We present a mathematical theory for a new type of quantum computer called a duality quantum computer that has recently been proposed. We discuss the nonunitarity of certain circuits of a duality quantum computer and point out a paradoxical situation that occurs when mixed states are considered. It is shown that a duality quantum computer can measure itself without needing a separate measurement apparatus to determine its final state.
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Gudder, S. Mathematical Theory of Duality Quantum Computers. Quantum Inf Process 6, 37–48 (2007). https://doi.org/10.1007/s11128-006-0040-3
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DOI: https://doi.org/10.1007/s11128-006-0040-3