Quantum computing exploits quantum mechanics to perform certain computations more efficiently than classical computers. Current quantum computers have performed carefully tailored computational tasks that would be difficult or impossible for even the fastest supercomputers in the world. This “quantum supremacy” result demonstrates that quantum computing is more powerful than classical computing in some computational regimes. At present, it is unknown if any computational problems related to the Earth's subsurface fall within these regimes. Here, we describe an approach to performing seismic inverse analysis that combines a type of quantum computer called a quantum annealer with classical computing. This approach improves upon past work on applying quantum computing to the subsurface (via subsurface hydrology) in two ways. First, the seismic inverse problem enables better performance from the quantum annealer because of the Earth's relatively narrow distribution of P-wave velocities compared to the broad distribution of hydraulic conductivities. Second, we develop an iterative approach to quantum-computational inverse analysis, which works with a realistic set of observations. By contrast, the previous method used an inverse method that depended on an impractically dense set of observations. In combination, these two advances significantly narrow the gap a quantum-computational advantage for a practical subsurface geoscience problem. Closing the gap completely requires more work, but has the potential to dramatically accelerate inverse analyses for subsurface geoscience.