The effect of data encoding on the expressive power of variational quantum machine learning models

Maria Schuld; Ryan Sweke; Johannes Jakob Meyer

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The effect of data encoding on the expressive power of variational quantum machine learning models

Maria Schuld, Ryan Sweke, Johannes Jakob Meyer

arXiv:2008.08605v2Aug 19, 2020quant-phstat.ML

TL;DR

The paper asks how data-encoding strategies determine the expressive power of parametrised quantum circuits as function approximators. It maps these models to partial Fourier series and shows that sufficiently rich accessible spectra can yield universal function approximation.

Problem
The paper examines whether a single quantum gate can encode data with an unrestricted frequency spectrum, addressing limits in understanding model expressivity.
Method
The paper systematically maps quantum models to partial Fourier series and relates accessible frequencies to data-encoding gate eigenvalues.
Results
Quantum models can realise all Fourier-coefficient sets and become universal function approximators when their accessible frequency spectra are asymptotically rich enough.
Takeaways & Limitations
The Fourier-series framework provides a foundation for theoretical analysis and a guide for searching for suitable quantum machine-learning applications.
Takeaways & Limitations
The universality result assumes exponentially deep trainable circuit blocks, whereas practical settings require depth-restricted blocks.

Abstract

from arXiv · show

Quantum computers can be used for supervised learning by treating parametrised quantum circuits as models that map data inputs to predictions. While a lot of work has been done to investigate practical implications of this approach, many important theoretical properties of these models remain unknown. Here we investigate how the strategy with which data is encoded into the model influences the expressive power of parametrised quantum circuits as function approximators. We show that one can naturally write a quantum model as a partial Fourier series in the data, where the accessible frequencies are determined by the nature of the data encoding gates in the circuit. By repeating simple data encoding gates multiple times, quantum models can access increasingly rich frequency spectra. We show that there exist quantum models which can realise all possible sets of Fourier coefficients, and therefore, if the accessible frequency spectrum is asymptotically rich enough, such models are universal function approximators.

I. QUANTUM MODELS AS PARTIAL FOURIER SERIES

Quantum models can be represented as partial Fourier series whose accessible frequency spectrum is determined solely by the eigenvalues of the data-encoding gates, while Fourier coefficients depend on the full circuit. Their expressive power is therefore governed by both spectrum size and degree and by the expressivity of the coefficients.

Model construction: A quantum model is defined as an observable’s expectation value after a parametrised circuit prepares an input-dependent quantum state.The circuit is built from repeated data-encoding blocks S(x) and trainable blocks W(θ), with encoding gates G(x) = e^-ixH.
Fourier representation: Diagonalising the encoding Hamiltonian shows that repeated encodings generate basis functions with frequencies formed from differences of sums of Hamiltonian eigenvalues.For multi-index j, the summed eigenvalue is Λ_j = λ_j1 + ··· + λ_jL, and grouped terms have frequency ω = Λ_k − Λ_j.
Spectrum properties: The resulting frequency spectrum is symmetric around zero, includes 0, and determines the number of independent non-zero frequencies and the spectrum degree.K = (|Ω| − 1)/2 measures independent non-zero frequencies, while D = max(Ω) is the degree.
Expressivity factors: The encoding-gate eigenvalues alone determine the frequency spectrum, whereas arbitrary gates and the measurement observable determine the Fourier coefficients.For integer-valued eigenvalues, the accessible frequencies are integer-valued as well.
Expressivity factors: Quantum-model expressivity depends on both the frequency spectrum’s size and degree and the model’s ability to control its Fourier coefficients.These properties characterize the function classes that different quantum models can learn, for integer or non-integer frequencies.

II. THE EXPRESSIVITY OF QUANTUM MODELS

This section uses Fourier-series formalism to investigate quantum-model expressivity. It analyzes single-qubit Pauli rotations in data encoding before characterizing expressivity limits for general encoding gates.

Fourier-series formalism is used to investigate the expressivity of quantum models.
The analysis begins with single-qubit Pauli rotations in the encoding subroutine S(x), a widely used strategy.
The section then characterizes the limits of quantum-model expressivity for a given data encoding gate in more general terms.

A. A single Pauli-rotation encoding can only learn a sine function

With a single Pauli-rotation encoding, the quantum model can represent only a degree-1 sine function, regardless of circuit width or depth. Numerical evidence further shows that fitting a single-frequency Fourier series requires exact matching between the frequency and data scaling.

