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Quantum machine learning in feature Hilbert spaces

Maria Schuld, Nathan Killoran

arXiv:1803.07128v1quant-ph

TL;DR

The paper investigates the shared logic of quantum computing and kernel methods: efficient computation in very large Hilbert spaces. It treats quantum input encoding as a nonlinear feature map, develops implicit kernel and explicit variational-classifier approaches, and illustrates them with squeezing-based continuous-variable simulations. The paper concludes that feature-Hilbert-space methods provide a promising avenue for quantum machine learning and pattern recognition.

  • Problem

    The paper addresses the limited exploration of the relationship between quantum computing, feature maps, and kernel methods in quantum machine learning.

  • Method

    The paper encodes inputs as quantum states in a feature Hilbert space, then uses quantum-state inner products for kernels or variational circuits as explicit linear classifiers.

  • Results

    Small-scale simulations with a squeezing feature map illustrate that both implicit kernel-based and explicit circuit-based approaches can produce interesting quantum machine-learning results.

  • Takeaways & Limitations

    Quantum feature maps connect quantum data encoding with kernel methods and provide hardware-independent approaches suitable for intermediate-term quantum technologies.

Abstract

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The basic idea of quantum computing is surprisingly similar to that of kernel methods in machine learning, namely to efficiently perform computations in an intractably large Hilbert space. In this paper we explore some theoretical foundations of this link and show how it opens up a new avenue for the design of quantum machine learning algorithms. We interpret the process of encoding inputs in a quantum state as a nonlinear feature map that maps data to quantum Hilbert space. A quantum computer can now analyse the input data in this feature space. Based on this link, we discuss two approaches for building a quantum model for classification. In the first approach, the quantum device estimates inner products of quantum states to compute a classically intractable kernel. This kernel can be fed into any classical kernel method such as a support vector machine. In the second approach, we can use a variational quantum circuit as a linear model that classifies data explicitly in Hilbert space. We illustrate these ideas with a feature map based on squeezing in a continuous-variable system, and visualise the working principle with $2$-dimensional mini-benchmark datasets.

I. INTRODUCTION

Quantum computing and kernel methods both enable efficient computation in very large Hilbert or feature spaces. The paper frames quantum input encoding as a feature map and derives implicit kernel-based and explicit linear-classifier approaches.

  • Quantum computing and kernel methods: Quantum algorithms manipulate rapidly growing Hilbert spaces with at most polynomially many operations, including formally infinite-dimensional spaces under continuous-variable operations.Squeezing is given as an example of one operation acting on an infinite-dimensional Hilbert space.
  • Quantum computing and kernel methods: Kernel methods embed inputs into higher-dimensional feature spaces where data may become linearly separable, while computing through kernels rather than explicit feature vectors.Support vector machines illustrate this approach by learning a separating decision boundary in feature space.
  • Motivation: Kernel methods have received comparatively little attention in quantum machine learning, despite the field’s focus on several other quantum learning approaches.The paper positions feature-space and kernel ideas as an underexplored connection in quantum machine learning.
  • Paper aim: The paper interprets quantum-state encoding as a feature map into the quantum Hilbert space, where inner products define kernels and changing encodings changes the kernel.This connects quantum data representations with the kernel trick and linear models in feature space.
  • Quantum classifier approaches: The proposed classifiers use either a quantum device to evaluate a kernel for a classical model or a variational circuit to learn a linear decision boundary directly in feature space.The first is implicit and kernel-based; the second is explicit and circuit-based.
  • Feature maps and kernels: Feature maps provide a metric space for inputs, and nonlinear maps can alter relative positions so that classification becomes easier.Every feature map induces a kernel through inner products of mapped vectors.

B. Reproducing kernel Hilbert spaces

A reproducing kernel Hilbert space links kernel functions to function spaces and linear models in feature space. The representer theorem further reduces regularized optimization over such spaces to finite kernel expansions.

