Quantum automated learning with provable and explainable trainability

TL;DR

This work introduces quantum automated learning, where no variational parameter is involved and the training process is converted to quantum state preparation, and establishes an unconventional quantum learning strategy that is gradient-free with provable and explainable trainability.

Abstract

Machine learning is widely believed to be one of the most promising practical applications of quantum computing. Existing quantum machine learning schemes typically employ a quantum-classical hybrid approach that relies crucially on gradients of model parameters. Such an approach lacks provable convergence to global minima and will become infeasible as quantum learning models scale up. Here, we introduce quantum automated learning, where no variational parameter is involved and the training process is converted to quantum state preparation. In particular, we encode training data into unitary operations and iteratively evolve a random initial state under these unitaries and their inverses, with a target-oriented perturbation towards higher prediction accuracy sandwiched in between. Under reasonable assumptions, we rigorously prove that the evolution converges exponentially to the desired state corresponding to the global minimum of the loss function. We show that such a training process can be understood from the perspective of preparing quantum states by imaginary time evolution, where the data-encoded unitaries together with target-oriented perturbations would train the quantum learning model in an automated fashion. We further prove that the quantum automated learning paradigm features good generalization ability with the generalization error upper bounded by the ratio between a logarithmic function of the Hilbert space dimension and the number of training samples. In addition, we carry out extensive numerical simulations on real-life images and quantum data to demonstrate the effectiveness of our approach and validate the assumptions. Our results establish an unconventional quantum learning strategy that is gradient-free with provable and explainable trainability, which would be crucial for large-scale practical applications of quantum computing in machine learning scenarios.

Figures (2)

• Figure 1: Comparison between two different quantum learning paradigms. Upper panel: a sketch of quantum learning and data encoding schemes. Lower left panel: an illustration of gradient-based quantum learning. In this paradigm, the input datum $\bm{x}$ is usually first encoded into a quantum state $\left|\phi(\bm{x})\right\rangle$ and then fed into a parametrized quantum circuit $U(\bm{\theta})$. For a $k$-class classification task, one measures about $\log k$ qubits and obtains the gradient of a predefined loss function on a classical computer based on the measurement results. With the obtained gradient, one updates the variation parameters $\bm{\theta}$. This process is iterated a number of times until the loss function no longer decreases (the training process). At the inference stage, one inputs $\left|\phi(\bm{x})\right\rangle$ for an unseen sample $\bm{x}$ into the variational circuit with optimized parameters and then does the same measurement on $\log k$ qubits to make predictions. Lower right panel: a sketch of quantum automated learning. In this scenario, the input datum $\bm{x}$ is encoded into $U(\bm{x})$ and the training process is transferred into quantum state preparation, which in turn can be accomplished by a dissipation process with guaranteed convergence. At the inference stage, we evolve the prepared state $\left|\psi^*\right\rangle$ with $U(\bm{x})$ and then do the measurement to make a prediction for the unseen sample $\bm{x}$. The quantum automated learning approach involves no variational parameter and is inherently gradient-free, thus it is more scalable to large-scale practical quantum learning applications.
• Figure 2: Classify images in the Fashion MNIST dataset xiaoFashionMNISTNovelImage2017 using QAL.a, Example images from the dataset belonging to the classes "trouser" and "ankle boot". The pixel values of an image are encoded into the rotation angles of variational single-qubit gates in a quantum circuit. Here, CNOT and CZ denote the controlled-NOT and controlled-Z gates, respectively. b, Testing and training accuracy during the training process. We also plot the accuracy amplified by majority vote with multiple trials, see Supplementary Sec. II D for the details of the majority vote. c, The trade-off between testing accuracy and the success probability of post-selection. The success probability is analytically calculated from Eq. \ref{['Eq:density Tsteps']}, where we omit the higher-order term $O(T\eta^2)$. d, Spectrum of the Hamiltonian $H_S$ associated with the training dataset. e, Training performance under depolarizing noise with different two-qubit gate noise strengths. The single-qubit noise rate is set to one-tenth of the two-qubit noise rate. The green line (5‰) corresponds to the noise level of real-world superconducting quantum devices jinObservationTopologicalPrethermal2025. f, Illustration of state reusability: whenever the training loss drops below a threshold of 0.15, a prediction is made while training continues. The x-axis represents the number of steps required to recover the performance.

Theorems & Definitions (3)

• Theorem 1
• Theorem 2
• Theorem 3