Sparse Quantum Voxel Encoding for Readout-Efficient Molecular Geometry Reconstruction on NISQ Devices

We propose a sparse computational-basis encoding of voxelized molecular geometries that converts molecular reconstruction from full-state tomography into support recovery by computational-basis sampling. To realize the encoding scheme, the molecular space is discretized into a 3D grid, and each atom's position and chemical species is mapped to a single computational basis state. This discretization introduces spatial quantization at the voxel-resolution scale. The molecule is then encoded as an equal superposition over this sparse set of occupied states, where we assume that a suitable state preparation method exists. In contrast to full state tomography, which requires on the order of $\mathcal{O}(3^n \times 10^{2\text{--}3})$ measurement shots, where $n$ is the number of qubits, our proposed encoding scheme reduces to a coupon-collector sampling problem in the computational basis. Complete recovery of an $A$-atom molecule requires $\mathcal{O}(A\log A)$ shots on noise-free hardware. On noisy hardware, the required number of shots increases. We demonstrate the method on the 156-qubit IBM Kingston device using 8-qubit circuits to reconstruct the discretized geometry of a 10-atom ethylamine molecule with high mean reconstruction recall using only $\mathcal{O}(10^2)$ shots despite substantial hardware noise. These results demonstrate that our proposed encoding scheme is a practical, readout-efficient representation for molecular geometries on near-term devices.