Optimizing the Phase Estimation Algorithm Applied to the Quantum Simulation of Heisenberg-Type Hamiltonians

The phase estimation algorithm is a powerful quantum algorithm with applications in cryptography, number theory, and simulation of quantum systems. We use this algorithm to simulate the time evolution of a system of two spin-1/2 particles under a Heisenberg Hamiltonian. The evolution is performed through both classical simulations of quantum computers and real quantum computers via IBM's Qiskit platform. We also introduce three optimizations to the algorithm: circular, iterative, and Bayesian. We apply these optimizations to our simulations and investigate how the performance improves. We also discuss the paradigms of iterative and update-based algorithms, which are attributes of these optimizations that can improve quantum algorithms generally.