A Logarithm Depth Quantum Converter: From One-hot Encoding to Binary Encoding

Within the quantum computing, there are two ways to encode a normalized vector $\{{\alpha }_i\}$. They are one-hot encoding and binary coding. The one-hot encoding state is denoted as $\mid {\psi }_O^{(N)}⟩={\sum }_{i=0}^{N-1}{\alpha }_i{\mid 0⟩}^{\otimes N-i-1}\mid 1⟩{\mid 0⟩}^{\otimes i}$ and the binary encoding state is denoted as $\mid {\psi }_B^{(N)}⟩={\sum }_{i=0}^{N-1}{\alpha }_i\mid b_i⟩$, where $b_i$ is interpreted in binary of $i$ as the tensor product sequence of qubit states. In this paper, we present a method converting between the one-hot encoding state and the binary encoding state by taking the Edick state as the transition state, where the Edick state is defined as $\mid {\psi }_E^{(N)}⟩={\sum }_{i=0}^{N-1}{\alpha }_i{\mid 0⟩}^{\otimes N-i-1}{\mid 1⟩}^{\otimes i}$. Compared with the early work, our circuit achieves the exponential speedup with $O({\log }^{2}N)$ depth and $O(N)$ size.