A novel quantum algorithm for converting between one-hot and binary encodings

In the domain of quantum computing, two widely employed techniques for encoding a normalized vector of length N, denoted as \(\{ \alpha _i \}\) $\{{\alpha }_i\}$ , are one-hot encoding and binary encoding. The one-hot encoding state is represented as \(\vert \psi _{OH}^{(N)} \rangle \) $|{\psi }_{\mathit{OH}}^{(N)}⟩$ and can be expressed as: \(\vert \psi _{OH}^{(N)} \rangle =\sum _{i=0}^{N-1} \alpha _i \vert 0 \rangle ^{\otimes N-i-1} \vert 1 \rangle \vert 0 \rangle ^{\otimes i}\) $|{\psi }_{\mathit{OH}}^{(N)}⟩={\sum }_{i=0}^{N-1}{\alpha }_i{|0⟩}^{\otimes N-i-1}{|1⟩|0⟩}^{\otimes i}$ . On the other hand, the binary encoding state is symbolized as \(\vert \psi _{BI}^{(N)} \rangle \) $|{\psi }_{\mathit{BI}}^{(N)}⟩$ and is defined as: \(\vert \psi _{BI}^{(N)} \rangle =\sum _{i=0}^{N-1} \alpha _i \vert b_i \rangle \) $|{\psi }_{\mathit{BI}}^{(N)}⟩={\sum }_{i=0}^{N-1}{\alpha }_i|b_i⟩$ , where \(b_i\) $b_i$ corresponds to the binary representation of i. In this paper, we introduce a method for converting between the one-hot encoding state and the binary encoding state, utilizing the Domain Wall state as an intermediary. The Domain Wall state, denoted as \(\vert \psi _{DW}^{(N)} \rangle \) $|{\psi }_{\mathit{DW}}^{(N)}⟩$ , is defined as: \(\vert \psi _{DW}^{(N)} \rangle =\sum _{i=0}^{N-1} \alpha _i \vert 0 \rangle ^{\otimes N-i-1} \vert 1 \rangle ^{\otimes i}\) $|{\psi }_{\mathit{DW}}^{(N)}⟩={\sum }_{i=0}^{N-1}{\alpha }_i{|0⟩}^{\otimes N-i-1}{|1⟩}^{\otimes i}$ . Our proposed circuit achieves a depth of \(O(\log ^2 N)\) $O({log}^{2}N)$ and a size of O(N).Kindly check and confirm that the corresponding author mail id is correctly identified.