We study variational quantum distribution learning through a hierarchy of Walsh--Fourier approximations on the Boolean cube. At each level, a selected set of target Fourier coefficients defines a spectral truncation, which is projected onto the probability simplex and used as the target of a quantum circuit Born machine. Parameters learned at one level initialize the next through a warm-start map. We prove an end-to-end expected learning guarantee where the approximation term is determined by the omitted Fourier mass, while a normalized unbiased estimator yields an explicit statistical bound for empirical truncations. We then instantiate the abstract discrepancy conditions for total variation distance and relate the resulting distributional error to quantum-state fidelity. The total-variation specialization incurs the explicit factor 2n−1 under our normalized ℓ2 convention and is therefore informative only for sufficiently concentrated Fourier tails. The framework does not establish global trainability or eliminate barren plateaus; rather, it identifies the conditions under which low-to-high spectral training admits a approximation--estimation--optimization analysis.
Figures & tables
Figure 1: The figure tests the implementation of the target hierarchy. The hierarchy level k on the horizontal axis and ∥pδ,K−Tkpδ,K∥2 on the vertical axis. Show the numerical evaluation and the exact curve for several values of δ , with n , K , and α fixed.
Figure 2: (Left) Convergence of the best achieved loss versus iteration for the Naive, ADAM, and Hierarchical methods. The faint lines represent the raw loss, while the solid lines indicate the cumulative minimum loss. (Top Right) Magnitude of the normalized gradient ∥∇new∥ at the expansion point, comparing the Naive and Hierarchical approaches. (Bottom Right) The relationship between the hierarchy level and the trap density, showing how the spectral truncation affects the density of the approximation space via the hierarchy level.
Figure 3: Mean normalized ℓ2 estimation error is shown as a function of the sample size m on logarithmic axes, with n , K , k , α , and δ held fixed. We compare the raw truncation error, ∥Tkp−pk∥2 , with the error after projection onto the probability simplex, ∥Tkp−ΠΔ(pk)∥2 . shaded intervals denote confidence intervals over independent data samples. The m−1/2 reference line highlights the expected Monte Carlo scaling.
Distributed quantum algorithms offer a promising pathway to scale variational quantum algorithms beyond the constraints of noisy intermediate-scale quantum hardware. However, existing approaches implicitly assume a trusted entanglement-sharing layer across quantum processors. We show that this assumption introduces a fundamental vulnerability: adversarial perturbations of shared entanglement induce structured gate-level noise that directly impacts quantum learning. We develop a framework that maps entanglement-level perturbations to gate-level noise via an explicit Kraus representation. To quantify their impact, we introduce Kraus expressibility, a metric that generalizes unitary expressibility to noisy quantum channels. We then establish a trade-off between Kraus expressibility and trainability of noisy quantum circuits through gradient variance analysis. Our analysis reveals that an adversary can manipulate Kraus expressibility to maintain sufficiently large cost gradients (avoiding barren plateaus) while systematically biasing optimization toward incorrect solutions. We validate these findings through numerical simulations, demonstrating adversarial degradation of expressibility and trainability.
Abhishek Sadhu, Sharu Theresa Jose
School of Computer Science, University of Birmingham · Edgbaston, Birmingham B15 2TT, United Kingdom
Variational quantum circuits have been central to many proposed near-term applications of quantum computing, but a growing body of evidence suggests that trainability and quantum advantage are fundamentally at odds: ansätze expressive enough to resist efficient classical simulation tend to exhibit barren plateaus, while structures that provably rule out barren plateaus typically render them classically simulable. We propose a stacked linear combination of unitaries (S-LCU) as a variational ansatz which provides a tunable trade-off between barren plateaus and classical simulability. Using a diagrammatic analysis, we bound the loss-landscape variance of the Free Fermion S-LCU, whose elements are fermionic Gaussian unitaries. We prove a variance lower bound of Ω(1/(nk3l)), with a simulation cost of O(k2ln3) using the best known classical algorithm, compared to a quantum gate complexity of only O(lkn2). The number of layers l serves as a single dial that trades computational complexity against the rate of cost concentration. This offers practitioners a systematic method for constructing ansätze with a complexity-trainability trade-off that best suits their application and hardware.
Nikhil Khatri, Stefan Zohren, Gabriel Matos
Machine Learning Research Group, University of Oxford, United Kingdom · 2Quantinuum, London, United Kingdom
Barren-plateau results have established exponential gradient suppression as a widely cited obstacle to the scalability of variational quantum algorithms. When and whether these results extend to a given objective has been addressed through loss-specific arguments, but a general structural characterization has remained open. We show that the objective itself admits a fixed-observable representation if and only if the loss is affine in the measured statistics, thereby identifying the exact boundary of the standard concentration-based proof template. Existing transfer results for non-affine losses achieve this reduction under additional assumptions; our characterization implies that such a reduction is not structurally available for a class of non-affine objectives, placing them outside the automatic reach of the existing proof template. Beyond the affine regime, a chain-rule decomposition reveals three governing factors -- model responsivity, loss-side signal, and transmittance -- and induces a loss-class dichotomy: bounded-gradient losses inherit suppression, while amplification-capable losses can in principle counteract it. In the exponentially wide setting, both classes fail, but for different structural reasons. When the interface is instead designed at polynomial width -- exposing coarse-grained statistics rather than individual bitstring probabilities -- the exponential-dimensional obstruction is relaxed and the dichotomy plays a genuine role. In a numerical demonstration on a charge-conserving quantum system, the amplification-capable objective produces resolved gradients several orders of magnitude larger than affine and inheriting baselines at comparable shot budgets. Over the tested interval, its scaling trend is statistically distinguished from the exponential trend of both alternatives. The boundary is affine; what lies beyond it is a representation-design problem.
Gordon Ma, Xiufan Li
Centre for Quantum Technologies, National University of Singapore, Singapore