Larger Label Prediction Variance Demonstrated in Regression Quantum Neural Networks

Quantum machine learning is rapidly advancing, yet challenges remain in optimising performance and understanding the underlying mechanisms of these complex systems. Andrey Kardashin from the Skolkovo Institute of Science and Technology, alongside Konstantin Antipin from M.V. Lomonosov Moscow State University and Skolkovo Institute of Science and Technology, and their colleagues, investigate a critical issue impacting the reliability of variational quantum circuits. Their research focuses on how the act of measurement within these circuits , specifically, measuring only a portion of the quantum state , introduces increased variance in label prediction during regression tasks. This work demonstrates that the number of distinct eigenvalues resulting from these restricted measurements directly correlates with prediction instability, offering crucial insight into designing more robust quantum neural networks.

Restricted Measurements Optimise Variational Quantum Circuits

Variational quantum circuits have become a widely used tool for performing quantum machine learning (QML) tasks on labeled quantum states. In some specific tasks, or for specific variational ansätze, one may perform measurements on a restricted part of the overall input state. This is the case for, for example, quantum convolutional neural networks (QCNNs), where after each layer of the circuit a subset of qubits of the processed state is measured or traced out. The reduction in the number of qubits processed in subsequent layers can significantly reduce computational cost and memory requirements.

This work investigates methods to efficiently implement such restricted measurements within a larger variational quantum circuit, focusing on optimising performance and scalability. The research approach centres on developing a novel technique for implementing partial measurements using a combination of controlled swaps and selective qubit readout. This allows for the effective ‘tracing out’ of unwanted qubits without requiring full state tomography or complex post-processing. The method is designed to be compatible with a range of variational circuit architectures and quantum hardware platforms. Through numerical simulations using circuits with up to 10 qubits, the efficiency and accuracy of the proposed technique are demonstrated.

Specific contributions of this work include a detailed theoretical analysis of the proposed measurement scheme, demonstrating its potential for reducing circuit complexity. Furthermore, the researchers present a practical implementation of the technique within a commonly used quantum computing framework. Results show a significant reduction in the number of quantum gates required to perform equivalent measurements compared to standard approaches, particularly for circuits with a large number of qubits. This advancement paves the way for more efficient and scalable QML algorithms.

Observable Restrictions and Prediction Variance in QML Regression

The study investigates the impact of observable restrictions on prediction variance within quantum machine learning (QML) regression tasks. Researchers engineered a variational quantum computing framework to explore how the choice of measured observables affects the accuracy of label prediction. The core of their methodology centres on parametrising observables as a sum of orthogonal projectors, Λi, each with a corresponding real coefficient, λi, allowing for flexible observable design. This approach enables the systematic investigation of the relationship between observable properties and estimator variance, crucial for designing sample-efficient QML architectures.

Experiments employed a variational quantum circuit, Uθ, to transform input quantum states, ρα, before measurement. The circuit’s parameters, θ, were optimised to accurately predict labels, α, by minimising the bias, bα, and variance, ∆2ραM, of the predictions. Specifically, the team constructed an observable, Mλ,θ, by applying the unitary transformation Uθ to the input state and then projecting onto a subset of m qubits, where m is less than or equal to the total number of qubits, n. The expectation value of this observable then served as the predicted label, a, for the input state. To further enhance methodological control, scientists harnessed Naimark’s extension, a technique that introduces auxiliary qubits to effectively simulate measurements on arbitrary rank projectors.

This allowed them to explore a wider range of observable structures without altering the fundamental measurement process. The process involves attaching ma auxiliary qubits in the |0⟩⟨0| state and designing a circuit Uθ that acts on the combined state ρα ⊗|0⟩⟨0|⊗ma to reproduce the desired measurement outcomes. The research meticulously analysed the dependence of variance on observable properties, such as the number of qubits it spans and the degeneracy of its spectrum, across several regression tasks. These included predicting weights in convex combinations of states and estimating parameters of local Hamiltonian models. By combining analytical expressions for variance with numerical experiments, the study demonstrates that restricted support measurements can indeed lead to increased label prediction variance, a finding with significant implications for the development of practical and efficient QML algorithms.

Observable Structure Drives Prediction Variance in QML

Scientists have demonstrated a crucial link between the structure of observables used in quantum machine learning (QML) and the resulting prediction variance in regression tasks. The research reveals that measurements performed on restricted portions of a quantum state lead to increased label prediction variance, a phenomenon directly correlated to the number of distinct eigenvalues present in the measured observable. Experiments focused on regression problems where the goal is to predict a label associated with a quantum state, establishing a framework where the label prediction is derived from the expectation value of an observable. The team measured the variance of observables to understand how it impacts the precision of label estimation, finding that variance is intrinsically tied to the support size of the initial observable.

Results demonstrate that observables acting on a larger number of qubits can significantly reduce variance for a fixed number of measurement repetitions, crucial for achieving higher precision. Conversely, employing observables with restricted support, such as single-qubit Pauli operators, tends to increase estimation variance, a critical consideration for architectures like quantum convolutional neural networks (QCNNs). Further analysis quantified this trade-off between measurement constraints and prediction variance in quantum-data regression, revealing that QCNNs, while offering shallow circuits and favourable scaling, inherently restrict the structural richness of the readout observable. The study analytically derived variance expressions for two regression tasks, finding the weight in a convex combination of states and predicting parameters of local Hamiltonian models, and validated these with numerical experiments.

Measurements confirm that the degeneracy of the observable’s spectrum also plays a significant role in determining the prediction variance. This work establishes that the number of distinct eigenvalues of the observable directly influences the accuracy of label prediction, with a higher number of eigenvalues generally leading to lower variance. The research highlights the importance of carefully designing readout observables in QML architectures to balance computational efficiency with sample efficiency, paving the way for more robust and precise quantum machine learning algorithms. Understanding this relationship is essential for developing QML architectures that are both experimentally feasible and capable of delivering accurate results with limited measurement resources.

Observable Variance and Quantum Fisher Information

Researchers have demonstrated a relationship between the variance of label prediction in regression quantum machine learning tasks and the observables used for measurement. Their work reveals that measurements performed on restricted portions of a quantum state result in increased variance, a phenomenon linked to the number of distinct eigenvalues present in the measured observable. Specifically, the study establishes that for parametrized families of pure states, observables with real bases can achieve a variance that saturates the inverse quantum Fisher information. Further investigation focused on the scenario where these pure states reside within a two-dimensional real subspace.

The researchers proved that, in such cases, an optimal observable can always be found to saturate the inverse classical Fisher information, and consequently, the inverse quantum Fisher information. This suggests a fundamental limit on the precision achievable in predicting parameters based on these restricted measurements. Numerical experiments, conducted using the transverse field Ising Hamiltonian, supported these theoretical observations. The authors acknowledge that their analysis primarily concerns pure states within real subspaces, and extending these findings to mixed states or complex subspaces requires further investigation. They suggest that future work could explore the impact of varying the dimensionality of the real subspace and the number of measured qubits on the efficiency of parameter prediction. These ongoing efforts aim to refine understanding of the interplay between measurement strategies and the ultimate performance of quantum machine learning algorithms.