Hamiltonian Learning Through Lie-Theoretic Methods

This paper presents a method for identifying unknown coefficients of basis elements in an effective Hamiltonian of single-qubit cases. This is done directly from the measurement statistics of a quantum system's unitary evolution, utilizing a unique method of Lie algebra coefficient recovery. The method exploits the fact that repeated application of a unitary operator amplifies the generator coefficient linearly in the exponent. The result produces measurement probabilities that constrain the unknown coefficient differently at each power tested. Inversion of any single measurement probability yields a multivalued set of candidates due to the periodic nature of the trigonometric functions involved. However, the true coefficient can be identified as the unique true value satisfying a consistency criterion across all powers simultaneously. The method is demonstrated numerically for the single-generator Hamiltonian (He=aX) and extended analytically to the two-generator case (He=aX+bZ), where geometric conjugation reduces the Hamiltonian to an effective single-generator form. Numerical simulations further verify the geometric reduction and the subsequent coefficient reconstruction for representative two-generator Hamiltonians, given a known rotation angle. In both cases, the correct coefficients are recovered from measurement statistics, without ancilla qubits, coherent control circuits, or prior knowledge of the system's spectrum; for the two-generator case, the rotation angle used to align the Hamiltonian is currently taken as given rather than estimated from data.