Simulating high-accuracy nuclear motion Hamiltonians using discrete variable representation and Walsh–Hadamard QROM on fault-tolerant quantum computers

Abstract We present a quantum algorithm for simulating rovibrational Hamiltonians on fault-tolerant quantum computers. The method integrates exact curvilinear kinetic energy operators and general-form potential energy surfaces expressed in a hybrid finite-basis/discrete-variable representation. The Hamiltonian is encoded as a unitary quantum circuit using a quantum read-only memory construction based on the Walsh–Hadamard transform, enabling high-accuracy quantum phase estimation of rovibrational energy levels and dynamics simulations. Our technique provides asymptotic reductions in both logical qubit count and T-gate complexity that are exponential in the number of atoms and at least polynomial in the total Hilbert-space size, relative to existing block-encoding techniques based on linear combinations of unitaries and variational basis representation. Compared with classical variational methods, it offers exponential memory savings and polynomial reductions in time complexity. The quantum volume required for computing the rovibrational spectrum of water can be reduced by up to 10 5 × compared with other quantum methods, increasing to at least 10 6 for a classically intractable 30-dimensional (12-atom) molecular system, with fewer than 300 logical qubits.