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Is quantum advantage the right goal for quantum machine learning?
Maria Schuld, Nathan Killoran
TL;DR
The paper asks why practical quantum advantage in machine learning is so difficult to assess when classical algorithms are powerful and learning theory remains incomplete. It analyzes the limitations of current theoretical and empirical tools, surveys alternative research directions, and concludes that fundamental questions may be more productive until large-scale quantum benchmarks become available.
Problem
Quantum machine learning lacks sufficient theoretical and realistic empirical evidence for judging practical quantum advantage against powerful classical machine-learning algorithms.
Method
The perspective examines quantum machine learning through learning theory, quantum-kernel analysis, quantum-gradient research, and alternative questions beyond beating classical algorithms.
Results
The paper argues that current approaches can restrict analysis to biased model, dataset, and theoretical subsets, while empirical advantages are difficult to interpret and scale reliably.
Takeaways & Limitations
The authors advocate broadening quantum machine-learning research toward quantum-model building blocks, theoretical tools, and software that supports scalable experiments.
Abstract
from arXiv · showhide
Machine learning is frequently listed among the most promising applications for quantum computing. This is in fact a curious choice: Today's machine learning algorithms are notoriously powerful in practice, but remain theoretically difficult to study. Quantum computing, in contrast, does not offer practical benchmarks on realistic scales, and theory is the main tool we have to judge whether it could become relevant for a problem. In this perspective we explain why it is so difficult to say something about the practical power of quantum computers for machine learning with the tools we are currently using. We argue that these challenges call for a critical debate on whether quantum advantage and the narrative of 'beating' classical machine learning should continue to dominate the literature the way it does, and highlight examples for how other perspectives in existing research provide an important alternative to the focus on advantage.
I. WHY MACHINE LEARNING IS SUCH A CHALLENGING PROBLEM
The paper formulates machine-learning tasks mathematically through empirical risk minimisation to make the challenges of quantum machine learning explicit.
- I. WHY MACHINE LEARNING IS SUCH A CHALLENGING PROBLEM: The paper uses empirical risk minimisation as a mathematical framework for formulating machine-learning tasks.This framework is introduced to make the subsequent argument more explicit.
- I. WHY MACHINE LEARNING IS SUCH A CHALLENGING PROBLEM: The formalisation examines how a machine-learning task can be expressed as a mathematical problem.
- I. WHY MACHINE LEARNING IS SUCH A CHALLENGING PROBLEM: The framework provides technical material for analysing why machine learning is challenging to access from a quantum-computing perspective.
A. How to formalize learning
Supervised learning formalizes learning as predicting target labels from finite labeled samples drawn from an unknown input distribution. Its theoretical difficulty arises because the distribution and target function are unknown, and the associated integral is generally hard to compute.
- A. How to formalize learning: Learning is described as acquiring skills from examples, with supervised learning using ground-truth information to define when a problem is solved.
- A. How to formalize learning: In high dimensions, finite samples leave parts of the data space unobserved, so learning requires substantial structure in the distribution, model, or selection strategy.
- A. How to formalize learning: A supervised task consists of an input domain, label domain, input distribution, ground-truth mapping, finite labeled dataset, and loss function.
- A. How to formalize learning: The goal is to choose a model that performs well on the expected loss over the full data distribution, not merely on observed samples.
- A. How to formalize learning: Because the input distribution and target function are usually unknown, even the basic formalization is generally mathematically unsolvable.
B. Solving the problem in practice
In practice, machine learning replaces population-risk minimization with empirical risk minimization on finite data and studies whether this proxy generalizes. Test-set evaluation is useful but difficult to make implementation-independent, especially for quantum models on limited hardware.
- B. Solving the problem in practice: Empirical risk minimization solves a proxy problem by evaluating model performance on the finite dataset D.
- B. Solving the problem in practice: Learning theory studies guarantees connecting solutions of the empirical proxy with performance on the original distribution.
