Reversible Logic Circuit Synthesis Â£ Vivek V. Shende, Aditya K. Prasad, Igor L. Markov, and John P. Hayes Advanced Computer Architecture Laboratory, University of Michigan, Ann Arbor, MI 48109-2122 vshende,akprasad,imarkov,jhayes @umich.edu ABSTRACT Reversible or information-lossless circuits have applications in digital signal processing, communication, computer graphics and cryptography. They are also a fundamental requirement in the emerging eld of quantum computation. We investigate the synthesis of reversible circuits that employ a minimum number of gates and contain no redundant input-output line-pairs (temporary storage channels). We prove constructively that every even permutation can be implemented without temporary storage using NOT, CNOT and TOFFOLI gates. We describe an algorithm for the synthesis of optimal circuits and study the reversible functions on three wires, reporting distributions of circuit sizes. Finally, in an application important to quantum computing, we synthesize oracle circuits for Grover s search algorithm, and show a signi cant improvement over a previously proposed synthesis algorithm. 1. INTRODUCTION In most computing tasks, the number of output bits is relatively small compared to the number of input bits. For example, in a decision problem, the output is only one bit (yes or no) and the input can be as large as
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