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arXiv:2610.17145

On the Computational Complexity of Blind Quantum Machine Learning

Rex Fleur, Tzula B. Propp, Sebastian Feld

quant-phSubmitted Oct 12, 2026• Announced Oct 13, 2026• 0 citations
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Abstract

Can quantum advantages in computational efficiency coexist with quantum advantages in privacy? In this paper, we construct and use MasonQ, a universal brickwork transpiler, to answer this question with a resounding yes. We approach this question by bridging two seemingly disparate paradigms: universal blind quantum computation (UBQC), based on measurement-based quantum computation, and quantum machine learning (QML) algorithms such as the Harrow-Hassidim-Lloyd (HHL) algorithm and quantum recommendation systems. We provide new upper bounds on the depth of UBQC brickwork graphs, establishing that $D = \mathcal{O}(ωn)$, where $ω$ is the circuit width and $n$ its computational gate complexity. This is enabled by our newly developed transpiler MasonQ, which maps arbitrary quantum circuits to computational brickwork graph states. Given an optimal Qiskit circuit as input, we prove optimality (up to logarithmic corrections) of the MasonQ transpiler in terms of asymptotic scaling of the total brickwork graph width and depth after decomposition into bricks, enabling systematic cost analyses of blind implementations in near-term experiments and asymptotic regimes. Leveraging MasonQ, we derive depth bounds for several representative blind quantum machine learning (B-QML) algorithms, including the quantum Fourier transform, the HHL algorithm, recommendation systems, and quantum transformers. Lastly, we analyze concrete resource estimates for near-term demonstrations of blind quantum machine learning. We propose a minimal B-QML implementation requiring only a $12$ qubit quantum server. To the best of our knowledge, MasonQ is the first functional and optimal universal circuit-to-brickwork-graph transpiler to implement the full brickwork transpilation process from first principles.

