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Conditions for a quadratic quantum speedup in nonlinear transforms with applications to energy contract pricing

Gabriele Agliardi, Corey O'Meara, Kavitha Yogaraj, Kumar Ghosh, Piergiacomo Sabino, Marina Fernández-Campoamor, Giorgio Cortiana, Juan Bernabé-Moreno, Francesco Tacchino, Antonio Mezzacapo, Omar Shehab

TL;DR

This paper advances a hybrid quantum-classical approach to evaluate nonlinear transforms in energy contract pricing by approximating nonlinear functions with polynomials and computing the resulting inner products via Quantum Hadamard Products. It analyzes end-to-end complexity and identifies the conditions under which a quadratic quantum speedup is achievable, notably requiring bilinear structure, degree-2 polynomial approximations, and efficient data loading. Among several variants, it argues that the main variant (c) delivers the optimal asymptotic speedup when $K\le 2$, provided loading costs are sublinear in $N$, and demonstrates a proof-of-principle on IBM Quantum devices for small $N$ with dynamic circuits and error-mitigated experiments. The study highlights practical considerations such as data encoding choices (amplitude encoding vs BOE), inner-product computation methods, and the integration of QAE to boost precision, offering a roadmap toward scaling quantum acceleration in energy-risk applications.

Abstract

Computing nonlinear functions over multilinear forms is a general problem with applications in risk analysis. For instance in the domain of energy economics, accurate and timely risk management demands for efficient simulation of millions of scenarios, largely benefiting from computational speedups. We develop a novel hybrid quantum-classical algorithm based on polynomial approximation of nonlinear functions, computed through Quantum Hadamard Products, and we rigorously assess the conditions for its end-to-end speedup for different implementation variants against classical algorithms. In our setting, a quadratic quantum speedup, up to polylogarithmic factors, can be proven only when forms are bilinear and approximating polynomials have second degree, if efficient loading unitaries are available for the input data sets. We also enhance the bidirectional encoding, that allows tuning the balance between circuit depth and width, proposing an improved version that can be exploited for the calculation of inner products. Lastly, we exploit the dynamic circuit capabilities, recently introduced on IBM Quantum devices, to reduce the average depth of the Quantum Hadamard Product circuit. A proof of principle is implemented and validated on IBM Quantum systems.

Conditions for a quadratic quantum speedup in nonlinear transforms with applications to energy contract pricing

TL;DR

This paper advances a hybrid quantum-classical approach to evaluate nonlinear transforms in energy contract pricing by approximating nonlinear functions with polynomials and computing the resulting inner products via Quantum Hadamard Products. It analyzes end-to-end complexity and identifies the conditions under which a quadratic quantum speedup is achievable, notably requiring bilinear structure, degree-2 polynomial approximations, and efficient data loading. Among several variants, it argues that the main variant (c) delivers the optimal asymptotic speedup when , provided loading costs are sublinear in , and demonstrates a proof-of-principle on IBM Quantum devices for small with dynamic circuits and error-mitigated experiments. The study highlights practical considerations such as data encoding choices (amplitude encoding vs BOE), inner-product computation methods, and the integration of QAE to boost precision, offering a roadmap toward scaling quantum acceleration in energy-risk applications.

Abstract

Computing nonlinear functions over multilinear forms is a general problem with applications in risk analysis. For instance in the domain of energy economics, accurate and timely risk management demands for efficient simulation of millions of scenarios, largely benefiting from computational speedups. We develop a novel hybrid quantum-classical algorithm based on polynomial approximation of nonlinear functions, computed through Quantum Hadamard Products, and we rigorously assess the conditions for its end-to-end speedup for different implementation variants against classical algorithms. In our setting, a quadratic quantum speedup, up to polylogarithmic factors, can be proven only when forms are bilinear and approximating polynomials have second degree, if efficient loading unitaries are available for the input data sets. We also enhance the bidirectional encoding, that allows tuning the balance between circuit depth and width, proposing an improved version that can be exploited for the calculation of inner products. Lastly, we exploit the dynamic circuit capabilities, recently introduced on IBM Quantum devices, to reduce the average depth of the Quantum Hadamard Product circuit. A proof of principle is implemented and validated on IBM Quantum systems.
Paper Structure (35 sections, 21 theorems, 98 equations, 17 figures, 8 tables)

This paper contains 35 sections, 21 theorems, 98 equations, 17 figures, 8 tables.