  • RKHS definition: Each kernel determines a reproducing kernel Hilbert space, whose functions satisfy a reproducing property through point evaluation.The reproducing kernel is unique for the associated Hilbert space, up to the stated Mercer-kernel qualification.
  • Feature maps and RKHS: A feature map induces a kernel and therefore a corresponding reproducing kernel Hilbert space.The paper states this relationship as a construction from feature map to kernel to RKHS.
  • Linear models: Functions formed from inner products with a vector in feature space act as linear models whose parameter vector defines a hyperplane.This provides the connection between RKHS functions and linear decision boundaries in feature space.
  • Representer theorem: Under the stated RKHS and regularization assumptions, an empirical-risk minimizer can be represented as an expansion of kernel functions evaluated on training inputs.The result applies to model functions in the RKHS with a strictly increasing regularization term.
  • Representer theorem: The representer theorem replaces explicit optimization over an infinite-dimensional RKHS with a finite convex optimization over expansion parameters.The finite representation uses the kernel functions associated with the training examples.

C. Input encoding as a feature map

Quantum input encoding is treated as a feature map implemented by a state-preparation circuit, allowing quantum states to define kernels and linear models. The paper surveys encodings whose induced kernels include delta, linear, polynomial, and cosine forms.

  • Quantum feature maps: Encoding an input x as a quantum state |φ(x)⟩ in Hilbert space fulfills the definition of a quantum feature map.The associated kernel is obtained from inner products of the encoded quantum states.
  • Classifier approaches: Quantum feature maps support two supervised-learning routes: implicit kernel evaluation for classical training and explicit variational-circuit classification in feature space.The implicit route evaluates kernels from quantum-state inner products; the explicit route trains a circuit directly.
  • Feature-embedding circuits: A feature-embedding circuit Uφ(x) prepares |φ(x)⟩ from a ground or vacuum state, while a model circuit prepares a state |w⟩ for inner-product models.The resulting models correspond to functions in the reproducing Hilbert space.
  • Input encodings: Basis encoding induces a Kronecker-delta kernel because inputs map to orthonormal computational-basis states.The resulting similarity is nonzero only for identical inputs.
  • Input encodings: Amplitude encoding produces a linear kernel, while taking copies of amplitude-encoded states implements polynomial kernels.These encodings associate normalized input vectors with quantum-state amplitudes.
  • Input encodings: Product encoding maps each input feature to a separate qubit state and implies a cosine kernel.The example encodes xi as cos(xi)|0⟩ + sin(xi)|1⟩.

B. Building a quantum classifier

The paper presents explicit classification in quantum feature Hilbert space using trainable variational circuits, while motivating quantum kernels for settings where classical evaluation is difficult.

  • Explicit approach: The explicit approach trains a parametrised circuit W(θ) to learn a model directly in feature Hilbert space.The circuit architecture restricts the possible decision boundaries and can act as regularisation.
  • Implicit approach: The implicit approach can use quantum-evaluated kernels with classical kernel methods when the desired kernel is classically intractable.The paper identifies runtime advantages and classically intractable kernels as motivations for quantum devices.
  • Feature-space construction: The examples use squeezing in continuous-variable systems, whose feature Hilbert space is an infinite-dimensional Fock space.The construction is intended to be implementable with optical quantum computers.

C. Squeezing as a feature map

The squeezing feature map encodes inputs into multimode Fock space by associating input coordinates with squeezing phases. Its strength hyperparameter controls the kernel’s variance, while the chosen encoding preserves that tunability.

  • Feature-map definition: A squeezed vacuum state uses a complex squeezing factor z = re^iϕ, with the input encoded through the phase in |φ(x)⟩ = |(c, x)⟩.Here c controls squeezing strength and x is associated with phase.
  • Classification illustration: The SVM experiments use the custom squeezing kernel on circles, moons, and blobs datasets, while varying c in a larger dataset experiment.Figure 5 reports training and test classification rates for these decision boundaries.
  • Feature-map definition: Multidimensional inputs are mapped to joint states of multiple squeezed vacuum modes in a multimode Fock space.The paper calls this construction the squeezing feature map with phase encoding.
  • Kernel behavior: The squeezing kernel is classically easy to compute, and its hyperparameter c determines the kernel function’s variance.Figure 4 visualizes the kernel shape for different c values with one input fixed at (0, 0).
  • Encoding choice: The phase encoding is used because absolute-value encoding does not allow the kernel variance to be varied.Absolute-value encoding instead maps x to |(x, c)|.

