On measurement-dependent variance in quantum neural networks

TL;DR

This work addresses how measurement locality in variational quantum regression affects label-estimation variance. By formulating the observable $M$ with tunable eigenstructure and optimizing over both $M$ and the variational circuit $U_\theta$, the authors connect variance to the eigenvalue degeneracy and to classical/quantum Fisher information, deriving analytical solutions for convex mixtures and identifying key regimes where variance can be minimized. They show that measuring more qubits (larger $m$) or using Naimark extensions to access higher-rank projectors can reduce variance, albeit at the cost of circuit locality and measurement resources, and they verify these findings with extensive numerical experiments across QCNN, HEA, Ising, Schwinger, and cluster Hamiltonians. The results provide concrete guidelines for designing quantum regression architectures that balance readout locality and sampling precision, with implications for QCNNs and other variational readout schemes.

Abstract

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, e.g., 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, and at the end of the network one typically measures a local observable. In this work, we demonstrate that measuring observables with restricted support results in larger label prediction variance in regression QML tasks. We show that the reason for this is, essentially, the number of distinct eigenvalues of the observable one measures after the application of a variational circuit.

Figures (9)

• Figure 1: Upper: Schematic representation of measuring the observable \ref{['eq:obs-par']} in an $n$-qubit state $\rho_\alpha$, with $m$ qubits being measured. Lower: Instead of measuring the $m$ qubits of $\rho_\alpha$, one can introduce $m_{\mathrm{a}}$ auxiliary qubits via the Naimark's extension as in \ref{['eq:naimark-variational']}, which allows obtaining the eigenprojectors of arbitrary ranks.
• Figure 2: Squared difference between the prediction $\mathsf{a}=\langle M_m^*\rangle_{\rho_\alpha}$ and the true parameter $\alpha$ (left) and the variance of the optimized observable $M_m^*$ (right) obtained via numerically solving \ref{['eq:ls_min']}. The training set is $\mathcal{T}=\{(\rho_{\alpha_i}, \alpha_i)\}_{i=1}^{10}$ with $\rho_\alpha$ being a state of $n=5$ qubits defined by \ref{['eq:lin_al']} and \ref{['eq:lincomb_ghz']}, and $\alpha$ are picked equidistantly in $[0, 1]$. Different colors indicate different numbers of measured qubits $m \in \{1,3,5\}$ in \ref{['eq:obs-par']}. The parametrized unitary $U_{\boldsymbol{\theta}}$ is represented by HEA described in Appendix \ref{['app:hea']}. In the right panel, the dashed lines show the analytical variances \ref{['eq:obs_opt-n']} and \ref{['eq:obs_opt-m']}, while the solid green line stands for the right-hand side of the bound \ref{['eq:var-qcrb']}.
• Figure 3: Numerical results for predicting the transverse field $h$ of the Ising Hamiltonian \ref{['fig:ising']} of $n\in\{3,4\}$ qubits. The observable $M_m^*$ is obtained via numerically solving \ref{['eq:ls_min']}. Left: Squared difference between the prediction $\mathsf{h}=\langle M_m^*\rangle_{\psi_h}$ and true $h$. Center: Variance of $M_m^*$ for the case of $n=3$ qubits with $m=1$ qubit measured. Right: Variance of $M_m^*$ for the case of $n=4$ qubits with $m \in \{1,4\}$ measured qubits. The training set is $\mathcal{T}=\{(|\psi_{h_i}\rangle, h_i)\}_{i=1}^{10}$ with $h$ picked equidistantly from $[0.05, 2]$. Different colors indicate different numbers of measured qubits $m$ in \ref{['eq:obs-par']}, as well as the number of qubits $n$ of the ground state. The dashed lines of the corresponding colors show the central part of the the bound \ref{['eq:var-qcrb']}, while the solid green line stands for right-hand side of it.
• Figure 4: An instance of two-layered HEA for $n=5$ qubits and $m\in\{1,3,5\}$ measured qubits. The operators in the blocks are Pauli rotations $R_\sigma(\theta_j) = e^{-i\theta_j\sigma}$ with $\sigma\in\{X,Z,ZZ\}$ being a Pauli string and the rotation angles $\theta_j$ omitted.
• Figure 5: Left: Quantum convolutional neural network (QCNN) used in this work, with convolutional and pooling blocks denoted as $C$ and $P$, respectivelly; note that there are convolutional blocks connecting the first and the last qubits within each layer. Right: Representation of the blocks in terms of quantum gates adapted from nagano2023quantum; here, $U_3$ are universal single-qubit rotations, and $R_\sigma(\theta_j) = e^{-i\theta_j\sigma}$ is a two-qubit rotation with $\sigma\in\{XX,YY,ZZ\}$. QCNN of this form is used for the Ising Hamiltonian in Appendix \ref{['app:sec:ising-numerics']}, and Schwinger Hamiltonian in Appendix \ref{['app:schwinger-numerics']}; for the latter, the convolutional blocks $C$ between the first and the last qubits are removed (except for the last one, before the measurement).

Theorems & Definitions (6)

• Remark
• Lemma 1: sagawa2020div
• Theorem 2: Bhatia_1997
• Theorem 3: Schur's Theorem Bhatia_1997
• Lemma 4
• proof