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Supervised quantum machine learning models are kernel methods

Maria Schuld

arXiv:2101.11020v2quant-phstat.ML

TL;DR

The paper examines how to find optimal quantum models without relying on variational ansatz search. It reframes these models through kernel methods, showing that kernel-based training reaches better or equal minima while data encoding determines the cost-function minima.

  • Problem

    Variational training searches a restricted, generally non-convex subspace and may miss the globally optimal quantum measurement.

  • Method

    The paper rephrases supervised quantum models as kernel methods, using quantum-state feature spaces, RKHS representations, and kernel expansions of measurements.

  • Results

    Kernel-based training is guaranteed to find better or equal minima than variational circuit training; with hinge loss, the problem becomes a support vector machine with a quantum kernel.

  • Takeaways & Limitations

    Data encoding is the central ingredient determining the kernel and the minima preferred by regularised quantum-model training, with Fourier structure offering a possible route to kernel design.

Abstract

from arXiv · show

With near-term quantum devices available and the race for fault-tolerant quantum computers in full swing, researchers became interested in the question of what happens if we replace a supervised machine learning model with a quantum circuit. While such "quantum models" are sometimes called "quantum neural networks", it has been repeatedly noted that their mathematical structure is actually much more closely related to kernel methods: they analyse data in high-dimensional Hilbert spaces to which we only have access through inner products revealed by measurements. This technical manuscript summarises and extends the idea of systematically rephrasing supervised quantum models as a kernel method. With this, a lot of near-term and fault-tolerant quantum models can be replaced by a general support vector machine whose kernel computes distances between data-encoding quantum states. Kernel-based training is then guaranteed to find better or equally good quantum models than variational circuit training. Overall, the kernel perspective of quantum machine learning tells us that the way that data is encoded into quantum states is the main ingredient that can potentially set quantum models apart from classical machine learning models.

I. MOTIVATION

Quantum models share kernel methods’ high-dimensional feature-space structure, motivating a systematic kernel reformulation of supervised quantum machine learning. This perspective makes data encoding central and replaces variational-search difficulties with kernel-based optimisation trade-offs.

  • Quantum computing and kernel methods both process data mapped into high-dimensional spaces accessed through inner products or measurements.
  • The manuscript addresses the lack of a comprehensive review of the theoretical link between quantum computation and kernel theory.
  • Expressivity, optimisation, and generalisation are largely determined by the kernel, which is fixed by the data-encoding strategy.
  • Quantum models are linear models in density-matrix feature vectors, while their equivalent RKHS representation preserves observable model behaviour.
  • Typical optimal measurements admit kernel expansions in the training data, reducing the relevant parameters from general-measurement scale to M coefficients.
  • Kernel-based training is guaranteed to find better or equal minima than variational training, while requiring pairwise data-point distances.

III. QUANTUM COMPUTING, FEATURE MAPS AND KERNELS

Quantum data encoding acts as a feature map into a density-matrix space equipped with the Hilbert-Schmidt inner product. Inner products of encoded states define quantum kernels, providing the representation used to analyse quantum models as kernel methods.

  • A data-dependent quantum circuit encodes an input x into a quantum state, implementing a machine-learning feature map.
  • The paper uses density matrices ρ(x), rather than state vectors, as feature vectors so measurements define linear models in a Hilbert space.
  • The data-encoding feature space is the complex matrix space equipped with the Hilbert-Schmidt inner product.
  • Quantum kernels are formed from inner products of data-encoding quantum states, represented through quantities such as tr[ρ(x′)ρ(x)].
  • For a single-qubit Pauli-X embedding, the example preserves interval structure but is periodic and produces a squared cosine distance measure.

B. The data-encoding feature map gives rise to a kernel

Quantum data-encoding states define a positive-definite kernel through their Hilbert-Schmidt inner products. An equivalent vectorised feature-space formulation produces the same kernel, although it cannot represent mixed states.

  • Quantum kernels are inner products between data-encoding density matrices.They use the Hilbert-Schmidt inner product on the feature space.
  • The quantum kernel is positive definite because it is the product of a complex kernel and its conjugate.Products of kernels remain kernels, and the conjugate kernel is also positive definite.
  • Scalar Pauli-X rotation encoding provides a concrete single-qubit example of a quantum kernel.The embedding maps x to a density matrix generated by RX(x), after which the kernel is computed from state overlaps.
  • The vectorised feature map uses a state tensor-product with its complex conjugate and yields the same quantum kernel.Its inner product equals the absolute square of the original Hilbert-space inner product.
  • Vectorised density-matrix notation is less powerful because it cannot capture mixed quantum states.

IV. EXAMPLES OF QUANTUM KERNELS

Quantum data embeddings generate kernels resembling familiar classical kernels, but their input assumptions, resource costs, and similarity structures differ. Some embeddings offer simple kernels while others require substantial state-preparation procedures or impose strict similarity measures.

