The results demonstrate a rigorous learning separation for a natural ML task based on Hamiltonian evolution, while building connections between quantum learning theory, quantum simulation, and QML.
Abstract
Given that quantum computers are naturally suited to simulate the behavior of quantum many-body systems, an immediate question arises: can one formulate physically motivated quantum machine learning (QML) tasks that exhibit learning separations? We address this problem by studying the learnability of quantum many-body dynamics from the perspective of probably approximately correct (PAC)-learning. Concretely, we devise a supervised learning problem where the training set consists of specifications of randomized stabilizer probe states, evolution times sampled uniformly from a polynomially large time interval $[0,T]$, coupled with expectation values of certain observables evaluated on the resulting time-evolved state under an unknown Hamiltonian. For this learning task, we provide an efficient quantum procedure whose training phase learns the underlying Hamiltonian from short-time training samples, and whose deployment phase combines Hamiltonian simulation with the classical shadows protocol to perform inference on a newly given data point. By contrast, the existence of $O(\mathsf{poly}(n))$-time instances ensures classical hardness: by embedding a $\mathsf{BQP}$-complete computation into the polynomially long time-dynamics of a low-intersection variant of the Feynman-Kitaev clock Hamiltonian construction, we show that, for a certain family of input distributions, no randomized classical polynomial-time algorithm can fulfill our learning condition, unless $\mathsf{BQP}\subseteq\mathsf{P/poly}$. Furthermore, we show that the classically hard instance maintains quantum learnability. We also give an interpretation of our results in learning-assisted certified quantum simulation. Taken together, our results demonstrate a rigorous learning separation for a natural ML task based on Hamiltonian evolution, while building connections between quantum learning theory, quantum simulation, and QML.
A quantitative resource-theoretic framework for delineating the boundary between classical simulation and learning, motivate resource measures linking quantum resources to learnability, and guide the design of learning-based algorithms for scalable quantum systems beyond the reach of direct classical simulation are established.
Xin-Biao Wang, Yuxuan Du, Dacheng Tao· 0 citations
We introduce Coherent Quantum Learning (CQL), a training framework for quantum learning models in which the model parameters are quantum degrees of freedom evolved under a Hamiltonian that encodes the loss function. Current quantum machine learning retains classical optimization: parameters are updated by a classical outer loop using gradient estimates from measurements, and quantum coherence has no role in the training dynamics, just as in any classical treatment of the same problem. In the quantum case, a parameter register initialized in superposition evolves unitarily, and probability amplitude concentrates near low-loss configurations through interference, without gradient computation or classical feedback. We give an explicit construction using block encodings and Hamiltonian simulation, applicable to arbitrary parameterized circuits. Numerical experiments on binary classification and interferometric phase estimation confirm that the evolved distribution peaks at the optimal parameters, matching gradient-based performance. The construction is compatible in principle with fault-tolerant implementations and extends to batched training via sequential Hamiltonian evolution.
Ignacio B. Acedo, Javier Gonzalez-Conde, Pablo Rodriguez-Grasa et al.· 0 citations
Learning continuous quantum dynamical trajectories—essential for understanding non-equilibrium phenomena in quantum chemistry and condensed matter physics—remains prohibitively expensive on quantum computers. Conventional data-driven surrogates treat observables as generic time series, ignoring the governing Schrödinger evolution, and consequently require an infeasible density of samples to resolve highly oscillatory dynamics. We first establish a fundamental information-theoretic lower bound: any incoherent learning protocol that measures independently prepared copies without quantum memory needs Ω(T/ϵ2) oracle queries, revealing a quadratic penalty in 1/ϵ that renders naive dense sampling impossible. To circumvent this barrier, we introduce physics-informed kernel ridge regression (PI-KRR), which encodes the Heisenberg equation as a differentiable constraint. By extracting time derivatives from Hamiltonian commutators at zero additional quantum cost, PI-KRR doubles the information density per simulation shot without violating quantum estimation limits. Furthermore, integrated with classical shadow tomography, our framework reconstructs trajectories for M local observables simultaneously with measurement overhead scaling only as O(logM). We establish robustness guarantees for NISQ devices: isolated measurement outliers are suppressed as 1/m with training size m, and systematic Hamiltonian miscalibrations yield only linearly bounded prediction errors. Numerical experiments on transverse-field Ising models demonstrate that PI-KRR resolves sharp features such as light-cone fronts with drastically fewer simulations than standard kernel methods, achieving up to two orders of magnitude lower mean absolute error. This establishes a practical protocol for compressing complex quantum dynamics into classical predictive models, bridging quantum simulation and machine learning for experimental quantum science.
You-Le Wang, Lei Zhang· Quantum Science and Technolo...· 0 citations
This work confronts the field's central question -- whether the quantum sampler beats classical selected configuration interaction -- and reports a carefully scoped negative, and flags learning from quantum experiments, whose classical sample-complexity lower bound is an unconditional theorem, as the one adjacent frontier where a quantum advantage is provable but not yet bridged to chemistry.
Quantum algorithms based on linear-system approaches for solving differential equations demand qubit and precision resources beyond near-term capabilities. To address these challenges, this work proposes a physics-informed quantum machine learning (PIQML) framework with hard constraint embedding, specifically designed for NISQ era. Within this framework, parameterized quantum circuits serve as machine learning models, where the input variable is encoded into a high-dimensional feature space via a Fourier feature map. Subsequently, to eliminate approximation errors in critical physical conditions, the solution is constructed through a rigorously designed function mapper that analytically enforces initial conditions as hard constraints. Crucially, we compute derivatives with respect to the input variable using the parameter-shift rule---a quantum native gradient evaluation technique that avoids classical discretization. Unlike generic loss functions that target abstract data patterns, our loss function focuses on the differential equation residual and reference data. This design ensures that the trained model not only approximates the data but also intrinsically satisfies the physical constraint expressed by the DE itself. Our method is validated on several differential equations, including highly oscillatory ones, demonstrating its capability to tackle challenging nonlinear dynamics. Results demonstrate that our quantum model successfully learns the solution, showing close agreement with a high-precision classical numerical benchmark.
A QNN based framework for black box Hamiltonian learning and quantum system emulation using full density matrix trajectory learning, which establishes a data-driven pathway from black box quantum system identification to physical quantum emulation, with potential applications in quantum digital twin modeling.
Ahmad Salmanogli· 0 citations
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