A. A single Pauli-rotation encoding can only learn a sine function: A single encoding gate produces functions of the form f(x) = A sin(2γx + B) + C.The constants A, B, and C are determined by the nonencoding part of the variational circuit.
A. A single Pauli-rotation encoding can only learn a sine function: The resulting sine function is equivalent to a truncated Fourier series of degree 1.Repeating the encoding gate is identified as the mechanism for systematically increasing the accessible degree.
A. A single Pauli-rotation encoding can only learn a sine function: For Pauli rotations, the model has a single non-zero frequency, and the encoding scale satisfies ˜x = γx = x.The underlying spectrum after rescaling the generator eigenvalues is Ω = {−2, 0, 2}.
A. A single Pauli-rotation encoding can only learn a sine function: Circuit width, depth, unitaries, and measurements do not remove the expressivity limit imposed by the data encoding strategy.The limitation holds even for very wide and deep circuits that may be classically intractable to simulate.
A. A single Pauli-rotation encoding can only learn a sine function: Numerical evidence shows that Pauli-X encoding fits a single-frequency Fourier series only when its frequency exactly matches the data scaling.The experiment uses a single-qubit model with a Pauli-X rotation between general rotation gates.

B. Repeated Pauli encodings linearly extend the frequency spectrum

Repeating single-qubit Pauli encodings, either in parallel or across layers, systematically enlarges the accessible frequency spectrum. With r repetitions, the resulting univariate quantum model is a truncated Fourier series of degree r.

Parallel repetitions: Parallel repetition of Pauli-rotation encodings yields the frequency spectrum {−r, …, 0, …, r}.These frequencies arise from differences between eigenvalues formed by sums of r values ±1/2.
Sequential repetitions: Repeating a single-qubit Pauli rotation sequentially across r layers produces the same frequency spectrum as r parallel repetitions.The sequential spectrum is explicitly found to satisfy Ωseq = Ωpar.
Sequential repetitions: A model with r sequential repetitions of a single-qubit Pauli encoding is likewise a truncated Fourier series of degree r.Parallel and sequential repetition provide the same spectrum-growth mechanism.

C. Limits of expressivity

The section bounds the accessible Fourier spectrum from data-encoding gates and shows how repetitions or continuous-variable phase shifts expand it. It also distinguishes spectral access from coefficient control, which depends on trainable circuit structure and available degrees of freedom.

Spectrum-size bounds: The maximum spectrum size K(L, d) provides an upper bound on the number of frequencies accessible with L repetitions of a d-dimensional encoding gate.The frequency spectrum is formed from sums and differences of encoding-gate eigenvalues.
Spectrum-size bounds: For repeated single-qubit Pauli encodings, the spectrum size is K = L, whereas using L different encoding gates can increase the bound to 2^2L − 1.The single-qubit case recovers a degree-2^2−1 result, while identical repetitions yield the tighter K = L bound.
Spectrum-size bounds: Continuous-variable phase shifts can support the full integer spectrum Ω∞ = {−∞, …, −1, 0, 1, …, ∞} because their generator is the harmonic oscillator number operator.The number operator has eigenvalues diag(0, 1, 2, …), producing all integer frequencies.
Coefficient control: Arbitrary control of K Fourier coefficients requires at least M ≥ 2K + 1 real circuit degrees of freedom, including 2L for a repeated Pauli encoding with spectrum size L.This scaling is considered realistic for shallow circuits that aim to use the full frequency spectrum.
Coefficient control: Simulations suggest shallow trainable blocks can produce rich coefficient subsets, although the ansatz may structurally force particular Fourier coefficients to zero.The systematic impact of the trainable ansatz on coefficient control remains outside the paper’s scope.

III. QUANTUM MODELS ARE ASYMPTOTICALLY UNIVERSAL

Quantum models become asymptotically universal when repeated data encodings provide increasingly rich frequency spectra and sufficiently flexible circuit blocks allow the Fourier coefficients to be chosen freely. For a universal Hamiltonian family, the resulting model family can approximate any square-integrable function on [0, 2π]^N to arbitrary accuracy as the available system size grows.