- B. Solving the problem in practice: Unseen-data performance is usually measured on a held-out test set, but high-quality benchmark results can depend on implementation details.
- B. Solving the problem in practice: Quantum machine-learning benchmarks are harder to interpret because current hardware supports only limited system sizes.
- B. Solving the problem in practice: The central aim of machine learning is generalization, which is non-trivial to formalize and measure, particularly when quantumness is included.
C. Deep learning turns learning theory upside down
Deep learning challenges the traditional link between perfect training fit and poor generalization. Explaining this evidence requires theories that account for model structure, optimization procedures, data distributions, and the additional role of quantum theory.
- C. Deep learning turns learning theory upside down: Large models can fit arbitrary functions perfectly while still generalizing beyond training data, even in the presence of noise.
- C. Deep learning turns learning theory upside down: This evidence challenges the older assumption that perfect interpolation necessarily indicates excessive information fitting.
- C. Deep learning turns learning theory upside down: A viable theory must describe trained solutions and data distributions, not only the model class, because training algorithms and data strongly affect observed phenomena.
- C. Deep learning turns learning theory upside down: Neural networks are mathematically unwieldy because they comprise long sequences of linear and nonlinear transformations.
- C. Deep learning turns learning theory upside down: Quantum machine-learning theory adds quantum theory as another moving part while providing little empirical access to realistic learning regimes.
II. A CRITICAL LOOK AT QUANTUM ADVANTAGE
Machine learning is difficult for quantum computers to improve because existing algorithms perform well, inputs are large, core problems are mathematically complex, and theory poorly explains model success. Consequently, current tools for investigating quantum advantage constrain and bias claims about practical quantum machine learning.
- Existing machine-learning algorithms perform well, making them challenging baselines for quantum algorithms.
- Large inputs, mathematically complex problems, and limited theoretical understanding make quantum improvements difficult to assess.
- Because benchmarks often guide machine-learning evidence rather than theory, current quantum-advantage tools substantially limit and bias claims about practical use.
A. Progress in quantum machine learning
Quantum machine learning gained momentum around 2013 and developed along two prominent paths: accelerating existing algorithms and using variational quantum circuits as models. Related work also studies quantum data, quantum classification, quantum agents, and quantum analogues of classical models.
- Quantum machine learning, defined here as using quantum computers for classical- or quantum-domain learning tasks, gained momentum around 2013.
- One major approach seeks speedups for subtasks such as matrix inversion, Gibbs sampling, singular value estimation, and search.
- A second approach uses parametrized or variational quantum circuits as machine-learning models trained with gradient-descent-type algorithms.
- Other research examines quantum-data sample complexity, quantum classification, learning agents, and quantum versions of Ising-based models.
B. Quantum advantage
Quantum machine-learning research is heavily framed around beating classical methods, but current theoretical and empirical tools do not support reliable conclusions about practical quantum advantage. Theoretical results often concern artificial settings or broad model families, while experiments use small datasets and specific comparisons whose scaling remains unclear.
- Quantum machine-learning work commonly frames advantage through speedups, sample counts, expressivity, generalization, optimization landscapes, or small-scale test errors.
- Published examples claim rigorous speedups, exponentially fewer experiments, exponential advantage, or up to a 68x enhancement over classical alternatives.
- Other results report comparable classical and quantum performance, zero average gradients for broad random-circuit classes, or equal sample complexity up to constant factors.
- The authors identify a deeper structural issue: available quantum-computing tools are insufficient for meaningful statements about practical quantum machine learning.
- Exponential speedups in artificially constructed settings may encode favorable problem structure into quantum circuits and say little about possible applications.
- Traditional speedup proposals require extreme assumptions about data loading and read-out, complicating fair comparisons with classical models.
- Average or worst-case properties of very large quantum model families do not preclude specific subclasses from having different statistical behavior.
