Quantum-Accelerated Gowers $U_2$ Norm for Bent Boolean Functions

TL;DR

This paper introduces a hybrid quantum-classical genetic algorithm utilizing a quantum circuit to evaluate the Gowers U2 norm, enabling efficient construction of bent Boolean functions for large variable counts.

Contribution

The work presents a complexity-theoretic separation showing quantum evaluation drastically reduces computational overhead compared to classical methods.

Findings

Validated on 6 and 8 variable systems, achieving exact bent threshold for n=8.

Quantum evaluation reproduces classical results with finite-sampling noise.

For n > 25, quantum evaluation offers a decisive computational advantage.

Abstract

Bent Boolean functions extremal objects that maximally resist affine approximation are notoriously hard to construct for large numbers of variables. We propose a hybrid quantum-classical genetic algorithm (GA) that uses a \emph{quantum circuit} to evaluate the Gowers $U_2$ norm as the evolutionary fitness function. Our central contribution is a complexity-theoretic separation: the quantum evaluation circuit requires only $3n$ qubits and $\bigO(n^2)$ two-qubit gates per function query, whereas the classical computation of the exact Gowers $U_2$ norm demands $\bigO(2^{2n})$ arithmetic operations an exponential overhead that renders it infeasible for $n \gtrsim 25$. We validate the framework on $n=6$ and $n=8$ variable systems. For $n=8$, our classical GA run extended to 1000 generations achieves best fitness $\Utwof = 0.250000$ \emph{exactly} the theoretical bent threshold $2^{-n/4}$ with average fitness $0.257267$, confirming that the Gowers $U_2$ norm is a superior fitness criterion over Walsh-Hadamard spectral flatness. Quantum-assisted evaluation faithfully reproduces the classical trajectory up to finite-sampling noise, and our complexity analysis demonstrates that for $n > 25$ the quantum evaluator provides a decisive computational advantage on fault-tolerant hardware.