  • A. Data encoding that relates to classical kernels: Quantum embeddings can reproduce classical polynomial, Gaussian, or cosine-like kernels, sometimes up to an absolute square.The examples are summarised in Table I and illustrated using simulated quantum computations.
  • Basis encoding: Basis encoding maps binary strings to computational-basis states, producing a Kronecker-delta kernel and requiring O(n) qubits.The resulting similarity measure is very strict for the input space.
  • Amplitude encoding: Amplitude encoding produces the absolute square of the linear kernel, requires O(n) qubits, and uses arbitrary state preparation taking O(2^n) time.Its identity feature map is mainly useful for theoretical investigations that remove feature-map effects.
  • Rotation encoding: Rotation encoding maps each real feature into a qubit rotation, yielding a kernel related to the cosine kernel while requiring O(n) qubits.
  • Coherent state encoding: Coherent-state encoding maps real inputs into an infinite-dimensional feature space and produces a Gaussian kernel.Real vectors can be encoded into joint coherent states.

B. Fourier representation of the quantum kernel

A broad class of quantum embeddings generated by Hamiltonian evolutions yields kernels with Fourier structure. The encoding Hamiltonians determine the available frequencies, while repetition can increase basis complexity and reduce smoothness.

  • Most quantum kernels in this class arise from encoding inputs through Hamiltonian evolutions of the form e^(-itG).The theorem allows arbitrary unitary evolutions between encoding gates and covers a fairly general class of quantum models.
  • For integer eigenvalue differences, the quantum kernel is a multidimensional Fourier series.The Fourier frequencies are determined by the spectrum of the generating Hamiltonian.
  • Quantum kernels are not necessarily translation invariant unless the data-encoding coefficients satisfy additional constraints.Separate physical subsystems, such as rotation encoding, provide a case where translation invariance holds.
  • The connection between embeddings and Fourier basis functions may support kernel design or control of smoothness properties.
  • Repeating input encoding increases the Fourier basis functions and decreases kernel smoothness.Figure 8 illustrates this effect for one, two, and three repetitions on a single qubit.

V. QUANTUM MODELS AND REPRODUCING KERNEL HILBERT SPACES

Quantum models map data-encoding states into a feature space where measurements act as linear models. Their measurement functions can be represented using data-encoding states, connecting quantum-model training to kernel methods.

  • A. Quantum models are linear models in feature space: A quantum model computes the expectation of a measurement applied to a data-encoding quantum state.The framework includes most near-term supervised quantum machine-learning algorithms and many fault-tolerant proposals.
  • A. Quantum models are linear models in feature space: Variational quantum models introduce trainable measurement parameters by rotating the basis of a fixed measurement.An arbitrary single-qubit rotation followed by Pauli-Z measurement gives an expressive example whose output is independent of the third parameter.
  • A. Quantum models are linear models in feature space: Quantum models are linear models in the feature space of the data-encoding map.The model f(x) = tr[ρM] is linear in the density-matrix feature representation.
  • A. Quantum models are linear models in feature space: Any measurement can be replaced, on the encoded-data image, by a linear combination of data-encoding states with identical model outputs.The component orthogonal to the feature space contributes zero to every encoded state.
  • A. Quantum models are linear models in feature space: Optimal measurements for typical machine-learning cost functions can be expanded in the training data only.

B. The RKHS of the quantum kernel and the space of quantum models are equivalent

The RKHS of the quantum kernel is an alternative function-space representation of quantum models, and Theorem 4 establishes equivalence with linear models in the data-encoding feature space.

  • The RKHS therefore reproduces the observable behavior of quantum models while expressing them through functions of the data.
  • The RKHS is constructed from kernel functions κ(x,⋅) and their linear combinations, completed to include limits of Cauchy series.
  • The size of common quantum-model spaces and their RKHS is fundamentally limited by the generators of the data-encoding gates.
  • The kernel-induced feature map sends each input x to the function κ(x,⋅), making the RKHS an alternative feature space to the data-encoding feature space.
  • The reproducing property makes kernel evaluation recover f(x), so the kernel acts as the bounded evaluation functional on the RKHS.
  • Theorem 4 states that functions in the quantum-kernel RKHS are exactly linear models in the data-encoding feature space, and vice versa.

VI. TRAINING QUANTUM MODELS

Training quantum models is an optimisation problem over measurements, and the kernel perspective supplies results about their trainability and optimisation.

  • The remaining analysis studies trainability and optimisation, including the degrees of freedom of optimal measurements and the process of finding optimal models.

A. Optimising quantum models is equivalent to optimising over the RKHS

Regularised empirical risk minimisation for quantum models is equivalent to minimising the corresponding loss and RKHS norm over functions in the quantum-kernel RKHS.

  • Training quantum models minimises a cost function over measurements, formulated as regularised empirical risk minimisation.
  • The quantum-model prediction term can be replaced by the corresponding RKHS function because quantum models and RKHS functions are equivalent.
  • The measurement norm is equivalent to the norm of the corresponding RKHS function, preserving the regularisation term under the reformulation.
  • The resulting optimisation problem minimises regularised risk over the quantum-kernel RKHS and characterises training and its solutions.

