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

TL;DR

This paper introduces a hybrid quantum-classical algorithm for nonlinear function computation over multilinear forms, identifying specific conditions for quadratic quantum speedup, and demonstrates its practical implementation on IBM Quantum devices.

Contribution

The paper develops a novel hybrid quantum-classical approach using polynomial approximation and Quantum Hadamard Products, with conditions for quadratic speedup and practical circuit optimizations.

Findings

Quadratic quantum speedup achievable for bilinear forms with degree-two polynomial approximations.

Enhanced bidirectional encoding allows tuning circuit depth and width for inner product calculations.

Dynamic circuit capabilities reduce average circuit depth on IBM Quantum hardware.

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.