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Quantum Circuit Encodings of Polynomial Chaos Expansions

Junaid Aftab, Christoph Schwab, Haizhao Yang, Jakob Zech

Abstract

This work investigates the expressive power of quantum circuits in approximating high-dimensional, real-valued functions. We focus on countably-parametric holomorphic maps $u:U\to \mathbb{R}$, where the parameter domain is $U=[-1,1]^{\mathbb{N}}$. We establish dimension-independent quantum circuit approximation rates via the best $n$-term truncations of generalized polynomial chaos (gPC) expansions of these parametric maps, demonstrating that these rates depend solely on the summability exponent of the gPC expansion coefficients. The key to our findings is based on the fact that so-called ``$(\boldsymbol{b},ε)$-holomorphic'' functions, where $\boldsymbol{b}\in (0,1]^\mathbb N \cap \ell^p(\mathbb N)$ for some $p\in(0,1)$, permit structured and sparse gPC expansions. Then, $n$-term truncated gPC expansions are known to admit approximation rates of order $n^{-1/p + 1/2}$ in the $L^2$ norm and of order $n^{-1/p + 1}$ in the $L^\infty$ norm. We show the existence of parameterized quantum circuit (PQC) encodings of these $n$-term truncated gPC expansions, and bound PQC depth and width via (i) tensorization of univariate PQCs that encode Chebyšev-polynomials in $[-1,1]$ and (ii) linear combination of unitaries (LCU) to build PQC emulations of $n$-term truncated gPC expansions. The results provide a rigorous mathematical foundation for the use of quantum algorithms in high-dimensional function approximation. As countably-parametric holomorphic maps naturally arise in parametric PDE models and uncertainty quantification (UQ), our results have implications for quantum-enhanced algorithms for a wide range of maps in applications.

Quantum Circuit Encodings of Polynomial Chaos Expansions

Abstract

This work investigates the expressive power of quantum circuits in approximating high-dimensional, real-valued functions. We focus on countably-parametric holomorphic maps , where the parameter domain is . We establish dimension-independent quantum circuit approximation rates via the best -term truncations of generalized polynomial chaos (gPC) expansions of these parametric maps, demonstrating that these rates depend solely on the summability exponent of the gPC expansion coefficients. The key to our findings is based on the fact that so-called ``-holomorphic'' functions, where for some , permit structured and sparse gPC expansions. Then, -term truncated gPC expansions are known to admit approximation rates of order in the norm and of order in the norm. We show the existence of parameterized quantum circuit (PQC) encodings of these -term truncated gPC expansions, and bound PQC depth and width via (i) tensorization of univariate PQCs that encode Chebyšev-polynomials in and (ii) linear combination of unitaries (LCU) to build PQC emulations of -term truncated gPC expansions. The results provide a rigorous mathematical foundation for the use of quantum algorithms in high-dimensional function approximation. As countably-parametric holomorphic maps naturally arise in parametric PDE models and uncertainty quantification (UQ), our results have implications for quantum-enhanced algorithms for a wide range of maps in applications.

Paper Structure

This paper contains 27 sections, 12 theorems, 104 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Contributions
  3. Example: Response Surface of Parametric, Elliptic PDEs
  4. Related Works
  5. Neural Network Literature
  6. Quantum Computing Literature
  7. Organization
  8. Preliminaries
  9. Notation and Terminology
  10. Standard Notation
  11. Multi-Index Notation
  12. Function Space Notation
  13. Generalized polynomial chaos approximation
  14. Holomorphy
  15. Taylor-gPC expansions
  16. ...and 12 more sections

Key Result

Lemma 2.3

Assume the index set $\emptyset \ne S\subseteq{\mathcal{F}}$ is finite and d.c. Then, for any univariate basis $\{ p_j \}_{j\geq 0}$ of polynomials of preciseI.e., $p_j(x) = c_jx^j + ...$ with $c_j\ne 0$. degree $j \geq 0$, there holds Here, for ${\bm \nu}\in {\mathcal{F}}$, $P_{\bm \nu}({\boldsymbol y}) = p_{\nu_1}(y_1) p_{\nu_2}(y_2)...$ with the dots indicating the product of all nonzero entr

Figures (1)

  • Figure 1: (Left) The quantum circuit implementing the linear combination of unitaries technique. (Right) The quantum circuit implementing the Hadamard test.

Theorems & Definitions (24)

  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • Definition 2.4
  • Proposition 2.5
  • Theorem 2.6: infinite-dimensional case
  • proof
  • Theorem 2.7
  • Proposition 2.8
  • Remark 2.9
  • ...and 14 more