Universality of Many-body Projected Ensemble for Learning Quantum Data Distribution

TL;DR

The paper proves a universal approximation theorem for the Many-body Projected Ensemble (MPE), showing that any $n$-qubit pure-state distribution ${\mathcal{Q}}_t$ can be approximated within a 1-Wasserstein distance $W_1$ by a parameterized distribution ${\mathcal{Q}}_{\boldsymbol{\theta}}$ constructed via discretization and ancilla-assisted projections. It combines an $\varepsilon/2$-net discretization of the target distribution with an MPE-based projection to produce a trainable ${\mathcal{Q}}_{\boldsymbol{\theta}}$ whose $W_1$ error can be made as small as $\varepsilon$; key bounds rely on covering numbers ${\mathcal{N}}(\mathcal{M}, d, \delta) \le 5 D \ln(D) (1/\delta)^{2(D-1)}$ and a two-step construction using Lemmas on discrete approximation and projected ensembles. To address practical challenges like training efficiency and noise sensitivity, the authors introduce Incremental MPE, a layer-wise training scheme that decomposes the learning into $K$ cycles with hardware-efficient ansätze and fidelity-based loss, improving trainability on near-term devices. Demonstrations on a synthetic multi-cluster quantum distribution and a QM9-derived quantum dataset show that Incremental MPE can closely match target distributions while reducing parameter counts and mitigating barren-plateau-like behavior. The work provides a rigorous expressivity guarantee for quantum generative modeling of state distributions and offers a scalable path toward learning complex quantum data pertinent to quantum chemistry and materials science, while outlining limitations in sample complexity and hardware noise as directions for future research.

Abstract

Generating quantum data by learning the underlying quantum distribution poses challenges in both theoretical and practical scenarios, yet it is a critical task for understanding quantum systems. A fundamental question in quantum machine learning (QML) is the universality of approximation: whether a parameterized QML model can approximate any quantum distribution. We address this question by proving a universality theorem for the Many-body Projected Ensemble (MPE) framework, a method for quantum state design that uses a single many-body wave function to prepare random states. This demonstrates that MPE can approximate any distribution of pure states within a 1-Wasserstein distance error. This theorem provides a rigorous guarantee of universal expressivity, addressing key theoretical gaps in QML. For practicality, we propose an Incremental MPE variant with layer-wise training to improve the trainability. Numerical experiments on clustered quantum states and quantum chemistry datasets validate MPE's efficacy in learning complex quantum data distributions.

Figures (6)

• Figure 1: A scheme to construct a parameterized quantum distribution ${\mathcal{Q}}_{{\bm{\theta}}}$ approximating a target ${\mathcal{Q}}_t$ within error $\varepsilon$. (a) Form an $\varepsilon/2$-net ensemble to approximate ${\mathcal{Q}}_t$ within $\varepsilon/2$, with probabilities from (b) Voronoi partitioning. (c) Perform partial measurements of a single many-body wave function $\ket{\boldsymbol{\Phi}}$ on ancilla system $A$, yielding a projected ensemble in main system $M$. (d) Implement the approximation using an IQP circuit and controlled unitary circuits for the projected ensembles.
• Figure 2: Variation of the evaluation metric with changing circuit ansatz layers using the Incremental MPE framework to learn quantum distributions for (a) multi-cluster states and (b) molecular quantum states from a QM9 subset. Circle markers indicate individual trials, dotted lines show the mean over 20 trials, and shaded areas represent one standard deviation.
• Figure 3: The 1-Wasserstein distance between the generated ensemble and the true ensemble, varying with the number of training states $K$ in the Incremental MPE framework for learning the multi-cluster quantum distributions of $n$-qubit quantum states ($n=4,6$). The solid lines represent the average accuracy over 10 trials (with error bars).
• Figure 4: Evaluation metric variation with the number of incremental steps $K$ in the Incremental MPE framework for learning the multi-cluster quantum distributions. The solid lines represent the average accuracy over 10 trials (with error bars) where $L=20$ layers are trained at each step. The cross markers represent the standard training with $L=100$ layers without incremental steps. We plot the positions of the cross markers at step $k=5$ to illustrate that training 20 layers over 5 steps (a total of 100 layers) is significantly better than training 100 layers at once.
• Figure 5: The training loss functions (1-Wasserstein distance) of the Incremental MPE (red) and standard training (blue) for learning the multi-cluster quantum distributions of $n=6$ qubits. Here, we compare training Incremental MPE with $L=10$ layers over 20 incremental steps, and standard training with $L=100$ layers. The solid lines represent the median values at each epoch over 20 trials.

Theorems & Definitions (12)

• Definition 3.1: 1-Wasserstein Distance
• Theorem 4.1: Universality Approximation of the Many-body Projected Ensemble
• proof
• Lemma 4.2: Finite Ensemble Approximation
• proof
• Lemma 4.3: Approximate Probability Distribution
• proof : Proof Sketch
• Lemma 4.4: Exact Probability Distribution
• Lemma 4.5: Wasserstein Distance Bound for Projected Ensemble
• proof