Hierarchy of discriminative power and complexity in learning quantum ensembles

TL;DR

The paper introduces MMD_k, a hierarchy of distance measures between quantum state ensembles that trade discriminative power for sample efficiency. It proves that increasing the moment order $k$ enhances discriminative power and identifies a threshold at $k=N$ for full discrimination, with corresponding sample complexities $\Theta(N^{2-2/k})$ (for fixed $k$) and $\Theta(N^3)$ (at $k=N$), contrasted with $\Theta(N^2\log N)$ for quantum Wasserstein distance. The authors provide SWAP-test-based estimators and matching lower bounds, and demonstrate practical impact by using $\mathrm{MMD}_2$ as a loss in training QuDDPM to learn circular-state ensembles, while also addressing near-pure noisy states. The work offers principled guidance for loss-function design in quantum machine learning under measurement constraints and highlights a fundamental quantum-classical divide in ensemble learning. Overall, it delineates when higher discriminative power justifies extra sampling cost and opens avenues for non-integer $k$ and deeper links to Wasserstein metrics.

Abstract

Distance metrics are central to machine learning, yet distances between ensembles of quantum states remain poorly understood due to fundamental quantum measurement constraints. We introduce a hierarchy of integral probability metrics, termed MMD-$k$, which generalizes the maximum mean discrepancy to quantum ensembles and exhibit a strict trade-off between discriminative power and statistical efficiency as the moment order $k$ increases. For pure-state ensembles of size $N$, estimating MMD-$k$ using experimentally feasible SWAP-test-based estimators requires $Θ(N^{2-2/k})$ samples for constant $k$, and $Θ(N^3)$ samples to achieve full discriminative power at $k = N$. In contrast, the quantum Wasserstein distance attains full discriminative power with $Θ(N^2 \log N)$ samples. These results provide principled guidance for the design of loss functions in quantum machine learning, which we illustrate in the training quantum denoising diffusion probabilistic models.

Figures (8)

• Figure 1: Conceptual plot of the relationship between sample complexity scaling and moment-based discriminative power. As shown by points of MMD-$k$ (blue star), the scaling of sample complexity is faster with higher discriminative power. For MMD-$k$, to reach the full discriminative power, the sample complexity needed is $N^3$, while Wasserstein (orange triangle) needs $N^2 \log N$. This hierarchy formalizes a fundamental tradeoff between what properties of a quantum ensemble can be detected and how efficiently they can be learned from measurements.
• Figure 2: The circuit of SWAP test. $H$ is the Hadamard gate, $Z$ represents Pauli-Z measurements in the computation basis, the connected dot and crosses represent a controlled-NOT gate, with the dot indicating the control qubit. See Appendix \ref{['app: quantum circuit']} for the definition of the above gates.
• Figure 3: Numerical simulation of sample complexity. (a) Number of samples needed to estimate MMD-$k$ and Wasserstein as $N$ increases. Both axes are in logarithmic scale. (b) Scaling coefficient of MMD-$k$ with different values of $k$. The corresponding slope for Wasserstein is $2.29>2$ due to finite-size effects from the extra $\log(N)$ in Eq. \ref{['eq: sample compelxity wasserstein']}.
• Figure 4: Analysis of effect of loss functions and training performance of QUDDPM. (a) Distance metrics (scaled by log) between ensemble $S_0$ and the ensemble through diffusion process at step $t$, $S_t$ in generation of the circular state ensemble. The data set size of the ensemble is $\lvert \mathcal{S}\rvert=1000$, and the total number of steps is $T=60$. Due to finite samples, the vanishment is not exact. (b) deviation of generated states from unit circle in X-Z plane. The deviation $\langle Y \rangle^2$ for forward diffusion (red), backward training (blue), and backward test (green) are plotted. The shaded area shows the sample standard deviation.(c)(d)(e) Bloch visualization of the forward (c1)–(c3) and backward (d1)–(d3),(e1)–(e3) process.
• Figure 5: Schematic of QUDDPM. Reprinted from Ref. QuDDPM_PhysRevLett.132.100602. Copyright (2024) American Physical Society. Used with permission. The forward noisy process is implemented by a quantum scrambling circuit (QSC) in (a), while in the backward denoising process is achieved via measurement enabled by ancilla and parametrized quantum circuit (PQC) in (d). Subplots (b1)-(b5) and (c1)-(c5) present the Bloch sphere dynamics in generation of states clustering around $|0 \rangle.$

Theorems & Definitions (57)

• Definition 2.1: Ensemble of quantum states
• Definition 2.2: Equivalence of ensembles
• Definition 2.3: Discriminative power
• Definition 2.4: Discriminative power based on moments
• Definition 3.1: MMD-$k$ pseudo distance metrics
• Proposition 3.2
• Theorem 3.3: The hierarchy of discriminative power of MMD-$k$
• Theorem 3.4: The threshold for MMD-$k$ to reach full discriminative power
• Definition 5.1: Sample complexity
• Theorem 5.2: Sample complexity of Wasserstein distance