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Quantum computing promises exponential speed-ups for important simulation and optimization problems. It also poses new CAD problems that are similar to, but more challenging, than the related problems in classical (nonquantum) CAD, such as determining if two states or circuits are functionally equivalent. While differences in classical states are easy to detect, quantum states, which are represented by complex-valued vectors, exhibit subtle differences leading to several notions of equivalence. This provides flexibility in optimizing quantum circuits, but leads to difficult new equivalence-checking issues for simulation and synthesis. We identify several different equivalencechecking problems and present algorithms for practical benchmarks, including quantum communication and search circuits, which are shown to be very fast and robust for hundreds of qubits.

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