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arXiv:2605.28690

Latent-Conditioned Parameterized Quantum Circuits as Universal Approximators for Distributions over Quantum States

Quoc Hoan Tran, Koki Chinzei, Yasuhiro Endo, Hirotaka Oshima

quant-phcs.LGSubmitted May 27, 2026• Announced May 28, 2026• 0 citations• Revised Jun 17, 2026 (v3)
v1 · May 27, 2026v2 · Jun 1, 2026v3 · Jun 17, 2026
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Abstract

Many applications in quantum simulation, quantum chemistry, and quantum machine learning require not a single quantum state but an ensemble of states characterizing the heterogeneity of a target system. Preparing such ensembles state-by-state is prohibitive in both variational and fault-tolerant settings, thereby motivating a generative modeling approach. We introduce latent-conditioned parameterized quantum circuits (LPQCs), a hybrid quantum-classical framework in which classical neural networks map a latent variable sampled from a prior distribution to the parameters of a parameterized quantum circuit. We prove that LPQCs are universal approximators for probability measures over density operators in the 1-Wasserstein distance, extending classical universal approximation theorems to the quantum-distribution setting. We additionally introduce a multimodal latent prior and a mixture-of-experts circuit architecture, and show empirically that the latent-conditioned parameterization alleviates the barren plateau problem during optimization, a behavior for which we provide rigorous partial guarantees. Numerical experiments validate the framework on a synthetic multi-cluster ensemble of mixed quantum states and on a QM9-derived ensemble of 3-D molecular structures. In these tasks, LPQC outperforms recent quantum generative baselines and matches the generation quality of a classical neural-network baseline, while requiring an output dimension that grows only linearly with the number of qubits rather than exponentially. By leveraging classical expressivity in the latent space, LPQCs offer a tractable route to quantum generative modeling.

Comments
21 pages, 11 figures (fix the proof and update appendix for barren plateaus analysis)

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Are multi-layer parameterized quantum circuits (MPQCs) more expressive than classical neural networks (NNs)? How, why, and in what aspects? In this work, we survey and develop intuitive insights into the expressive power of MPQCs in relation to classical NNs. We organize available sources into a systematic proof on why MPQCs are able to generate probability distributions that cannot be efficiently simulated classically. We first show that instantaneous quantum polynomial circuits (IQPCs), are unlikely to be simulated classically to within a multiplicative error, and then show that MPQCs efficiently generalize IQPCs. We support the surveyed claims with numerical simulations: with the MPQC as the core architecture, we build different versions of quantum generative models to learn a given multi-dimensional, multi-modal discrete data distribution, and show their superior performances over a classical Generative Adversarial Network (GAN) equipped with the Gumbel Softmax for generating discrete data. In addition, we address practical issues such as how to efficiently train a quantum circuit with only limited samples, how to efficiently calculate the (quantum) gradient, and how to alleviate modal collapse. We propose and experimentally verify an efficient training-and-fine-tuning scheme for lowering the output noise and decreasing modal collapse. As an original contribution, we develop a novel loss function (MCR loss) inspired by an information-theoretical measure -- the coding rate reduction metric, which has a more expressive and geometrically meaningful latent space representations -- beneficial for both model selection and alleviating modal collapse. We derive the gradient of our MCR loss with respect to the circuit parameters under two settings: with the radial basis function (RBF) kernel and with a NN discriminator and conduct experiments to showcase its effectiveness.
Abstract A hybrid quantum–classical approach to model continuous classical probability distributions using a variational quantum circuit is proposed. The architecture of this quantum generator consists of a quantum circuit that encodes a classical random variable into a quantum state and a parameterized quantum circuit trained to mimic the target distribution. The model allows for easy interfacing with a classical function, such as a neural network, and is trained using an adversarial learning approach. It is shown that the quantum generator is able to learn using either a classical neural network or a variational quantum circuit as the discriminator model. This implementation takes advantage of automatic differentiation tools to perform the optimization of the variational circuits employed. The framework presented here for the design and implementation of the variational quantum generators can serve as a blueprint for designing hybrid quantum–classical models for other machine learning tasks.
Quantum Neural Networks (QNNs) represent a compelling convergence of quantum mechanics and machine learning, yet their practical deployment remains constrained by noise, limited qubit counts, and the cost of full quantum execution. Hybrid quantum-classical (HQC) training offers a pragmatic path forward: parameterized quantum circuits (PQCs) process information in exponentially large Hilbert spaces while classical optimizers update parameters via gradient methods accessible on today's hardware. We present a modular HQC training framework, formalising the parameter-shift rule for hardware-compatible gradient estimation, characterising the barren-plateau phenomenon through a variance-decay analysis, and introducing noise-aware training strategies for NISQ devices. Empirical results from state-vector and shot-based simulation confirm measurable quantum advantage on quantum-structured tasks, with the phase classification benchmark improving +12.1% over a classical MLP baseline. The accompanying open-source library provides a clean abstraction between quantum simulation and classical optimisation, serving as a foundation for near-term quantum machine learning research.
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