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arXiv:2504.15410 · Physical Review Research

Verifiable end-to-end delegated variational quantum algorithms

Matteo Antonio Inajetovic, Petros Wallden, Anna Pappa

Published May 9, 2026• 0 citations
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Abstract

Variational quantum algorithms (VQAs) have emerged as promising candidates for solving complex optimization and machine learning tasks on near-term quantum hardware. However, executing quantum operations remains challenging for small-scale users because of several hardware constraints, making it desirable to delegate parts of the computation to more powerful quantum devices. In this work, we introduce a framework for delegated variational quantum algorithms, where a client with limited quantum capabilities delegates the execution of a VQA to a more powerful quantum server. In particular, we introduce a protocol that enables a client to delegate a variational quantum algorithm to a server while ensuring that the input, the output, and also the computation itself remain secret. Additionally, if the protocol does not abort, the client can be certain that the computation outcome is indeed correct. This work builds on the general verification protocol introduced by Fitzsimons and Kashefi [], tailoring it to VQAs. Our approach first proposes a verifiable protocol for delegating the quantum computation at each optimization step of a VQA and then combines the iterative steps into an error-resilient optimization process that offers end-to-end verifiable algorithm execution. We also simulate the performance of our protocol tackling the transverse field Ising model. Our results demonstrate that secure delegation of variational quantum algorithms is a realistic solution for near-term quantum networks, paving the way for practical quantum cloud computing applications.

