Abstract
This paper introduces a new concept of reversible circuits based on EXOR-sum of Products-of-EXOR-sums (EPOE). Two algorithms are introduced that synthesize reversible functions using these new EPOE structures. The motivation for this work is to reduce the number of multiple controlled Toffoli gates and their number of inputs. To achieve these reductions the paper generalizes from existing 2-level AND-EXOR structures (ESOP) commonly used in reversible logic to a mixture of 3-level EXOR-AND-EXOR structures and ESOPs. Our approach can be applied to reversible and permutative quantum circuits to synthesize single output functions with one ancilla line. In addition, a variant of the algorithm with garbage lines is presented. A comparison of the ESOP minimizer EXORCISM-4 and variants of the new EPOE minimizer, called EPOEM-1s and EPOEM-1f, is presented. The results show that EPOE circuits do in fact achieve the above-stated cost reductions, in particular when expressed in terms of Maslov’s quantum cost model commonly used in quantum circuit synthesis.
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Notes
- 1.
A quantum permutative function is a binary reversible function where binary gates such as Toffoli and Feynman are internally realized with quantum primitives such as Controlled-V. The costs for reversible circuits and quantum permutative circuits differ considerably. This difference affects the choice of the structures to which functions are mapped and their respective synthesis algorithms.
- 2.
There are different definitions of quantum cost. For instance, cost can be set to the number of primitive two-qubit quantum gates from which reversible gates like multiple controlled Toffoli are used [19]. Alternatively cost can be the total number of quantum pulses to realize the circuit.
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Tran, L., Schaeffer, B., Gronquist, A., Perkowski, M., Kerntopf, P. (2014). Synthesis of Reversible Circuits Based on EXORs of Products of EXORs. In: Gavrilova, M., Tan, C., Thapliyal, H., Ranganathan, N. (eds) Transactions on Computational Science XXIV. Lecture Notes in Computer Science(), vol 8911. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-45711-5_7
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