On the Computational Complexity of Blind Quantum Machine Learning

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 $D = \mathcal{O}(\omega n)$, where $\omega$ 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.