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Quantum computing is poised to dramatically change the computational landscape, worldwide. Quantum computers can solve complex problems that are, at least in some cases, beyond the ability of even advanced future classical-style computers. In addition to being able to solve these classical computer-unsolvable problems, quantum computers have demonstrated a capability to solve some problems (such as prime factoring) much more efficiently than classical computing. This will create problems for encryption techniques, which depend on the difficulty of factoring for their security. Security, scientific, and other applications will require access to quantum computing resources to access their unique capabilities, speed and economic (aggregate computing time cost) benefits. Many scientific applications, as well as numerous other ones, use grid computing to provide benefits such as scalability and resource access. As these applications may benefit from quantum capabilities - and some future applications may require quantum capabilities - identifying how to integrate quantum computing systems into grid computing environments is critical. This paper discusses the benefits of grid-connected quantum computers and what is required to achieve this.