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Computing nonlinear functions over multilinear forms is a general problemwith applications in risk analysis. For instance in the domain of energyeconomics, accurate and timely risk management demands for efficient simulationof millions of scenarios, largely benefiting from computational speedups. Wedevelop a novel hybrid quantum-classical algorithm based on polynomialapproximation of nonlinear functions, computed through Quantum HadamardProducts, and we rigorously assess the conditions for its end-to-end speedupfor different implementation variants against classical algorithms. In oursetting, a quadratic quantum speedup, up to polylogarithmic factors, can beproven only when forms are bilinear and approximating polynomials have seconddegree, if efficient loading unitaries are available for the input data sets.We also enhance the bidirectional encoding, that allows tuning the balancebetween circuit depth and width, proposing an improved version that can beexploited for the calculation of inner products. Lastly, we exploit the dynamiccircuit capabilities, recently introduced on IBM Quantum devices, to reduce theaverage depth of the Quantum Hadamard Product circuit. A proof of principle isimplemented and validated on IBM Quantum systems.

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