Computational mechanics, since its inception in the ‘60s, has tracked developments in both theoretical and applied mechanics, as well as scientific computing. It started with the basics of finite-element development and solvers, with scant attention given to material modeling and physics of solids. However, early in the game a more ambitious plan emerged that encompassed nonlinear material behavior, history dependence and dissipation, coupled thermodynamics and phase transitions, extreme conditions of temperature, pressure and rate of deformation, and generally the full gamut of constitutive behavior of solids. In that way, computational mechanics tracked sweeping trends in the field of solid mechanics, including rational mechanics, micromechanics and multiscale analysis, uncertainty quantification and, more recently, data science and quantum computing. Thus, epochal advances in ab initio methods and experimental science have effectively changed solid mechanics from a data-poor and mostly empirical field to an increasingly data-rich and physics-based field. This paradigm shift raises many fundamental challenges concerned with how best to generate, manage and use material data to enable discovery, prediction and design. Deep mathematical and practical questions arise regarding how to generate data, how to build data into boundary value problems, how to ensure predictive accuracy and convergence of the solutions, and others. Recently, a new potentially game-changing paradigm has arrived in computational mechanics, namely, quantum computing. Quantum computers, operating on entirely different physical principles and abstractions from those of classical digital computers, have the unique ability to simultaneously evolve the state of an entire quantum system, which leads to quantum parallelism and interference. In addition, quantum entanglement enables the representation of systems of enormous dimensionality with a modicum of quantum bits, or qubits. Despite these prospects, opportunities to bring quantum computing to bear on problems of computational mechanics in general and, specifically, on data-driven computational mechanics, remain largely unexplored. Exploratory work to date suggests that quantum computing can indeed accelerate exponentially stubborn long-standing bottlenecks in classical multiscale and datadriven computational mechanics, thus bringing them closer to the realm of feasibility and practicality.