Frequency spectra: Repeated data-encoding gates produce truncated Fourier series whose accessible frequencies are determined by the encoding Hamiltonians.This applies to parallel repetition with L = 1 and serial repetition with L > 1.
Universality condition: With sufficiently flexible trainable circuit blocks, quantum models can realise arbitrary Fourier coefficients in addition to accessing the required frequencies.The analysis assumes trainable blocks capable of implementing arbitrary global unitaries, which may require exponential circuit depth in primitive gates.
Universality theorem: As the Hilbert-space dimension or number of finite-dimensional subsystems tends to infinity, the model family can approximate any square-integrable function on [0, 2π]^N to arbitrary accuracy.The proof reduces approximation of the target function to a truncated Fourier series and then constructs a corresponding quantum model.
Frequency spectra: Multivariate single-layer models realise Fourier series with frequencies set by the data-encoding Hamiltonians and coefficients set by the rest of the circuit.The model extends the univariate L = 1 construction to multiple input variables.
Universality condition: A universal Hamiltonian family asymptotically contains every integer frequency range ZK = {−K, . . . , 0, . . . , K}.For every K ∈ N, some family member has ZK contained in its associated frequency spectrum.

IV. PRACTICAL IMPLICATIONS FOR QUANTUM MACHINE LEARNING

This section discusses the practical relevance of the results, showing how the framework can cover models encoding classically pre-processed features and providing design guidelines for quantum machine learning algorithms.

Scope of the framework: Many quantum models outside the base model can still be analysed by assuming they encode classically pre-processed features φ(x) rather than original features x.The section extends the framework’s scope beyond models that immediately fit Eq. (3).
Algorithm design: The section summarises guidelines for designing quantum machine learning algorithms.These guidelines address the practical relevance of the paper’s results.

A. Classical pre-processing

Many quantum data-encoding strategies implicitly classically pre-process inputs into angles before applying the time-evolution encoding analyzed here. Consequently, expressivity claims depend on both the quantum algorithm and the specific pre-processing strategy.

Classical pre-processing: Implicit pre-processing maps original data into features used by the time-evolution encoding, so the paper’s results apply to those resulting features.The base encoding uses gates G(x) = e−ixH, while other strategies may first transform the data.
Classical pre-processing: Binary basis-state encoding maps each scalar feature to rotation angles representing its binary digits.For an n-bit representation, the angles are φ(x) = (φ1(x), . . . , φn(x)), with rotations of π or 0.
Classical pre-processing: Amplitude encoding classically maps an input vector x to angles φ(x) that parameterize an arbitrary state-preparation routine.The angles are computed from the original input before preparing the quantum state.
Classical pre-processing: Rescaling inputs or constructing higher-order features before Pauli rotations can alter the effective features available to the quantum model.Making rescaling hyperparameters trainable could enable adaptive frequency matching and potentially increase the expressivity of small circuits.
Classical pre-processing: The expressive power of a quantum machine-learning algorithm must be attributed to the quantum circuit together with its pre-processing strategy.Comparisons with classical models should identify the pre-processing and use the same pre-processed inputs.

B. Practical insights

The paper translates encoding and architecture choices into practical guidance for controlling quantum-model frequency spectra and expressivity. It also cautions that model selection should prioritize generalization capacity, not expressivity alone.

Practical design guidance: Hamiltonian time-evolution encoding yields partial Fourier-series models, with available frequencies set by the Hamiltonian and Fourier coefficients by non-encoding gates.This provides a natural representation for analyzing the functions quantum models can learn.
Practical design guidance: Repeating single-qubit Pauli encoding rotations increases the accessible frequency spectrum and can therefore improve quantum-model expressivity.The number of encoding rotations limits the number of accessible frequencies.
Practical design guidance: Quantum models naturally learn periodic functions, motivating data rescaling into the function class’s period and suggesting time-series and signal-processing applications.The Fourier representation may also indicate regularizing properties that exclude higher-order frequencies, although the passage is truncated.
Practical design guidance: Classical preprocessing, including feature creation, can enrich the frequency spectrum and give small quantum models greater expressivity.The benefit comes from adding structure to the encoded data before the quantum model processes it.
Practical design guidance: Fixing the observable limits applicability, whereas freely adjusting its entries—and potentially parametrizing it—supports more flexible quantum models.Adjustable observable entries were key to the paper’s universality proof in Section III.
Practical design guidance: Model selection should consider expected generalization performance through capacity metrics such as VC-dimension or Rademacher complexity, rather than relying purely on expressivity.The passage notes that such capacities are calculable for very simple function classes, while the discussion is truncated.

V. CONCLUSION · Appendix A: Partial Fourier Series Representation of Multivariate Functions

The paper establishes a systematic correspondence between a broad class of quantum machine-learning models and partial Fourier series, clarifying how data encoding determines expressivity. Its multivariate extension realizes partial Fourier series whose frequencies come from encoding Hamiltonians and coefficients from trainable components, while leaving generalization and model-selection questions open.

