Scientists are increasingly focused on the challenge of generating complex data distributions. This is an important task for advancing the understanding of large numbers of systems. Fujitsu Research’s Quoc Hoan Tran, Koki Chinzai, and Yasuhiro Endo, along with Hirotaka Oshima and others, demonstrate the universality of the many-body projective ensemble (MPE) framework for quantum machine learning. Their work proves that MPE can approximate any distribution of pure states with a guaranteed level of accuracy, filling a major theoretical gap in the field and providing a rigorous foundation for expressive quantum models. Additionally, the team proposes an improved incrementally trainable variant of MPE, validated through numerical experiments on complex datasets, suggesting a path to practical application of this powerful technique.
Additionally, the team proposes an improved incrementally trainable variant of MPE, validated through numerical experiments on complex datasets, suggesting a path to practical application of this powerful technique. Scientists are investigating the universality of approximations in quantum machine learning (QML). They address the question of whether parameterized QML models can approximate arbitrary quantum distributions. Numerical experiments support these findings.
Quantum data generation using generative models
Scientists are increasingly turning to quantum machine learning (QML) to process quantum data obtained from quantum systems. The fundamental task of QML is to generate quantum data by learning the underlying distribution. This is essential for understanding quantum systems, synthesizing new samples, and advancing applications of quantum chemistry and materials science. However, extending classical generative approaches to quantum data poses significant challenges, as quantum distributions exhibit superposition, entanglement, and nonlocality, which are difficult to reproduce efficiently with classical models. Quantum generative models, such as quantum adversarial generative networks and quantum variational autoencoders, can be used to prepare a fixed single quantum state, but are inefficient for generating ensembles of quantum states because they require deep training of parameterized quantum circuits (PQCs).
Quantum denoising diffuse stochastic models offer a promising approach to smoothly interpolate between the target distribution and the noise using intermediate training steps, thereby enabling efficient training. However, the diffusion process requires high-fidelity scrambled random unitary circuits, which poses the implementation challenge of precise spatiotemporal control. In the noisy intermediate-scale quantum (NISQ) era, learning quantum data distributions faces significant hurdles, such as noise-induced errors, limitations in qubit connectivity, and optimization difficulties such as barren plains where gradients vanish exponentially with system size. Furthermore, achieving universality, which requires the ability of a model to approximate arbitrary quantum distributions to arbitrary precision, remains a significant theoretical and practical challenge.
These limitations highlight the need for innovative frameworks that combine theoretical universality guarantees with scalable and noise-tolerant training strategies. Although practical QML methods face problems such as classical simulability, barren plateaus, and high resource requirements in training (Appendix A 3), the universality theorem can be seen as complementary, ensuring that the model can in principle capture any distribution before optimizing to the hardware. Numerical experiments on clustered quantum states and computational chemistry datasets verify the effectiveness of the framework. Generative models are powerful tools for generating samples from a target distribution and estimating the likelihood of specific data points.
Researchers address the problem of learning the unknown quantum data distribution Qt over the pure states of n qubits, given a training dataset S = {|ψ0⟩, ., |ψN−1⟩} of N independent states sampled from Qt. The generative model is defined by a parameterized probability distribution Qθ implemented via PQC. Here, θ represents a trainable parameter (e.g., gate angle). The goal of training is to optimize θ such that Qθ approximates Qt as measured by the distance metric D(Qθ, Qt). Since it is often infeasible to directly compute D(Qθ, Qt), a dataset S = {| ψj⟩}j is sampled from Qθ to minimize the empirical distance D(S, S).
In the inference phase, the optimized parameters θopt are fixed and a new quantum state |ψ⟩∼Qθopt is generated for use in quantum simulations and data analysis. For a density operator ρ acting on a Hilbert space (see Appendix A 1), the trace norm is defined as ∥ρ∥1 = Tr p ρ†ρ. Here Tr denotes the trace operation and ρ† is the Hermitian conjugate of ρ. The trace distance between the two density operators ρ and σ is d(ρ, σ) = 1 2∥ρ−σ∥1. This metric captures the distinguishability of two quantum states and serves as a fundamental measure of quantum information theory. The 1-Wasserstein distance is used to compare ensembles of quantum states. This extends the trace distance to the probability distribution of the density operator.