- Empirical studies use specific classical models and necessarily small datasets, making structural advantages difficult to distinguish from choices of hyperparameters, benchmarks, and comparisons.
III. ALTERNATIVE RESEARCH AGENDAS
Quantum machine learning is not at a dead end despite the difficulty of demonstrating performance improvements over classical methods. Alternative perspectives can broaden investigation through quantum models, theory, and software for scalable experiments.
- Research beyond advantage includes quantum perceptrons, connections between quantum circuits and kernel methods, and gradient-based circuit training with automatic differentiation software.
A. Quantum perceptrons or the search for building blocks of quantum models
The search for quantum perceptrons need not center on reproducing classical learning success or proving quantum advantage. Alternative goals emphasize trainability, implementability, theoretical tractability, and identifying non-classicality.
- A perceptron is a basic neural-network building block defined by an input vector, trainable weights, and a nonlinear scalar function.
- Quantum perceptrons have largely been motivated by the success of classical perceptrons and the desire to transfer that success to quantum computing.
- An advantage-focused study would compare quantum and classical versions by runtime or learning-task performance, but convincingly enabling such an advantage has not yet been achieved.
- Alternative figures of merit include natural quantum analogues of nonlinear activations, efficient trainability, hardware simplicity, theoretical tractability, and identifiable quantumness.
- Rather than automatically adopting Pauli rotations as universal building blocks, quantum researchers may need abstractions that distill useful perceptron properties for quantum hardware.
B. Quantum kernels as a bridge between quantum computing and learning theory
Quantum kernel methods connect quantum data encoding and measurement models to classical kernel theory. This bridge supports both advantage studies and broader analyses of optimization, learning, and generalization.
- Quantum kernel methods interpret data encoding as a feature map, allowing many quantum circuits to be viewed as linear models in a data feature space.
- Encoding inputs as quantum states and measuring observables yields a model whose quantum states can be distinguished by measurement-defined hyperplanes.
- The trace is a Hilbert-Schmidt inner product, so the quantum model becomes a linear model with a feature map, weight vector, and inner product.
- In variational quantum models, optimizing a parametrized circuit before fixed measurement selects a measurement basis and therefore the discriminating hyperplane.
- Kernel theory rewrites these models using distances between encoded training states and the state of the input being classified.
- Quantum-kernel advantage must arise in kernel evaluation, while kernel theory also connects quantum circuits with neural tangent kernels, random Fourier features, and generalization analysis.
C. Quantum gradients and making quantum software ready for machine learning applications
Quantum-gradient research has expanded the ability to train parametrized circuits and integrate them into machine-learning software. Its value is framed around enabling applications and cross-pollination, not immediate algorithmic superiority.
- Parametrized quantum circuits represent task-specific function families whose parameters can be adjusted to minimize a loss function.
- Training quantum circuits is closely linked to gradient descent, which iteratively updates free parameters using loss-function gradients.
- Parameter-shift rules estimate derivatives by running the same circuit with each parameter shifted forward and backward by a fixed amount.
- Parameter-shift gradients are analytically exact in the stated setting and hardware-friendly, although less efficient than classical backpropagation.
- Quantum-gradient research prioritizes enabling machine-learning applications while supporting quantum-aware optimizers and exchanges between quantum computing and deep learning.
- Automatic differentiation lets differentiable quantum subroutines plug into larger hybrid pipelines and be optimized with deep-learning tools such as Momentum or Adam.
IV. MOVING FORWARD
The perspective advocates moving quantum machine learning away from an exclusive focus on beating classical algorithms. Until large-scale benchmarks exist, it recommends pursuing more fundamental questions despite pressures favoring quantum supremacy narratives.
- The authors argue that seeking quantum advantage restricts analysis to the few machine-learning problems tractable with current tools.
- Until quantum computers support large-scale benchmarks, asking more fundamental questions may be a productive use of research effort.
- A research-agenda shift requires adjustments in supervision, science journalism, company deliverables, and publication standards.