D. An implicit quantum-assisted classifier

The implicit classifier evaluates a squeezing-state overlap as a custom kernel for a classical SVM, while related experiments test explicit linear separation and variational classification in Fock space.

  • Implicit classifier: The implicit approach feeds the squeezing kernel, computed from quantum-state overlaps, into a classical support vector machine.For squeezing, the kernel can be computed classically and is used as a custom SVM kernel.
  • Implicit classifier: The custom-kernel SVM learns decision boundaries on two-dimensional mini-benchmark datasets, including circles, moons, and blobs.Figure 5 varies the squeezing hyperparameter c and reports training/test classification rates.
  • Fock-space separability: The perceptron experiment asks whether squeezing makes data linearly separable in Fock space, using the blobs dataset as an example.The perceptron is used because it finds a separating hyperplane when one exists.
  • Fock-space separability: The squeezing feature map achieves perfect training fit on the illustrated data, but the non-increasing test accuracy shows that this fit is not useful by itself.The paper states that Appendix B proves linear separability in feature space.
  • Kernel limitations: More sophisticated kernels require non-Gaussian feature-map elements because Gaussian squeezed states are efficiently classically simulable.The paper points to cubic-phase gates or photon-number measurements as possible non-Gaussian ingredients.
  • Explicit classifier: The explicit classifier maps inputs into Fock space, applies a variational circuit, and uses measurements to produce class probabilities.Its restricted ansatz is intended to provide candidate decision boundaries that generalize better than unrestricted perfect fitting.
  • Explicit classifier: For the moons dataset, the explicit classifier’s training loss converges to almost zero after about 200 stochastic-gradient iterations.The experiment uses four repetitions of the gate block and 32 parameters in total.

IV. CONCLUSION

The conclusion frames quantum-state encodings as feature maps for kernel evaluation or explicit variational classification, illustrated with squeezing and small-scale simulations. It also identifies classically intractable kernels, variational-circuit training, and nonlinear encoding as important follow-up issues and opportunities.

  • Contributions: Quantum-state encodings define feature maps into quantum Hilbert spaces, whose inner products can evaluate kernels for classical models.The paper also trains variational circuits as explicit classifiers that learn decision boundaries in feature space.
  • Contributions: Squeezing serves as the example feature map, with small-scale simulations motivating both implicit and explicit classification approaches.The conclusion describes the simulations as producing interesting results rather than establishing broad performance claims.
  • Future directions: An open question is which quantum feature-map circuits yield powerful kernels while requiring classically intractable state preparation or inner-product estimation.The conclusion specifically highlights kernels associated with classically intractable state preparation.
  • Future directions: Another open question concerns the design and training of variational circuits for hybrid training schemes.The paper notes that this topic had only recently begun to be investigated by the quantum machine-learning community.
  • Nonlinearity: Feature-map encoding places nonlinearity in state preparation, offering a solution to nonlinear transformations that are difficult for amplitude-encoded quantum data.The paper contrasts this with nondeterministic postselection or repeat-until-success workarounds whose failure probability grows with architecture size.

Appendix A: Reproducing kernels of quantum systems

Quantum Hilbert spaces can induce reproducing kernels through inner products, but ordinary continuous orthogonal bases expose a mismatch with RKHS requirements. Generalised coherent states avoid this issue and form an RKHS for their input set.

