Figure 2

Complex conjugation circuits. (A) Quantum circuit implementation of the conditional phase shift operation \({\hat{{\rm{\Phi }}}}_{k=6}\) for a component \(\mathrm{|0110}\rangle \). The circuit involves three types of gates: 1-qubit NOT gate \(\hat{X}|b\rangle =|b\oplus 1\rangle \), 1-qubit unitary rotation \(\hat{T}(\,-\,2{\phi }_{k})[|0\rangle +a|1\rangle ]=|0\rangle +a{e}^{-2i{\phi }_{k}}|1\rangle \), and 3-qubit Toffoli gate which reverts the state of the last target qubit if and only if two first control qubits are both set to \(\mathrm{|1}\rangle \): \({\hat{{\rm{\Lambda }}}}_{2}|11\rangle \otimes |b\rangle =|11\rangle \otimes |b\oplus 1\rangle \). The first three Toffoli gates set the ancillary qubit c₂ into \(\mathrm{|1}\rangle \) if and only if the qubit register is set to the \(\mathrm{|0110}\rangle \) state and the last three Toffoli gates restore the original state \(|{b}_{0}{b}_{1}{b}_{2}{b}_{3}\rangle \otimes |000\rangle \). (B) The quantum circuit with the optimal Toffoli gate arrangement which conjugates four components: \(|1111\rangle \), \(\mathrm{|1110}\rangle \), \(\mathrm{|1101}\rangle \) and \(\mathrm{|1100}\rangle \). The circuit is partitioned into several nested blocks (subroutines) \({A}_{11??}\supset {A}_{111?}\), the question marks standing for an unknown bit value. The first-level block (blue) A_111? conjugates only computational states where three senior qubits \(|{b}_{0}{b}_{1}{b}_{2}\rangle \) are all set to \(\mathrm{|1}\rangle \). The next-level block (red) A_11?? contains as a subroutine the block A_111? and conjugates all components \(|{b}_{0}{b}_{1}\rangle =|11\rangle \).