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arXiv:2304.03787 · Physical Review A

Fourier expansion in variational quantum algorithms

Nikita A. Nemkov, Evgeniy O. Kiktenko, Aleksey K. Fedorov

Published Sep 6, 2023• 27 citations
arXivHTMLPDF

Abstract

The Fourier expansion of the loss function in variational quantum algorithms (VQAs) contains a wealth of information yet is generally hard to access. We focus on the class of variational circuits where constant gates are Clifford gates and parametrized gates are generated by Pauli operators, which covers most practical cases while allowing much control due to the properties of stabilizer circuits. We give a classical algorithm that, for an $N$-qubit circuit and a single Pauli observable, computes coefficients of all trigonometric monomials up to a degree $m$ in time bounded by $O(N{2}^{m})$. Using the general structure and implementation of the algorithm, we reveal several aspects of Fourier expansions in Clifford plus Pauli VQAs such as (i) reformulating the problem of computing the Fourier series as an instance of the multivariate Boolean quadratic system, (ii) showing that the approximation given by a truncated Fourier expansion can be quantified by the ${L}^{2}$-norm and evaluated dynamically, (iii) the tendency of Fourier series to be rather sparse and Fourier coefficients to cluster together, and (iv) the possibility to compute the full Fourier series for circuits of nontrivial sizes, featuring tens to hundreds of qubits and parametric gates.

DOI
10.1103/physreva.108.032406

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The Fourier expansion of the loss function in variational quantum algorithms (VQAs) contains a wealth of information yet is generally hard to access. We focus on the class of variational circuits where constant gates are Clifford gates and parametrized gates are generated by Pauli operators, which covers most practical cases while allowing much control due to the properties of stabilizer circuits. We give a classical algorithm that, for an NNN-qubit circuit and a single Pauli observable, computes coefficients of all trigonometric monomials up to a degree mmm in time bounded by O(N2m)O(N{2}^{m})O(N2m). Using the general structure and implementation of the algorithm, we reveal several aspects of Fourier expansions in Clifford plus Pauli VQAs such as (i) reformulating the problem of computing the Fourier series as an instance of the multivariate Boolean quadratic system, (ii) showing that the approximation given by a truncated Fourier expansion can be quantified by the L2{L}^{2}L2-norm and evaluated dynamically, (iii) the tendency of Fourier series to be rather sparse and Fourier coefficients to cluster together, and (iv) the possibility to compute the full Fourier series for circuits of nontrivial sizes, featuring tens to hundreds of qubits and parametric gates.
Variational quantum algorithms (VQAs) have shown advantages over classical approaches in solving many problems, yet their quantum neural networks (QNNs) models still perform poorly in complex tasks like multi-class classification and high-frequency time-series prediction, raising questions about their feasibility. While existing studies have characterized their performance from several perspectives, but research on their learning performance and visualization remains scarce. This work develops the universal approximation theory of variational quantum algorithms for multivariate function fitting, supported by a verification in visual training analysis on two-input binary datasets. We first visualize how a single-qubit VQA with single-encoding process four types of binary data, revealing inherent limitations, and then formulate the measurement expectation of VQA as a partial Fourier series, establishing its universal approximation capacity and showing that the Fourier spectrum is determined by the data encoding strategy: multiple encoding layers broaden the spectrum and enhance learning performance. Guided by this theory, we adjust the circuit structure and achieve nearly 100% accuracy across all four binary classification tasks.
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What makes a class of quantum circuits efficiently classically simulable on average? I present a framework that applies harmonic analysis of groups to circuits with a structure encoded by group parameters. Expanding the circuits in a suitable truncated multi-path operator basis gives algorithms to evaluate the Fourier coefficients of output distributions or expectation values that are viewed as functions on the group. Under certain conditions, a truncated Fourier series can be efficiently estimated with guaranteed mean-square convergence. For classes of noisy circuits, it leads to algorithms for sampling and mean value estimation under error models with a spectral gap, where the complexity increases exponentially with the gap's inverse and polynomially with the circuit's size. This approach unifies and extends existing algorithms for noisy parametrised or random circuits using Pauli basis paths. For classes of noiseless circuits, mean values satisfying Lipschitz continuity can be on average approximated using efficient sparse Fourier decompositions. I also discuss generalisations to homogeneous spaces, qudit systems and a way to analyse random circuits via matrix coefficients of irreducible representations.
In the domain of variational quantum algorithms, quantum Fourier models (QFMs) provide a mathematically well defined structure for quantum machine learning (QML). There has been a substantial amount of work on the scalability and trainability of such models showcasing the potential but also the limitations for the prospective application of QFMs. However, much less is known in the context of pulse-level quantum computing, where the microwave parameters that implement unitary operations on the hardware are used to perform computations directly instead of through the interface of quantum circuits. In this work, we evaluate QFMs through the lens of pulse parameters and link metrics such as expressibility and Fourier coefficient correlation (FCC) to this extended set of variational parameters. We show that while control over pulse shapes does not significantly alter the global expressibility or structural correlations of the Ansatz, it fundamentally alters the local optimisation landscape. For composite gates, independent pulse scalings replace a single logical angle by multiple independently tunable sub-angles. This relaxes the rigid monomial couplings induced by the gate-level parameterisation, and provides gradient descent with higher-dimensional escape routes, decoupling local parameter constraints and significantly boosting performance during training. Following an analytical proof, we show numerical results validating our theory on training a QFM with an exponential (ternary) feature map on a Fourier series with the same frequencies.
Variational quantum circuits (VQCs) can be understood through the lens of Fourier analysis. The function space represented by any circuit architecture can be described through a truncated Fourier sum. We show that the spectrum available to that truncated Fourier sum is not entirely determined by the encoding gates of the circuit, since the variational part of the circuit can constrain certain coefficients to zero, effectively removing that frequency from the spectrum. To the best of our knowledge, we give the first description of the functional dependence of the Fourier coefficients on the variational parameters as trigonometric polynomials. This allows us to provide an algorithm that computes the exact spectrum of any given circuit and the corresponding Fourier coefficients. Finally, we demonstrate that by comparing the Fourier transform of the dataset to the available spectra, it is possible to predict which VQC of a given list of choices will best fit the data.