B. The measurements of optimal quantum models are expansions in the training data

The representer theorem restricts optimal quantum models to expansions over the finite training set, providing a kernel-based form for optimal measurements.

  • The representer theorem expresses the risk-minimising RKHS function as a weighted sum of kernel evaluations between a new input and training data.
  • The optimal quantum model is therefore represented using only the finite training set rather than arbitrary points from the input domain.
  • For quantum models, the optimal measurement can be expanded in the training data, and this expansion determines the corresponding optimal model.
  • Variational circuits typically optimise only a subspace of the RKHS because their measurement operators are constrained by a circuit ansatz.
  • The kernel-based form guarantees a measurement with lower regularised empirical risk than the best variational-training solution.
  • Fully implementing the optimal measurement may require inverting operators on the feature space, while the coefficients can alternatively be computed classically and the kernel measured quantumly.

C. The kernel defines which models are punished by regularisation

The quantum kernel determines both the quantum-model function space and the functions penalised during regularised empirical risk minimisation. Kernel-based training expresses this optimisation as a finite-dimensional convex problem when the loss is convex.

  • Regularisation properties: Each kernel fixes which functions are penalised in regularised empirical risk minimisation through its associated regularisation operator.The operator extracts information from a model that is integrated in the norm and penalised during optimisation.
  • Regularisation properties: The data-encoding strategy determines the quantum kernel and therefore the regularisation properties of the resulting quantum models.Which encoding produces which regularisation property remains an open research question.
  • Convex optimisation: Quantum-model optimisation can be reduced to a finite-dimensional problem involving coefficients associated with the training data.The representer theorem supplies the finite-dimensional representation used in this reduction.
  • Convex optimisation: For convex losses, the resulting optimisation is convex and has a unique global minimum with established optimisation guarantees.The optimisation involves M degrees of freedom, where M is the number of training examples.
  • Support vector machines: A support vector machine is kernel-based training with hinge loss, so a quantum-kernel SVM finds the quantum model that minimises hinge loss.The quantum support vector machine is presented as a generic kernel-based training blueprint for quantum models.

VII. SHOULD WE SWITCH TO KERNEL-BASED QUANTUM MACHINE LEARNING?

Kernel-based training is guaranteed to access the globally optimal measurement, whereas variational training searches an ansatz-dependent subspace. Its main disadvantage is scaling with the number of training samples, although hardware overhead can make kernels faster for small applications.

  • Comparison with variational training: Kernel-based training is guaranteed to find the globally optimal measurement, while variational training may miss it because its ansatz subspace need not overlap the training-data subspace.This guarantee follows because the training-data subspace contains the globally optimal measurement.
  • Scaling: O(M^2) kernel training arises from computing pairwise distances between all training samples in the Gram matrix.The same distance computations are also required for predictions involving a new input.
  • Scaling: O(|θ|M) variational training can nearly match neural-network scaling when the parameter count grows sufficiently slowly with dataset size.If the ansatz parameter count grows linearly with the data, variational training also has quadratic circuit-evaluation scaling.
  • Practical comparison: Experiments with 10–20 parameters and about 100 data samples found kernel-based training faster because hardware gradient calculations add constant overhead.The reported comparison concerns small-scale applications.
  • Practical comparison: Variational training lacks a guarantee of optimality and involves high-dimensional non-convex landscapes plus the burden of choosing an effective ansatz.The manuscript concludes that the kernel perspective may significantly speed up current quantum machine-learning methods.
  • Fault-tolerant prospects: Quantum algorithms for training kernel models may offer lower scaling, but they likely require fully error-corrected quantum computers beyond near-term devices.The manuscript presents this as an in-principle possibility rather than a near-term capability.

Appendix A: Proof of Theorem 1

The appendix proves the data-encoding analysis by diagonalising encoding generators, absorbing basis changes into surrounding circuits, and organising terms by eigenvalue differences. It then relates quantum-kernel training to quantum linear-algebra routines.

  • Proof strategy: The encoding generator can be diagonalised without loss of generality, with the resulting basis changes absorbed into arbitrary circuits before and after encoding.The remaining derivation expands the circuit matrix products in the computational basis.
  • Proof strategy: The appendix organises circuit terms using multi-indices that select generator eigenvalues and their differences.The frequency set contains vectors formed from differences between eigenvalue vectors.
  • Example: The Pauli-X rotation example illustrates the general encoding construction after the spectral decomposition.The example sets the surrounding unitaries to identity before absorbing the decomposition basis changes.
  • Quantum training procedure: Quantum convex-optimisation routines can prepare a quantum state representing the M × M kernel Gram matrix in O(MN) time.Here N denotes the input size for data x ∈ R^N.
  • Quantum training procedure: The procedure prepares label and coefficient states, then estimates the S nonzero support-vector coefficients in time linear in the number of data points.The coefficient state is obtained by applying an inverse Gram-matrix-related operation to the label state.
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