V. CONCLUSION: The framework maps a large class of quantum machine-learning models to partial Fourier series and quantifies how data-encoding mechanisms affect expressivity.It is intended as a foundation for further theoretical analysis and as guidance when seeking applications for these models.
V. CONCLUSION: The framework connects quantum machine learning with classical ideas including periodic-activation neural networks and parametrised Fourier series.
V. CONCLUSION: Important open questions concern whether the framework can quantify quantum-model generalization capacity and support meaningful model-selection guidelines.The paper specifically asks whether partial-Fourier representations can yield modern generalization measures for model selection.
Appendix A: Partial Fourier Series Representation of Multivariate Functions: For L = 1 quantum models, encoding features into different quantum subsystems naturally generalizes the univariate construction to multivariate Fourier series.The described multivariate model has asymptotic universality stated in Section III and proved in Appendix C.
Appendix A: Partial Fourier Series Representation of Multivariate Functions: In the multivariate construction, arbitrary unitaries can be absorbed into the initial state and measurement, leaving an equivalent model with an arbitrary state and observable.The derivation also assumes, without loss of generality, that the Hamiltonians are diagonal.
Appendix A: Partial Fourier Series Representation of Multivariate Functions: The resulting model is a partial multivariate Fourier series whose accessible frequencies are determined by the encoding Hamiltonian spectra and whose coefficients are determined by trainable unitaries.Equivalently, the coefficients are determined by the state and observable.

Appendix B: Non-integer frequencies

Quantum models with non-integer frequencies can be analyzed using Fourier-series methods when all frequencies are integer multiples of a basic frequency. This requires rescaling the data or changing the interval, but very close frequencies can make the resulting spectrum sparse and approximation quality poor.

Fourier decomposition: Non-integer frequency functions generally contribute to infinitely many integer-valued Fourier coefficients.The decomposition uses sinc(z) = sin(πz)/πz.
Fourier decomposition: If all frequencies are integer multiples of a basic frequency ω0, the model can be treated equivalently using Fourier-series techniques.This condition is equivalent to all frequencies being mutually commensurable, meaning every pair has a rational ratio.
Rescaling: The resulting Fourier-type sum is periodic on [0, 2π/ω0] and can be interpreted as a partial Fourier series on that interval.Equivalently, the data can be rescaled by x̃ = x/ω0.
Limitations: Very close frequencies require a large data-rescaling factor, which can produce sparse Fourier coefficients and poor approximation quality.ω0 is at least as small as the smallest difference between frequencies in Ω.

Appendix C: Proof of the universality theorem · Appendix D: CO2 Emission Table

Appendix C proves universality by combining Fourier truncation with a universal Hamiltonian family whose accessible frequencies contain the required multivariate spectrum and whose state and observable can realize the coefficients. Appendix D reports approximately 7.5 kg of CO2 emissions for numerical simulations and zero for transport.

Appendix C: Proof of the universality theorem: Any target function g can first be approximated arbitrarily closely in L2 norm by a truncated Fourier series.The proof introduces a finite frequency set and coefficients for the truncated series.
Appendix C: Proof of the universality theorem: A universal Hamiltonian family can be chosen so that the required integer frequency range ZK = {−K, . . . , 0, . . . , K} lies within its spectrum.This supplies the frequencies needed by the truncated target series.
Appendix C: Proof of the universality theorem: In the multivariate case, accessible frequency vectors contain all combinations formed by the Cartesian product of N copies of the Hamiltonian spectrum.The frequency differences λj − λk independently combine the available component frequencies.
Appendix C: Proof of the universality theorem: Because the generated spectrum contains K, the selected model includes every Fourier term necessary to construct the truncated target series.The proof then matches the model output to that series.
Appendix C: Proof of the universality theorem: The initial state and observable can adjust all Fourier coefficients freely, subject only to the complex-conjugation symmetry required for a real-valued output.The proof fixes the initial state as an equal superposition and uses observable entries to set the coefficients, with Hermiticity determining the lower-triangular elements.
Appendix D: CO2 Emission Table: Approximately 7.5 kg of CO2 is reported for numerical simulations, while transport contributes 0 kg.The table lists approximately 100 total kernel hours, 50 W thermal design power per kernel, 5 kWh total energy consumption, and 1.5 kg/kWh average emissions in South Africa.