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Interpolating Parametrized Quantum Circuits using Blackbox Queries

Lars Simon, Holger Eble, Hagen-Henrik Kowalski, Manuel Radons

TL;DR

This work addresses constructing classical surrogates for parametrized quantum circuits using blackbox queries to approximate the observable expectation $f(\theta)=\langle\psi(\theta)|\mathcal{M}|\psi(\theta)\rangle$. It develops two interpolation-based approaches: a Taylor-polynomial surrogate leveraging parameter-shift rules for arbitrary-order derivatives and a trigonometric (Fourier) surrogate rooted in a reproducing-kernel Hilbert space with a multivariate Dirichlet-like kernel. Each method comes with explicit sample-complexity and error guarantees that scale polynomially in the parameter dimension $m$ for fixed order $L$, ensuring accurate local approximation near a grid point, which is particularly relevant for tasks like VQE and barren-plateau mitigation. The authors validate the surrogates on a representative 8-qubit circuit, illustrate local versus global behavior, and discuss connections to quantum circuit simulation and classical surrogates for quantum ML models, highlighting practical significance and avenues for future work, including denoising under realistic hardware noise.

Abstract

This article focuses on developing classical surrogates for parametrized quantum circuits using interpolation via (trigonometric) polynomials. We develop two algorithms for the construction of such surrogates and prove performance guarantees. The constructions are based on circuit evaluations which are blackbox in the sense that no structural specifics of the circuits are exploited. While acknowledging the limitations of the blackbox approach compared to whitebox evaluations, which exploit specific circuit properties, we demonstrate scenarios in which the blackbox approach might prove beneficial. Sample applications include but are not restricted to the approximation of VQEs and the alleviaton of the barren plateau problem.

Interpolating Parametrized Quantum Circuits using Blackbox Queries

TL;DR

This work addresses constructing classical surrogates for parametrized quantum circuits using blackbox queries to approximate the observable expectation . It develops two interpolation-based approaches: a Taylor-polynomial surrogate leveraging parameter-shift rules for arbitrary-order derivatives and a trigonometric (Fourier) surrogate rooted in a reproducing-kernel Hilbert space with a multivariate Dirichlet-like kernel. Each method comes with explicit sample-complexity and error guarantees that scale polynomially in the parameter dimension for fixed order , ensuring accurate local approximation near a grid point, which is particularly relevant for tasks like VQE and barren-plateau mitigation. The authors validate the surrogates on a representative 8-qubit circuit, illustrate local versus global behavior, and discuss connections to quantum circuit simulation and classical surrogates for quantum ML models, highlighting practical significance and avenues for future work, including denoising under realistic hardware noise.

Abstract

This article focuses on developing classical surrogates for parametrized quantum circuits using interpolation via (trigonometric) polynomials. We develop two algorithms for the construction of such surrogates and prove performance guarantees. The constructions are based on circuit evaluations which are blackbox in the sense that no structural specifics of the circuits are exploited. While acknowledging the limitations of the blackbox approach compared to whitebox evaluations, which exploit specific circuit properties, we demonstrate scenarios in which the blackbox approach might prove beneficial. Sample applications include but are not restricted to the approximation of VQEs and the alleviaton of the barren plateau problem.
Paper Structure (16 sections, 11 theorems, 52 equations, 3 figures, 1 table, 2 algorithms)

This paper contains 16 sections, 11 theorems, 52 equations, 3 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Content and structure
  3. Algorithms
  4. Technical preliminaries
  5. Algorithm based on Taylor polynomials
  6. Algorithm based on trigonometric polynomials
  7. Comparison with existing methods
  8. Quantum circuit simulation
  9. Classical surrogates of quantum machine learning models
  10. Experiments
  11. Behavior along curves in parameter space
  12. Approximation error in $L^2$-norm
  13. Conclusion
  14. Theoretical background
  15. Proof of Theorem \ref{['thm:performance_guarantees_taylor']}
  16. ...and 1 more sections

Key Result

Lemma 2

For all multiindices $\alpha\in (\mathbb{Z}_{\geq 0})^m$ we have: where we index the entries of $\mathfrak{i}\in\{-1,1\}^{|\alpha |}$ as and In particular, all terms appearing in the above sum can be evaluated efficiently, assuming that the number of non-zero coefficients in the decomposition of $\mathcal{M}$ is small.

Figures (3)

  • Figure 1: This figure shows the circuit used in the experiments in Section \ref{['sec:experiments']}.
  • Figure 2: These plots show the function $f$ along several curves, where $f$ is defined as in Section \ref{['sec:algorithms']} with respect to the circuit described in Section \ref{['sec:experiments']}, with $n=8$, $d=2$, and the observable $Z^{\otimes 8}$. The plots also show the approximation $\tilde{f}$ obtained by the respective algorithm for various values of $L$. The plots show the behavior close to the grid point $0\in\mathbb{R}^{16}$.
  • Figure 3: These plots show the function $f$ along several curves, where $f$ is defined as in Section \ref{['sec:algorithms']} with respect to the circuit described in Section \ref{['sec:experiments']}, with $n=8$, $d=2$, and the observable $Z^{\otimes 8}$. The plots also show the approximation $\tilde{f}$ obtained by the respective algorithm for various values of $L$. For Algorithm \ref{['algorithm:taylor']}

Theorems & Definitions (26)

  • Remark
  • Remark
  • Lemma 2
  • proof
  • Theorem 4
  • Lemma 5
  • proof
  • Theorem 7
  • Remark
  • Lemma 8
  • ...and 16 more