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 $\left | \psi_O^{ (N)} \right \rangle=\sum_{ i=0}^{ N-1} \alpha_i \left |0 \right \rangle^{ \otimes N-i-1} \left |1 \right \rangle \left |0 \right \rangle ^{ \otimes i}$ and the binary encoding state is denoted as $\left | \psi_B^{ (N)} \right \rangle=\sum_{ i=0}^{ N-1} \alpha_i \left |b_i \right \rangle$, 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 $\left | \psi_E^{ (N)} \right \rangle=\sum_{ i=0}^{ N-1} \alpha_i \left |0 \right \rangle^{ \otimes N-i-1} \left |1 \right \rangle ^{ \otimes i}$. Compared with the early work, our circuit achieves the exponential speedup with $O(\log^2 N)$ depth and $O(N)$ size.
Categories: quant-ph