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Quantum phase estimation (QPE) serves as a building block of many differentquantum algorithms and finds important applications in computational chemistryproblems. Despite the rapid development of quantum hardware, experimentaldemonstration of QPE for chemistry problems remains challenging due to itslarge circuit depth and the lack of quantum resources to protect the hardwarefrom noise with fully fault-tolerant protocols. In the present work, we take astep towards fault-tolerant quantum computing by demonstrating a QPE algorithmon a Quantinuum trapped-ion computer. We employ a Bayesian approach to QPE andintroduce a routine for optimal parameter selection, which we combine with a$[[ n+2,n,2 ]]$ quantum error detection code carefully tailored to the hardwarecapabilities. As a simple quantum chemistry example, we take a hydrogenmolecule represented by a two-qubit Hamiltonian and estimate its ground stateenergy using our QPE protocol. In the experiment, we use the quantum circuitscontaining as many as 920 physical two-qubit gates to estimate the ground stateenergy within $6\times 10^{-3}$ hartree of the exact value.

Demonstrating Bayesian Quantum Phase Estimation with Quantum Error Detection