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Can quantum advantages in computational efficiency coexist with quantum advantages in privacy? In this paper, we construct and use MasonQ, a universal brickwork transpiler, to answer this question with a resounding yes. We approach this question by bridging two seemingly disparate paradigms: universal blind quantum computation (UBQC), based on measurement-based quantum computation, and quantum machine learning (QML) algorithms such as the Harrow-Hassidim-Lloyd (HHL) algorithm and quantum recommendation systems.
We provide new upper bounds on the depth of UBQC brickwork graphs, establishing that D=O(ωn)D = \mathcal{O}(ωn)D=O(ωn), where ωωω is the circuit width and nnn its computational gate complexity. This is enabled by our newly developed transpiler MasonQ, which maps arbitrary quantum circuits to computational brickwork graph states. Given an optimal Qiskit circuit as input, we prove optimality (up to logarithmic corrections) of the MasonQ transpiler in terms of asymptotic scaling of the total brickwork graph width and depth after decomposition into bricks, enabling systematic cost analyses of blind implementations in near-term experiments and asymptotic regimes.
Leveraging MasonQ, we derive depth bounds for several representative blind quantum machine learning (B-QML) algorithms, including the quantum Fourier transform, the HHL algorithm, recommendation systems, and quantum transformers. Lastly, we analyze concrete resource estimates for near-term demonstrations of blind quantum machine learning. We propose a minimal B-QML implementation requiring only a 121212 qubit quantum server.
To the best of our knowledge, MasonQ is the first functional and optimal universal circuit-to-brickwork-graph transpiler to implement the full brickwork transpilation process from first principles.
Simulating the Universal Blind Quantum Computation (UBQC) protocol on gate-based platforms requires updating each measurement basis from earlier outcomes within the same execution. Existing Measurement-Based Quantum Computation (MBQC) simulation tools typically handle this feed-forward outside a single executable quantum program, so they do not natively expose the fixed-graph measurement-by-measurement update order studied here. We address this gap by implementing the flow-based Pauli corrections and a simulation-level UBQC angle-update rule as in-circuit dynamic-circuit primitives in Qiskit. Applied to two-qubit Grover's algorithm on a custom 2×92\times 92×9 flow of 18 qubits, the resulting program achieves deterministic success under both plain MBQC and the UBQC layer. Under the depolarizing-noise setting considered here, the angle-blinding layer remains close to plain MBQC, with only a modest additional gap relative to the dominant MBQC-over-circuit overhead. To show that the construction is not tied only to this Grover-specific pattern, we also verify a G2,5G_{2,5}G2,5​ brickwork-state fragment with representative universal-gate primitives under the same client/server blinding view. Structural blindness for a full algorithmic brickwork resource remains future work, but within this scope the construction provides a concrete dynamic-circuit route toward fixed-graph, privacy-preserving delegated-quantum-computing simulation at the protocol-update level.
Blind Quantum Computation (BQC) enables clients to delegate quantum computations to a quantum server while maintaining the privacy of their data and algorithms, even when the server is untrusted. In this work, we extend BQC frameworks to Quantum Machine Learning (QML) by implementing a network of entangled clients and a quantum server. Specifically, we explore the integration of Variational Quantum Classifiers (VQC) and Quantum Convolutional Neural Networks (QCNNs) within this paradigm. Our proposed model allows clients to perform classical preprocessing and optimization locally while leveraging the quantum server for computationally expensive quantum tasks. The entanglement-based network is managed by a controller that dynamically allocates resources according to the BQC protocol, allowing secure and efficient execution. We present simulation results indicating the feasibility of this approach, including an analysis of network efficiency and resource consumption, alongside the F1 score on QML benchmark datasets.
Blind quantum computation is a cryptographic primitive that allows a limited-capability client to delegate its complex computation to a remote server without revealing its data and/or computation. This branch of quantum cryptography has been bifurcated into two distinct primitives, quantum homomorphic encryption (concerning the security of only data) and universal blind quantum computation (concerning the security of data and the computing algorithm). These primitives have immense applicability in problems like secure cloud computing, secure quantum variational algorithms, quantum federated learning, and secure multiparty computation. However, no software tools exist for the rapid prototyping of such protocols, hindering the academic interrogation for potential applications. In this paper, we describe the development of the first such library for transpiling circuits written in Qiskit to its blind counterpart, which can then be delegated in a client-server architecture without revealing the client's data and/or computation. The proposed library is designed in modular and reusable component layers, enabling easier scalability to newer BQC primitives and robustness against changes in underlying primitives. We show the implementation of these primitives to a blind variational quantum classifier for the IRIS dataset.
Universal blind quantum computation allows a client who has limited quantum abilities to delegate his or her private computation to an untrusted quantum server. The first universal blind quantum computation protocol was proposed by Broadbent, Fitzsimons, and Kashefi. In their work the computation resource is the so-called brickwork state, which can be constructed by a number of specific single qubits. The number of single qubits is in theory linearly related to the size of the quantum circuit. However, due to the fixed structure of the brickwork states, the actual number of qubits is usually far more than the linear size. In this work we mainly construct three improved brickwork states whose structures are no longer fixed, thus they can efficiently reduce the client's qubit consumption. Using those improved brickwork states we propose a class of efficient universal blind quantum computation protocols. In our basic protocol, to implement a single-qubit basic gate the client needs to prepare only two ancillary qubits, while in the original protocol it will consume four ancillary qubits. Making use of our improved brickwork states the qubit consumption of implementing a nonadjacent two-qubit gate such as a control-XXX gate can be sharply reduced.
Blind quantum computation (BQC) is a recent proposition in which a computation is performed on a server by a client in a manner that the server is kept blind about the input, the algorithm and the output of the computation. This model possesses a natural security, which makes it suitable for applications where privacy matters. In this paper we propose a framework for BQC which can be implemented by circuit models as well as measurement-based models of quantum computation. It is based on gate teleportation, a process in which by storing one-qubit gates in entangled pairs one can apply it later via teleportation. We elaborate on examples of such universal BQC (UBQC) protocols. Specially, we elaborate on UBQC for correlation-space measurement-based quantum computation.
Current cloud-based quantum processors offer access to advanced hardware hosted on a remote server, but do not guarantee data or algorithm privacy. Blind quantum computation provides information-theoretic privacy by enabling a client to execute an algorithm without disclosing information about either the task or the final result. Here, we execute a measurement-based blind quantum computation protocol on a superconducting processor comprising two flip-chip-bonded modules, one acting as a server and the other as a client. The server generates a two-dimensional cluster state and forwards it to the client. Using this resource, the client implements a universal gate set with only adaptive single-qubit rotations and measurements. To illustrate this approach, we execute a three-qubit instance of the Deutsch-Jozsa algorithm. We analyze the server's quantum state after each rotation of a measurement-based single-qubit gate to verify that negligible information about the computation is revealed to the server, consistent with the one-way flow of information that guarantees blindness. This proof-of-principle demonstration establishes key elements of blind quantum computation in superconducting-circuit architectures, indicating that intermediate-scale implementations of blind protocols may become feasible with realistic near-term improvements in gate fidelities.