  Quantum phase estimation (QPE) serves as a building block of many different
quantum algorithms and finds important applications in computational chemistry
problems. Despite the rapid development of quantum hardware, experimental
demonstration of QPE for chemistry problems remains challenging due to its
large circuit depth and the lack of quantum resources to protect the hardware
from noise with fully fault-tolerant protocols. In the present work, we take a
step towards fault-tolerant quantum computing by demonstrating a QPE algorithm
on a Quantinuum trapped-ion computer. We employ a Bayesian approach to QPE and
introduce a routine for optimal parameter selection, which we combine with a
$[[ n+2,n,2 ]]$ quantum error detection code carefully tailored to the hardware
capabilities. As a simple quantum chemistry example, we take a hydrogen
molecule represented by a two-qubit Hamiltonian and estimate its ground state
energy using our QPE protocol. In the experiment, we use the quantum circuits
containing as many as 920 physical two-qubit gates to estimate the ground state
energy within $6\times 10^{-3}$ hartree of the exact value.
