controlled and parameterized operations relevant to variational quantum algorithms and delegated computation. We provide a formal specification of the proposed encryption model, detail its implementation procedure, and report the results obtained from both simulated environments and real quantum processors. Experimental validation demonstrates the correctness of the homomorphic operations and the preservation of key secrecy under circuit-level noise and real-device constraints.
This work takes a step toward bridging the gap between theoretical quantum homomorphic encryption and practical realization on near-term quantum hardware, offering a scalable and symmetric cryptographic primitive for privacy-preserving quantum computation.
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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.
Quantum homomorphic encryption (QHE) can allow clients to directly perform quantum computation on encrypted data with the assistance of a remote quantum server. However, existing QHE schemes often overlook the crucial property of verifiability which enables clients to validate the correctness of computation results provided by the server. In this paper, we propose a verifiable QHE scheme based on the universal quantum gate set {H,P,Toffoli}. At first, a specialized gadget is designed to eliminate the errors that may arise during the homomorphic evaluation of non-Clifford Toffoli gates in a non-interactive manner. Furthermore, the designed gadget is versatile and can be seamlessly integrated into an existing QHE scheme that implements quantum gates in another universal quantum gate set {H,T,CNOT} to homomorphically evaluate more quantum gates. Subsequently, a verifiable method is introduced to the QHE scheme based on {H,P,Toffoli} for the client to detect whether the server is honest during the homomorphic computation by employing three types of indistinguishable quantum circuits.
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.
Homomorphic encryption has giant advantages in the protection of privacy information. In this paper, we present a new kind of probabilistic quantum homomorphic encryption scheme for the universal quantum circuit evaluation. Firstly, the pre-shared non-maximally entangled states are utilized as auxiliary resources, which lower the requirements of the quantum channel, to correct the errors in non-Clifford gate evaluation. By using the set synthesized by Clifford gates and T gates, it is feasible to perform the arbitrary quantum computation on the encrypted data. Secondly, our scheme is different from the previous scheme described by the quantum homomorphic encryption algorithm. From the perspective of application, a two-party probabilistic quantum homomorphic encryption scheme is proposed. It is clear what the computation and operation that the client and the server need to perform respectively, as well as the permission to access the data. Finally, the security of probabilistic quantum homomorphic encryption scheme is analyzed in detail. It demonstrates that the scheme has favorable security in three aspects, including privacy data, evaluated data and encryption and decryption keys.
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.