Quantum Machine Learning for Complex Systems

TL;DR

A structured overview of recent advances that bridge foundational quantum learning principles with real-world applications is provided, including variational quantum algorithms, quantum kernel methods, and neural-network quantum states, with emphasis on their applicability to complex quantum systems.

Abstract

Quantum machine learning (QML) is rapidly transitioning from theoretical promise to practical relevance across data-intensive scientific domains. In this Review, we provide a structured overview of recent advances that bridge foundational quantum learning principles with real-world applications. We survey foundational QML paradigms, including variational quantum algorithms, quantum kernel methods, and neural-network quantum states, with emphasis on their applicability to complex quantum systems. We examine neural-network quantum states as expressive variational models for correlated matter, non-equilibrium dynamics, and open quantum systems, and discuss fundamental challenges associated with training and sampling. Recent advances in quantum-enhanced sampling and diagnostics of learning dynamics, including information-theoretic tools, are reviewed as mechanisms for improving scalability and trainability. The Review further highlights application-driven QML frameworks in drug discovery, cancer biology, and agro-climate modeling, where data complexity and constraints motivate hybrid quantum-classical approaches. We conclude with a discussion of federated quantum machine learning as a route to distributed, privacy-preserving quantum intelligence. Overall, this Review presents a unified perspective on the opportunities and limitations of QML for complex systems.

Figures (14)

• Figure 1: A schematic of a Restricted Boltzmann Machine (RBM), a widely used neural-network quantum state. The learner network is represented by a bipartite graph $G = (V, E)$, consisting of a set of hidden neurons $\{ h_j \}_{j=1}^{p}$, shown in gray and associated with bias vector $\vec{b}$, and a set of visible neurons $\{ v_i \}_{i=1}^{n}$, shown in red and associated with bias vector $\vec{a}$. The inter-layer connections are encoded by the weight matrix $\vec{W}$, with elements $W_{ij}$, shown in blue. The full parameter set $\vec{X} = (\vec{a}, \vec{b}, \vec{W})$ is optimized during the training of the network.
• Figure 2: Trotterized quantum circuit implementing the quantum-enhanced proposal distribution. The proposal transition probability is defined as $P_{\mathrm{prop}}\!\left(\vec{v}^{(i+1)} \mid \vec{v}^{(i)}\right) = \left|\langle \vec{v}^{(i+1)} | U(\tau,\gamma) | \vec{v}^{(i)} \rangle\right|^2 ,$ where the unitary evolution operator is $U(\tau,\gamma) = e^{-i\tau\left[\gamma, h_1 + (1-\gamma), h_2\right]} .$ The surrogate Hamiltonian consists of an Ising term $h_1 = \sum_i \ell_i(\vec{X}),\sigma_z^{(i)} + \sum_{i,j} J_{ij}(\vec{X}),\sigma_z^{(i)}\sigma_z^{(j)},$ whose Gibbs state encodes the approximate probability distribution of the neural-network quantum state, and a transverse-field mixer $h_2 = \sum_i \sigma_x^{(i)},$ which induces transitions between configurations. The evolution is implemented via a first-order Trotter decomposition, alternating applications of $e^{-i h_1 \delta t}$ and $e^{-i h_2 \delta t}$ for $N_T$ Trotter steps. Measurements in the computational basis yield the proposed configuration $\vec{v}^{(i+1)}$.
• Figure 3: (left) Absolute spectral gap $\delta = \lambda_0 - \lambda_1$ of the transition matrix $T(\vec{v}^{(i+1)} \mid \vec{v}^{(i)})$ as a function of the number of visible neurons (n), for different proposal distributions labeled A–H. Classical proposals include (A) local single-spin-flip updates, (B) uniform random proposals, and (C) Haar-random proposals. Quantum-assisted proposals include (D) a quantum-averaged proposal, and (E–G) time-homogeneous (TH) quantum proposals with different evolution times $\tau$ at fixed mixing parameter $\gamma$. Case (H) corresponds to the same time-homogeneous quantum proposal implemented using a first-order Trotterized circuit. Inset shows the fitted decay slopes of $\delta$ versus (n) for each proposal. Quantum proposals (D–H) exhibit a markedly slower decay of the spectral gap compared to classical proposals (A–C), indicating improved mixing time and faster convergence to the stationary distribution. (right) Spectral gap of the transition matrix as a function of inverse temperature $\beta$ for different proposal distributions. Shaded regions indicate statistical uncertainty from averaging over multiple instances. The quantum proposal maintains a substantially larger spectral gap over a broad range of $\beta$, particularly in the low-temperature regime, indicating improved mixing properties compared to classical proposals.
• Figure 4: Benchmarking our algorithm for ground state learning in (a) Li , H molecule as a function of distortion in bond length between the Li and H atom (b) H2O molecule as a function of the distortion in bond angle between O and the two H atoms. For computations without ZVE (orange), the minimum energy value from the last few data points in the training protocol is used. For RBM + ZVE (blue), the y-intercept from the Zero variance extrapolation scheme is used. Both methods, especially RBM + ZVE, show good agreement with the exact CASSCI results (dashed gray line) across different bond lengths and bond angles.
• Figure 5: The upper bound(UB) and lower bound (LB) in the convex $I$–$\eta$ space for network G as described in Eqs. \ref{['eq:UB_eta']} and \ref{['eq:LB_eta']} respectively. Provided alongside in green is the conventionally known lower bound for any general bipartition of an arbitrary system wolf2008area. The lower bound (LB) for G is thus stricter than the known general bound. Representative learning trajectories of the network G are shown.