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Lost in Translation

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We consider the question of forward and backward translation between measurement-based quantum computing, called patterns, and quantum circuit computation. It is known that the class of patterns with a particular properties, having flow, is in one-to-one correspondence with quantum circuits. However we show that a more general class of patterns, those having generalised flow, will sometime translate to imaginary circuits, quantum circuits with time-like curves. Extending this approach, we first present the semantics of quantum circuits with time-like curves in terms of post-selection quantum computing and then characterise the class of curves with unitary or completely-positive semantic. Finally we present the re-write rules for opening the loops to transform an imaginary circuit to a normal circuit and discuss the connection between time-like curves and depth complexity.