Accessing the geometric origin of shift current in noncentrosymmetric quantum materials is essential for understanding nonlinear photoresponses governed by band geometry and Berry connections. We present a quantum-algorithmic approach, implemented on a classical simulator of quantum circuits, to compute the shift current in a phase-coherent and gauge-consistent manner. By expressing the geometric ingredients of the bulk photovoltaic effect as observables accessible within a quantum-circuit framework, the method enables the evaluation of the second-order dc photocurrent without explicit gauge fixing. Using the one-dimensional Rice–Mele model as a benchmark, we obtain shift-current spectra in quantitative agreement with classical calculations and exhibiting smooth behavior across the Brillouin zone. These results position quantum algorithms as a promising framework for addressing shift-current responses and, more broadly, nonlinear optical phenomena in quantum materials.