Quantum Machine Learning: Restricted Measurements and Prediction Variance

Andrey Kardashin (Skolkovo Institute of Science and Technology), Konstantin Antipin (M.V. Lomonosov Moscow State University & Skolkovo Institute of Science and Technology), and colleagues

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Quantum machine learning is rapidly advancing, yet challenges remain in optimising performance and understanding the underlying mechanisms of these complex systems. The researchers investigate a critical issue impacting the reliability of variational quantum circuits. Their work 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. They demonstrate 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 example, for 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 include:

• A detailed theoretical analysis of the proposed measurement scheme, showing its potential for reducing circuit complexity.

• A practical implementation of the technique within a commonly used quantum‑computing framework.

• Numerical results indicating a significant reduction in the number of quantum gates required to perform equivalent measurements compared with standard approaches, especially for circuits with many qubits.

These advancements pave 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 regression tasks. Researchers engineered a variational quantum‑computing framework to explore how the choice of measured observables affects label‑prediction accuracy. The methodology centres on parametrising observables as a sum of orthogonal projectors ( \Lambda_i ), each with a real coefficient ( \lambda_i ), allowing flexible observable design. This enables systematic investigation of the relationship between observable properties and estimator variance—crucial for designing sample‑efficient QML architectures.

Key elements of the approach:

• A variational quantum circuit ( U_\theta ) transforms input quantum states ( \rho_\alpha ) before measurement.

• Circuit parameters ( \theta ) are optimised to predict labels ( \alpha ) by minimising bias ( b_\alpha ) and variance ( \Delta^2_{\rho_\alpha M} ).

• An observable ( M_{\lambda,\theta} ) is constructed by applying ( U_\theta ) to the input state and projecting onto a subset of ( m \le n ) qubits. The expectation value of this observable serves as the predicted label ( a ).

To broaden the range of observable structures without altering the fundamental measurement process, the team employed Naimark’s extension, introducing auxiliary qubits to simulate measurements on arbitrary‑rank projectors. This involves attaching ( m_a ) auxiliary qubits in the (|0\rangle\langle0|) state and designing a circuit that acts on the combined state ( \rho_\alpha \otimes |0\rangle\langle0|^{\otimes m_a} ) to reproduce the desired measurement outcomes.

The research analytically and numerically examined how variance depends on observable properties such as support size and spectral degeneracy across several regression tasks (e.g., predicting weights in convex combinations of states and estimating parameters of local Hamiltonian models). The findings confirm that restricted‑support measurements can increase label‑prediction variance, a result with significant implications for practical, 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 and the resulting prediction variance in regression tasks. 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.

Key observations:

• Observables acting on a larger number of qubits can significantly reduce variance for a fixed number of measurement repetitions, enabling higher precision.

• Conversely, observables with restricted support (e.g., single‑qubit Pauli operators) tend to increase estimation variance—an important consideration for architectures like QCNNs.

• The trade‑off between measurement constraints and prediction variance was quantified for regression tasks such as predicting convex‑combination weights and local Hamiltonian parameters, with analytical variance expressions validated by numerical experiments.

The study also shows that the degeneracy of an observable’s spectrum plays a significant role: a higher number of distinct eigenvalues generally leads to lower variance. These insights highlight 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.

Observable Variance and Quantum Fisher Information

Researchers have established a relationship between label‑prediction variance in regression QML tasks and the observables used for measurement. Measurements on restricted portions of a quantum state increase variance, a phenomenon linked to the number of distinct eigenvalues of the measured observable.

For parametrised families of pure states, observables with real bases can achieve a variance that saturates the inverse quantum Fisher information. When these pure states reside within a two‑dimensional real subspace, an optimal observable can always be found that saturates both the inverse classical and quantum Fisher information, indicating a fundamental limit on the precision achievable with restricted measurements.

Numerical experiments using the transverse‑field Ising Hamiltonian support these theoretical observations. While the analysis currently focuses on pure states within real subspaces, extending the results to mixed states or complex subspaces remains an open challenge. Future work may explore how varying the dimensionality of the real subspace and the number of measured qubits affects parameter‑prediction efficiency, further refining our understanding of the interplay between measurement strategies and quantum‑machine‑learning performance.