Garbage in Reversible Designs of Multiple Output Functions By: Dmitri Maslov Gerhard Dueck

Definitions Definition 1. Multiple output Boolean function is called reversible iff: 1. 2. - is a bijection. Examples. 1. - (NOT) is reversible. 2. - (Feynman gate) is reversible. 3. - is not reversible.

Definitions … NOT Toffoli Generalized Toffoli CNOT (Feynman) Definition 2. Garbage is the number of outputs added to make an n-input k-output function reversible.

Minimal garbage 24 325 209 8 25 vg2 9 45 60 1 9sym 28 38 10 7 5xp1 6 Theorem 1. For an n-input k-output function the minimal garbage to be added to make it reversible is , where M – maximum number of times an output pattern is repeated in the truth table. 24 325 209 8 25 vg2 9 45 60 1 9sym 28 38 10 7 5xp1 6 43 3 rd73 4 15 19 5 rd53 Min G RPGAG WCG out in name

Current systematic approaches 1. Reversible cascades: realizes a multiple output function in PLA-like fashion. 2. Reversible programmable gate array: realizes a multiple output symmetric function. 3. Spectral techniques for Toffoli family reversible synthesis.

New model What do we want from the new model? Constraints on the reversible logic design: no fan-outs; no feed-backs. Leaves us with the cascade structures as the only one satisfying these conditions. What do we want from the new model? 1. Minimal garbage. 2. Systematic synthesis method. 3. A structure, which can be easily implemented in a technology.

New model - different line types. Network design: Make a multiple output function reversible; realize it as a cascade of these new gates. Sample gate.

Regular synthesis Our approach: - build cascade by adding one gate at a time (start with empty cascade); - start building the cascade from its end; - try to add the gates so that the cascade applied to the output vector produces a function close to the input vector; - suppose patterns are in the order - while work with a pattern, do not apply gates that affect patterns with lower order.

Regular synthesis 000 000 011 101 110 111 000 001 010 100 011 101 110 111 000 001 010 100 101 111 110 000 001 010 100 011 011 101 110 100 000 001 010 111 111 100 110 101 000 001 010 011 001 001 010 010 100 100 011 101 101 110 110 111 011 111 Can use a single gate to interchange Hamming distance-1 patterns.

Some theoretical results Theorem 1. There exists a reversible function which requires at least gates. Theorem 2. Every reversible function can be realized with no more than gates. The theoretical algorithm usually creates expensive circuits, so the actual design method is different.

Quantum cost analysis Minimizing NOTs procedure (post processing). A new gate has NOT array in front of the generalized Toffoli gate. Note, the circuit design procedure stays the same.

Quantum cost analysis controls garbage Toffoli gate cost New gate cost Relative cost Average rel. cost 2 7 9 1.286 1.143 3 13 16 1.231 1.115 4 29 33 1.138 1.069 5 61 66 1.082 1.041 6 125 131 1.048 1.024 112 118 1.054 1.027 253 260 1.028 1.014 124 1.056 8 509 517 1.016 1.008 172 180 1.047 1.023 Introduction of the new gates brings only marginal gate cost increase.

Acknowledgements We would like to thank Dr. Perkowski for his help and valuable comments on the early stage of the research.

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