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Prospects for quantum advantage in machine learning from the representability of functions

Sergi Masot-Llima, Elies Gil-Fuster, Carlos Bravo-Prieto, Jens Eisert, Tommaso Guaita

TL;DR

The paper introduces a representability framework that maps parametrized quantum circuits (PQCs) to the function spaces they can realize in machine learning tasks, emphasizing two key properties: efficient evaluation and efficient identifiability. It classifies PQCs into three nested classes reflecting classical simulability and surrogation, and analyzes how data-encoding, circuit depth, and architectural constraints influence this classification. By tying well-known dequantization techniques (tensor networks, stabilizer methods, Pauli back-propagation, and free-fermionic approaches) to function representability, the work shows that many practical PQCs fall into classes that admit efficient classical handling, while true quantum advantage is likely confined to the most general, non-dequantizable architectures (Class 3). It also discusses approximate randomized dequantization and average-case considerations, arguing that practical advantage depends on task-model pairs and that average-case concentration can undermine worst-case quantum advantages. Overall, the paper reframes quantum learning advantage in terms of the representability of the hypothesis space, guiding future PQC design toward architectures that resist dequantization and emphasizing the role of average-case analysis and surrogate models in realistic settings.

Abstract

Demonstrating quantum advantage in machine learning tasks requires navigating a complex landscape of proposed models and algorithms. To bring clarity to this search, we introduce a framework that connects the structure of parametrized quantum circuits to the mathematical nature of the functions they can actually learn. Within this framework, we show how fundamental properties, like circuit depth and non-Clifford gate count, directly determine whether a model's output leads to efficient classical simulation or surrogation. We argue that this analysis uncovers common pathways to dequantization that underlie many existing simulation methods. More importantly, it reveals critical distinctions between models that are fully simulatable, those whose function space is classically tractable, and those that remain robustly quantum. This perspective provides a conceptual map of this landscape, clarifying how different models relate to classical simulability and pointing to where opportunities for quantum advantage may lie.

Prospects for quantum advantage in machine learning from the representability of functions

TL;DR

The paper introduces a representability framework that maps parametrized quantum circuits (PQCs) to the function spaces they can realize in machine learning tasks, emphasizing two key properties: efficient evaluation and efficient identifiability. It classifies PQCs into three nested classes reflecting classical simulability and surrogation, and analyzes how data-encoding, circuit depth, and architectural constraints influence this classification. By tying well-known dequantization techniques (tensor networks, stabilizer methods, Pauli back-propagation, and free-fermionic approaches) to function representability, the work shows that many practical PQCs fall into classes that admit efficient classical handling, while true quantum advantage is likely confined to the most general, non-dequantizable architectures (Class 3). It also discusses approximate randomized dequantization and average-case considerations, arguing that practical advantage depends on task-model pairs and that average-case concentration can undermine worst-case quantum advantages. Overall, the paper reframes quantum learning advantage in terms of the representability of the hypothesis space, guiding future PQC design toward architectures that resist dequantization and emphasizing the role of average-case analysis and surrogate models in realistic settings.

Abstract

Demonstrating quantum advantage in machine learning tasks requires navigating a complex landscape of proposed models and algorithms. To bring clarity to this search, we introduce a framework that connects the structure of parametrized quantum circuits to the mathematical nature of the functions they can actually learn. Within this framework, we show how fundamental properties, like circuit depth and non-Clifford gate count, directly determine whether a model's output leads to efficient classical simulation or surrogation. We argue that this analysis uncovers common pathways to dequantization that underlie many existing simulation methods. More importantly, it reveals critical distinctions between models that are fully simulatable, those whose function space is classically tractable, and those that remain robustly quantum. This perspective provides a conceptual map of this landscape, clarifying how different models relate to classical simulability and pointing to where opportunities for quantum advantage may lie.

Paper Structure

This paper contains 32 sections, 4 theorems, 39 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Framework
  3. Supervised learning
  4. A taste of dequantization in learning
  5. Abstract classification of quantum circuits
  6. Classical representations of PQCs
  7. Quantum circuit families for machine learning
  8. Computational and resource-based constraints.
  9. Architectural constraints.
  10. Classical simulation of PQCs
  11. Tensor networks
  12. Stabilizer states
  13. Pauli back-propagation
  14. Dynamical Lie groups and free fermions
  15. Combinations of approaches
  16. ...and 17 more sections

Key Result

Proposition 1

The ERM task with respect to the larger hypothesis family $\mathcal{F}'$ can be solved efficiently on a classical computer.

Figures (6)

  • Figure 1: Overview of the proposed framework. (a) We establish a mapping between the structure of PQCs and the hypothesis families $f$ they generate. (b) Based on the classical evaluatability of $f$ and its identifiability from the circuit parameters, we categorize PQCs into three distinct classes. (c) This classification provides a perspective for addressing key open questions in the field, such as the potential for quantum advantage and the limits of efficient classical simulation.
  • Figure 2:
  • Figure 3: Hierarchy of efficient output function classes for PQCs. The nested structure illustrates the increasing generality of each class, denoted by varying color intensity.
  • Figure 4: Classification of specific PQC architectures within the functional landscape. Circuit types are mapped onto the background of output functions established in Fig. \ref{['fig:func_dec']}, with color coding corresponding to the distinct classes defined in Section \ref{['ss:classes']}. Note that circuits in Class \ref{['class:3']} give rise to classically-inefficient functions, and thus do not fit this Venn diagram.
  • Figure 5: A tensor diagram representation of the expression \ref{['eq:MPS-result']} in terms of the MPS/MPOs for $\rho_{\boldsymbol{i}}$, $S^{\alpha}_{\boldsymbol{i}, \boldsymbol{j}}$ and $O_{\boldsymbol{j}}$.
  • ...and 1 more figures

Theorems & Definitions (9)

  • Definition 1: Hypothesis family of parametrized quantum circuits
  • Proposition 1: Informal
  • proof : Proof sketch:
  • Proposition 2: Informal
  • proof : Proof sketch:
  • Proposition 1: Rigorously restated
  • proof
  • Proposition 2: Rigorously restated
  • proof