We present a fault-tolerant quantum algorithm for implementing the Discrete
Variable Representation (DVR) transformation, a technique widely used in
simulations of quantum-mechanical Hamiltonians. DVR provides a diagonal
representation of local operators and enables sparse Hamiltonian structures,
making it a powerful alternative to the finite basis representation (FBR),
particularly in high-dimensional problems. While DVR has been extensively used
in classical simulations, its quantum implementation, particularly using
Gaussian quadrature grids, remains underexplored. We develop a quantum circuit
that efficiently transforms FBR into DVR by following a recursive construction
based on quantum arithmetic operations, and we compare this approach with
methods that directly load DVR matrix elements using quantum read-only memory
(QROM). We analyze the quantum resources, including T-gate and qubit counts,
required for implementing the DVR unitary and discuss preferable choices of
QROM-based and recursive-based methods for a given matrix size and precision.
This study lays the groundwork for utilizing DVR Hamiltonians in quantum
algorithms such as quantum phase estimation with block encoding.

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