Quantum computation of multifractal exponents through the quantum wavelet transform

TL;DR

This paper demonstrates how quantum wavelet transforms can efficiently compute multifractal exponents of quantum states, offering exponential speedups over classical methods, with implications for quantum simulations of complex systems.

Contribution

It introduces quantum algorithms utilizing wavelet transforms to estimate multifractal exponents with polynomial speedup over classical approaches.

Findings

Quantum wavelet transform enables efficient extraction of multifractal exponents.

Numerical results suggest exponential speedup for rough fractality estimates.

Applicable to quantum maps and Anderson model at metal-insulator transition.

Abstract

We study the use of the quantum wavelet transform to extract efficiently information about the multifractal exponents for multifractal quantum states. We show that, combined with quantum simulation algorithms, it enables to build quantum algorithms for multifractal exponents with a polynomial gain compared to classical simulations. Numerical results indicate that a rough estimate of fractality could be obtained exponentially fast. Our findings are relevant e.g. for quantum simulations of multifractal quantum maps and of the Anderson model at the metal-insulator transition.