Cellular automata (CA) are abstract computational models of dynamic systems that change some features with space and time. Music is the art of organising sounds in space and time, and it can be modelled as a dynamic system. Hence, CA are of interest to composers working with generative music. The art of generating music with CA hinges on the design of algorithms to evolve patterns of data and methods to render those patterns into musical forms. This paper introduces methods for creating original music using partitioned quantum cellular automata (PQCA). PQCA consist of an approach to implementing CA on quantum computers. Quantum computers leverage properties of quantum mechanics to perform computations differently from classical computers, with alleged advantages. The paper begins with some explanations of background concepts, including CA, quantum computing, and PQCA. Then, it details the PQCA systems that we have been developing to generate music and discusses practical examples. PQCA-generated materials for Qubism, a professional piece of music composed for London Sinfonietta, are included. The PQCA systems presented here were run on real quantum computers rather than simulations thereof. The rationale for doing so is also discussed.