Limits of quantum generative models with classical sampling hardness

TL;DR

This work investigates the trainability of quantum generative models through the lens of output-distribution properties, showing that anticoncentration—central to many quantum advantages—drives exponential loss concentration and renders models untrainable on average. By analytically characterizing three distribution families (product, pseudo-independent, and peaked pseudo-independent) and validating with numerical experiments, it demonstrates that only sparse or biased distributions can be trained effectively, while anticoncentrated models resist learning and verification. The study clarifies the tension between classical sampling hardness and practical trainability, and it highlights verification as a guiding constraint for identifying viable quantum-generated data channels. Overall, quantum advantage in generative modeling can still be achieved, but its source must be distinct from anticoncentration, with peaked and sparse distributions offering the most promising paths for practical impact.

Abstract

Sampling tasks have been successful in establishing quantum advantages both in theory and experiments. This has fueled the use of quantum computers for generative modeling to create samples following the probability distribution underlying a given dataset. In particular, the potential to build generative models on classically hard distributions would immediately preclude classical simulability, due to theoretical separations. In this work, we study quantum generative models from the perspective of output distributions, showing that models that anticoncentrate are not trainable on average, including those exhibiting quantum advantage. In contrast, models outputting data from sparse distributions can be trained. We consider special cases to enhance trainability, and observe that this opens the path for classical algorithms for surrogate sampling. This observed trade-off is linked to verification of quantum processes. We conclude that quantum advantage can still be found in generative models, although its source must be distinct from anticoncentration.

Figures (7)

• Figure 1: Schematic description for the proof of classical hardness on sampling tasks. The initial step is assuming the existence of a classical machine outputting samples from the distribution generated by the quantum device (to $\epsilon$ error). The samples are used as an oracle to approximate $p_C(x) = \left\vert \bra{x} C \ket 0\right\vert^2$, which encodes a #P-hard problem. Multiplicative estimation of this quantity would imply the collapse of the polynomial hierarchy, which is conjectured to be false. Hence, classical sampling cannot be possible.
• Figure 2: Concentration of the product distribution. The solid lines depict the quantity $\operatorname{Prob}_{}\left( p_C(x) \geq y \right)$, for different numbers of qubits $n \in [1, 26]$, the red circles highlight the values at $y = 2^{-(n + 1)}$, and the blue line is the upper bound from \ref{['prop.concentrationproduct']}. The relevant observation for concentration is that these red circles render exponentially vanishing values as $n$ increases.
• Figure 3: Connection between sampling tasks with quantum advantage and pseudo-independent distributions. Quantum models with classical sampling hardness have approximated analytical descriptions matching our pseudo-independent probability distributions with underlying Gamma distribution. IQP circuits exhibit similar behavior.
• Figure 4: Concentration of the pseudo-independent distribution for random states, i.e., those following a Beta distribution. The solid lines depict the quantity $\operatorname{Prob}_{}\left( p_C(x) \geq y \right)$, for different numbers of qubits $n \in [1, 26]$, the red circles highlight the values at $y = 2^{-(n + 1)}$, and the blue line is the approximation from \ref{['prop.anticoncentrationindependent']}. The relevant observation for concentration is that these red circles appear at constant values of $y$ independently of $n$.
• Figure 5: Average of the SD for the considered models. As expected, all of the averages decay exponentially with the number of qubits, with the exception of the peaked distributions. The analytical bounds are satisfied in all the considered examples. For the Pareto distribution, exponential decay is also observable, even though the analytical bound is meaningless.

Theorems & Definitions (66)

• Definition 1: Product distribution
• Proposition 1: Concentration of product distribution
• Definition 2: Pseudo-independent distributions
• Corollary 1
• Proposition 2: Anticoncentration of pseudo-independent distribution
• Definition 3: Pseudo-independent peaked distributions
• Proposition 3: Concentration of peaked pseudo-independent distribution
• Definition 4: Squared distance (SD) over probability distributions
• Definition 5: Squared population ${\operatorname{MMD}}^2$ JMLR:v13:gretton12a
• Proposition 4: SD of product distribution