Implementing Quantum Rational Transformations via Real-Time Evolution

Rational functions are invaluable in scientific computing, offering rapid convergence where polynomial methods often fall short. However, their potential to enhance quantum algorithms remains underexplored. In this talk, we present efficient implementations of rational transformations of target operators on quantum hardware. By employing integral representations of the operator resolvent, we demonstrate that quantum rational transformations can be performed efficiently using Hamiltonian simulations, specifically via the linear-combination-of-unitaries (LCU) method. We introduce two complementary LCU-based strategies — discrete-time and continuous-time approaches — for approximating integral representations of the resolvent. These approaches provide versatile pathways for implementing quantum rational transformations. To illustrate the power of this framework, we develop a rational approach for resolving many-body spectra using only real-time quantum dynamics. Our numerical simulations on spin systems show that this approach is both compact and highly accurate, yielding accurate estimates of low-lying energies.