We present a quantum annealing-based solution method for topology optimization (TO). In particular, we consider TO in a more general setting, i.e., applied to structures of continuum domains where designs are represented as distributed functions, referred to as continuum TO problems. According to the problem’s properties and structure, we formulate appropriate subproblems that can be solved on an annealing-based quantum computer. The methodology established can effectively tackle continuum TO problems formulated as mixed-integer nonlinear programs. To maintain the resulting subproblems small enough to be solved on quantum computers currently accessible with small numbers of qubits and limited connectivity, we further develop a splitting approach that splits the problem into two parts: the first part can be efficiently solved on classical computers, and the second part with a reduced number of variables is solved on a quantum computer. By such, a practical continuum TO problem of varying scales can be handled on the D-Wave quantum annealer. More specifically, we concern the minimum compliance, a canonical TO problem that seeks an optimal distribution of materials to minimize the compliance with desired material usage. The superior performance of the developed methodology is assessed and compared with the state-of-the-art heuristic classical methods, in terms of both solution quality and computational efficiency. The present work hence provides a promising new avenue of applying quantum computing to practical designs of topology for various applications.