DOI
10.1103/9yqy-8ch9

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Variational quantum algorithms (VQAs) have emerged as promising candidates for solving complex optimization and machine learning tasks on near-term quantum hardware. However, executing quantum operations remains challenging for small-scale users because of several hardware constraints, making it desirable to delegate parts of the computation to more powerful quantum devices. In this work, we introduce a framework for delegated variational quantum algorithms, where a client with limited quantum capabilities delegates the execution of a VQA to a more powerful quantum server. In particular, we introduce a protocol that enables a client to delegate a variational quantum algorithm to a server while ensuring that the input, the output, and also the computation itself remain secret. Additionally, if the protocol does not abort, the client can be certain that the computation outcome is indeed correct. This work builds on the general verification protocol introduced by Fitzsimons and Kashefi [], tailoring it to VQAs. Our approach first proposes a verifiable protocol for delegating the quantum computation at each optimization step of a VQA and then combines the iterative steps into an error-resilient optimization process that offers end-to-end verifiable algorithm execution. We also simulate the performance of our protocol tackling the transverse field Ising model. Our results demonstrate that secure delegation of variational quantum algorithms is a realistic solution for near-term quantum networks, paving the way for practical quantum cloud computing applications.
Variational quantum algorithms (VQAs) are considered as one of the most promising candidates for achieving quantum advantages on quantum devices in the noisy intermediate-scale quantum (NISQ) era. They have been developed for numerous applications such as image processing and solving linear systems of equations. The application of VQAs can be greatly enlarged if users with limited quantum capabilities can run them on remote powerful quantum computers. But the private data of clients may be leaked to quantum servers in such a quantum cloud model. To solve the problem, a novel quantum homomorphic encryption (QHE) scheme which is client-friendly and suitable for VQAs is constructed for quantum servers to calculate encrypted data. Then delegated VQAs are proposed based on the given QHE scheme, where the server can train the ansatz circuit using the client's data even without knowing the real input and the output of the client. Furthermore, a delegated variational quantum classifier to identify handwritten digit images is given as a specific example of delegated VQAs and simulated on the cloud platform of Original Quantum to show its feasibility.
Generative machine learning has shown remarkable results across a wide range of interdisciplinary tasks, motivating significant efforts toward Quantum Generative Models as a path to beyond-classical performance, given their capability of generating classically intractable distributions. However, comparatively little attention has been paid to securely delegating such computations to a powerful remote quantum server. In this work, we propose a blind and verifiable protocol for sampling that can be used for the inference phase of such quantum models. Such a protocol inherits blindness from Universal Blind Quantum Computing and achieves verifiability by exploiting known approaches based on trapification schemes. Alongside, to delegate the training phase, we discuss an extension of the previously proposed verifiable end-to-end Delegated Variational Quantum Algorithms (Inajetovic et al., Phys. Rev. Research, 2026). Beyond the usual Accept/Abort outcome characterizing verification protocols, the sampling protocol for inference also provides, even against a malicious server, an estimation of the amount of corrupted generated samples, i.e., a fine-grained feature describing the quality of the outcome connecting verification protocols and application-level benchmarking.
Delegated quantum computation enables a client with limited quantum capabilities to outsource computations to a more powerful quantum server while preserving correctness and privacy. Verification is crucial in this setting to ensure that the untrusted quantum server performs the computation honestly and returns correct results. A common verification method is the quantum cut-and-choose technique. Inspired by classical verification methods for two-party computation, the client uses the majority of the delegated rounds to test the server's honesty, while keeping the remaining ones for the actual computation. Combining this technique with other methods, such as quantum error correction, could help achieve negligible cheating probabilities for the server; however, such methods can impose significant overheads making implementations unfeasible for the near-term future. In this work, we investigate whether cut-and-choose can yield efficient and secure verifiable quantum computation without additional costly techniques. We find that verifiable delegated quantum computation protocols relying solely on cut-and-choose techniques cannot be secure and efficient at the same time.
Variational quantum algorithms (VQAs) can train parameterized quantum circuits via classical optimizers to find approximate solutions to some important problems. They can overcome the limitations of existing quantum technologies only allowing for a few qubits and small circuit depth and are considered as one of the most promising methods for achieving quantum advantages in the noisy intermediate-scale quantum (NISQ) era. In this paper, we propose secure delegated VQAs by utilizing quantum homomorphic encryption (QHE) for users with limited quantum power to delegate the task of running VQAs to remote quantum servers while still keeping the training data private. Firstly, a client-friendly QHE scheme that allows quantum servers to perform calculation on encrypted data is proposed to be suitable for VQAs. Then, delegated VQAs based on the given QHE scheme are presented, where servers can train the ansatz circuit using the encrypted data. Finally, a delegated variational quantum classifier to identify handwritten digit images is given as a specific example of delegated VQAs and simulated on the cloud platform of Original Quantum to show the feasibility. Secure delegated VQAs will provide significant technical support for future quantum cloud services.
Variational quantum algorithms (VQAs) have been considered to be useful applications of noisy intermediate-scale quantum (NISQ) devices. Typically, in VQAs, a parametrized ansatz circuit is used to generate a trial wave function, and the parameters are optimized to minimize a cost function. On the other hand, blind quantum computing (BQC) has been studied in order to provide a quantum algorithm with security by using cloud networks. A client with a limited ability to perform quantum operations hopes to have access to a quantum computer of a server, and BQC allows the client to use the server's computer without leakage of the client's information (such as input, running quantum algorithms, and output) to the server. However, BQC is designed for fault-tolerant quantum computing, and this requires many ancillary qubits, which may not be suitable for NISQ devices. Here, we propose an efficient way to implement the NISQ computing with guaranteed security for the client. In our architecture, only N+1N+1N+1 qubits are required, under an assumption that the form of ans\"atze is known to the server, where NNN denotes the necessary number of the qubits in the original NISQ algorithms. The client only performs single-qubit measurements on an ancillary qubit sent from the server, and the measurement angles can specify the parameters for the ans\"atze of the NISQ algorithms. The no-signaling principle guarantees that neither parameters chosen by the client nor the outputs of the algorithm are leaked to the server. This work paves the way for new applications of NISQ devices.
Abstract With the advent of cloud-based quantum computing, it has become vital to provide strong guarantees that computations delegated by clients to quantum service providers have been executed faithfully. Secure—blind and verifiable—delegated quantum computing (SDQC) has emerged as one of the key approaches to address this challenge, yet current protocols lack at least one of the following three ingredients: composability, noise-robustness and modularity. To tackle this question, our paper lays out the fundamental structure of SDQC protocols, namely mixing two components: the computation which the client would like the server to perform and tests that are designed to detect a server’s malicious behaviour. Using this abstraction, our main technical result is a set of sufficient conditions on these components which imply the security and noise-robustness of generic SDQC protocols in the composable Abstract Cryptography framework. This is done by establishing a correspondence between these security properties and the error-detection capabilities of the test computations. Changing the types of tests and how they are mixed with the client’s computation automatically yields new SDQC protocols with different security and noise-robustness capabilities. This approach thereby provides the desired modularity as our sufficient conditions on test computations simplify the steps required to prove the security of the protocols and allows to focus on the design and optimisation of test rounds to specific situations. We showcase this by systematising the search for improved SDQC protocols for bounded-error quantum polynomial-time ( BQP ) computations. The resulting protocols do not require more hardware on the server’s side than what is necessary to blindly delegate the computation without verification, and they outperform all previously known results.