Definition III0.1 (1-Wasserstein distance). Let P and Q be two probability measures (or ensembles) on the space of density operators. The 1-Wasserstein distance between P and Q is defined as the expected minimum trace distance between a pair of states sampled from the union of P and Q: W1(P, Q) = inf π∈Π(P,Q) E(ρ,σ)∼π 1 2∥ρ −σ∥1, (1) where Π(P, Q) is the joint with limits P and Q (joint probability measure) indicates the set of Ensembles of states can be generated from a single wavefunction by performing local measurements on parts of the entire system. Researchers consider dividing a many-body system into a subsystem M (containing nm qubits) and its complement system A (containing na qubits).
For integration, A is considered an auxiliary system [Fig0.1(c)]. Given a generator state |Φ⟩ that is a pure many-body wavefunction on the entire system A + M, local measurements are performed on A, usually on a computational basis. This produces different pure states |φ(zA)⟩M on M, each corresponding to a distinct measurement zA on A, for example, a bit string of the form |ψj⟩. These cells form a Voronoi partition that assigns each state to the nearest center in the δ-net. qj = Qt(Cj) is defined. This is the probability that Qt assigns to every pure state. The MPE framework (Lemma IV0.3 or Lemma IV0.4) constructs a class of projection ensembles P = {(pj, |ψj⟩)}N−1 j=0, identified as Qθ, such that there is a parameter θ∗ such that W1(Qθ∗, Q) ≤ε.
From the trigonometric inequality, W1(Qt, Qθ*) ≤ W1(Qt, Q) + W1(Q, Qθ*) ≤ε 2 + ε 2 = ε. The scaling of N and the number of auxiliary qubits are explained in more detail in the following lemmas. Lemma IV0.2 (Finite Ensemble Approximation). For any target quantum data distribution Qt over pure n qubit states and any ε 0, there exists a finite ensemble Q = {(qj, |ψj⟩)}N−1 j=0 such that the 1-Wasserstein distance satisfies W1(Qt, Q) ≤ε. N = N(Qt, d, δ) ≤5 · D ln(D) · (1/δ)2(D−1). (2) where D is a D-dimensional subspace of the complete Hilbert space C2n.
MPE ensures universal expressivity of state design and
This breakthrough provides a fundamental understanding of the capabilities of parameterized quantum models in approximating arbitrary distributions. The researchers then defined a Voronoi partition, creating measurable cells that assign each state to the nearest center in the δ-net, and defined qj as the probability assigned to each cell by the target distribution Qt. This split facilitated the construction of a discrete approximation to Qt, denoted by the ensemble Q = {(qj, |ψj⟩)}. The data show that the 1-Wasserstein distance between Qt and Q is bounded by W1(Qt, Q) ≤ ε/2, completing the proof of the universality of MPE.
The team then utilized the ε/2 cover method to construct a δ-net and applied MPE to scenarios where the target distribution Qt is unknown but samples are available for training. Specifically, Lemma IV0.3 establishes that δ(p, q) ≤ ε/2 with nm = na + ⌈log2(1/ε)⌉ qubits in the hidden system M. where na is the number of qubits in attached system A. A further refinement detailed in Lemma IV0.4 achieves exact probability matching (p = q) with the same number of parameters, albeit using complex values. The resulting projection ensemble P = {(pj, |ψj⟩)} is designed to approximate Q = {(qj, |ψj⟩)} such that the Wasserstein distance between P and Q is bounded by W1(P, Q) ≤ ε/2, as demonstrated in Lemma IV0.5. This study provides a rigorous theoretical basis and practical implementation for generating data from underlying distributions, with potential applications in various fields requiring accurate data modeling and simulation.
The universality and practical learning of MPE has shown great progress
Numerical experiments performed on both cluster states and the QM9 molecular dataset confirm the effectiveness of MPE in learning complex quantum data distributions. These discoveries lay a strong foundation for advances in quantum production modeling and may impact fields such as quantum chemistry and materials science. The authors acknowledge that the sample complexity of this method can scale exponentially depending on the inherent dimensionality of the data, and future work could focus on reducing this scaling through assumptions about the smoothness or symmetry of the target distribution. Future studies could also address the effects of incomplete gating on auxiliary aid measurements and consider extending the framework to encompass mixed-state distributions.
