Analysis of converting \(\mathfrak{C^{0}}\)-circuit into \(\mathfrak{C^{*}}\)-circuit

Abstract

A \(\mathfrak{C^{*}}\)-circuit, which was proposed in Asiacypt 2022 by Huang and Sun (Advances in cryptology – ASIACRYPT 2022, pp. 614–644, 2022), can directly perform calculations with the existing quantum states, thereby reducing the use of quantum resources in quantum logic synthesis. We theoretically prove how to convert a \(\mathfrak{C^{0}}\)-circuit into the corresponding \(\mathfrak{C^{*}}\)-circuit through two lemmas and one theorem. The first lemma proves the interchangeability of CNOT gates and NOT gates by using the equivalence of quantum circuits. The second lemma proves that adding CNOT gates to the front of a quantum circuit whose initial states are all \(|0\rangle \)s will not change the output states of the circuit. The theorem is used to describe what kind of \(\mathfrak{C^{0}}\)-circuit can be transformed into \(\mathfrak{C^{*}}\)-circuit, and the correctness of this transformation is proved. Our work will provide a theoretical basis for converting \(\mathfrak{C^{0}}\)-circuit to \(\mathfrak{C^{*}}\)-circuit. Then applying the theoretical analysis results to the multiplication over \(\text{GF}(2^{8})\), the constructed quantum circuit needs 27 Toffoli gates and 118 CNOT gates, which is 15 fewer Toffoli gates and 43 CNOT gates than the current best result. This shows that the method of constructing quantum circuits by using the conversion of \(\mathfrak{C^{0}}\)-circuit to \(\mathfrak{C^{*}}\)-circuit is very efficient.