  • Quantum Hilbert spaces: Quantum systems associate inputs with Hilbert spaces of complex-valued functions, but the resulting space need not be an RKHS.For continuous orthogonal bases, the Dirac delta is not square integrable, so the reproducing-property requirements fail.
  • Quantum Hilbert spaces: For discrete orthonormal bases, the reproducing kernel is the Kronecker delta κ(s_i, s_j) = δ_i,j.The kernel follows by identifying basis-state inner products with the reproducing kernel.
  • Generalised coherent states: Generalised coherent states provide a continuous-labelled family with a resolution of identity and finite overlaps between basis states.These properties yield a functional representation in which ψ(l) = ⟨l|ψ⟩ and the inner product acts as a reproducing kernel.
  • Generalised coherent states: The coherent-state reproducing kernel is κ(l, l′) = ⟨l|l′⟩, making the coherent-state space an RKHS for the input set {l}.Finite overlap prevents the kernel from reducing to the Dirac delta function.
  • Optical coherent states: For optical coherent states, the associated kernel has a squared magnitude corresponding to a radial basis function or Gaussian kernel.The passage identifies this kernel form as a consequence of the optical coherent-state construction.

Appendix B: Linear separability in Fock space

The squeezing feature map sends distinct inputs to linearly independent Fock-space states, so the mapped data become linearly separable and can support arbitrary binary label assignments.

  • Linear separability: A dataset mapped into feature space becomes linearly separable when its mapped vectors are linearly independent.This follows from the stated proposition that M vectors in R^N are linearly separable if M − 1 are linearly independent.
  • Multimode extension: Therefore, phase-encoded squeezing maps any dataset to states that can be separated by a hyperplane in Fock space.The multidimensional feature map preserves the relevant linear-independence property.
  • Squeezing feature map: Distinct squeezing phases with a shared real hyperparameter produce linearly independent squeezed-vacuum Fock states.The result is stated for the single-mode squeezing map and also confirmed for absolute-value encoding.
  • Verification: Symbolic rank calculations confirm the linear-independence result for randomly selected squeezing factors up to M = 10 with a 40-dimensional Fock-space cutoff.This is a finite-dimensional computational check of the stated proposition.

Appendix C: Proof of Proposition 1

The proof establishes linear separability by converting arbitrary binary labels into a linear system and applying rank conditions. Linear independence provides the key dimensional guarantee, including a boundary case for one additional dependent point.

  • Dataset and separability: The dataset contains M vectors in R^N with binary labels, and separability is guaranteed for every class assignment under the proposition’s rank argument.The separating hyperplane uses parameters w_1, ..., w_N and b.
  • Proof strategy: Showing the stronger condition for some parameters automatically satisfies the original sign-based condition.This bypasses the difficulty of handling the sign function directly.
  • Rank criterion: Equation C2 forms M linear equations in N + 1 unknowns, with solvability determined by equality between coefficient and augmented-matrix ranks.The coefficient and augmented matrices are represented using the data coordinates, a constant column, and the labels.
  • Independent data points: For M linearly independent data points, N ≥ M and rank(X) = M; augmenting X with columns preserves this rank and yields a solution.The argument uses rank as the number of linearly independent rows or columns.
  • Boundary case: After adding dependent points until M = N, exactly one further dependent data point can still preserve guaranteed separability, whereas two additional points exceed the available rank structure.The boundary follows from the dimensions of the augmented system [X|1] and the effect of adding the label column.

Appendix D: Proof of Proposition 2

The proof constructs a matrix from squeezed-state feature vectors, rescales it into Vandermonde form, and shows its determinant is positive. Therefore, distinct encoded data points are linearly independent in Fock space for the stated squeezing maps.

  • Matrix construction: The squeezed states in the Fock basis are arranged as the rows of a matrix M.This matrix collects the feature representations whose column independence is studied.
  • Matrix transformation: Auxiliary diagonal matrices D1 and D2 rescale M into V := D1MD2.The transformed matrix is used to expose a structure whose determinant can be analyzed.
  • Vandermonde argument: V has Vandermonde structure, allowing the determinant to certify full rank after analyzing the feature-map parameters.The proof focuses on the determinant of V and its relation to det(M).
  • Distinctness condition: With phase encoding ri = rj = c and c > 0, equal feature parameters require ϕi = ϕj, which occurs only for the same datapoint.Because distinct datapoints are excluded from this equality, det(V) > 0 and hence det(M) > 0.
  • Independence result: The full-rank matrix M has linearly independent feature-vector columns; the same proof extends this conclusion to absolute-value encoding.The conclusion applies to distinct data points represented in Fock space.
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