Table of Contents

  1. Introduction
  2. Technical overview
  3. Prior art on nonlinear transformations
  4. Background on portfolio risk analysis in the energy industry
  5. A hybrid quantum-classical approach
  6. A hybrid quantum-classical approach
  7. Data encoding and loading
  8. Non-linear transformation with QHP
  9. Computing inner products
  10. Quantum Amplitude Estimation techniques
  11. Assembling the main variant and adapting the circuit for QAE
  12. Quantum complexity analysis
  13. Conditions for method correctness.
  14. Assumptions in complexity analysis.
  15. Conditions for asymptotic speedup.
  16. ...and 20 more sections

Key Result

Lemma A.1

Let $\bar{X}_S$ be the mean of $S$ i.i.d. random variables with mean $\mu >0$ and variance $\sigma^2$, and let $Y_S := \sqrt{\max\{a X _S + b, 0\}}$, for some real constants $a, b$ with $a \neq 0$ and $a\mu+b > 0$. Then $\frac{ Y_S -\sqrt{a\mu+b} }{\abs{a} \sigma/\sqrt{4S(a\mu+b)}}$ is asymptoticall asymptotically when $\epsilon \to 0$, where $\Phi$ is the CDF of the standard normal distribution.

Figures (17)

  • Figure 1: A conceptual representation of the paper, detailed in Subsec. \ref{['subsec:tech-overview']}. We propose a hybrid approach for the problem resolution. While tuning and implementing the algorithm, we derive five intermediate findings, which contribute to different implementation variants. Variant (c) is finally selected for providing the desired speedup, which constitutes our final result. The following terms are used in the picture: QHP for Quantum Hadamard Product (see Subsec. \ref{['subsec:qhp']}), *QAE for Quantum Amplitude Estimation or equivalent alternatives, BOE for Bidirectional Orthogonal Encoding (see Subsec. \ref{['subsec:data-encoding']}).
  • Figure 2: Characteristics of the four proposed implementation variants. Each row is a variant. See Table \ref{['tab:algo-compare-simplified']} for quantitative time scaling analysis. Variant (c) is highlighted as it provides the desired quantum speedup. Terms used in the table are defined in Section \ref{['sec:quantum-approach']}: see Subsec. \ref{['subsec:data-encoding']} for encoding ($U_T$ and $U_E$ are the loading unitaries for temperatures and prices, BOE stands for Bidirectional Orthogonal Encoding), Subsec. \ref{['subsec:qhp']} for powers (QHP is the Quantum Hadamard Product), Subsec. \ref{['subsec:inner']} for inner product, Subsec. \ref{['subsec:qae']} for the circuit design (*QAE is the Quantum Amplitude Estimation, or any equivalent technique). Refer to Appendix \ref{['appendix:complexity']} for a detailed description of the variants and for the extensive complexity analysis comparison.
  • Figure 3: Algorithm for the estimation of the contract value through the polynomial approximation of $f$. The figure instantiates the approach represented in the first column of Fig. \ref{['fig:contrib']}. Blue boxes represent classical pre- and post-processing, while green boxes are the core quantum processing.
  • Figure 4: Circuit implementation of the Quantum Hadamard Product producing the state $\ket{\psi_0 \odot \psi_1}$ when the bottom register is measured in the state $\ket{0}^{\otimes n}$. The success probability is $a^{-2}$, where $a$ is the normalization constant appearing in Eq. \ref{['eq:qhp']}. The CNOT notation on registers indicates pairwise application on belonging qubits (namely, a CNOT between the first qubits of both register, a CNOT between the second qubits of both registers, etc.)
  • Figure 5: QHP of $k=4$ vectors with mid-measurements and mid-resets, requiring $k-1=3$ iterations.
  • ...and 12 more figures

Theorems & Definitions (54)

  • Lemma A.1
  • proof
  • Proposition A.2: Sampling complexity of the swap test in amplitude encoding
  • proof
  • Remark A.3
  • Remark A.4
  • Proposition A.5: Sampling complexity of the ancilla-free method
  • proof
  • Remark A.6
  • Proposition A.7: Algorithm QHP + swap test in amplitude encoding
  • ...and 44 more