Arbitrary state creation via controlled measurement

Abstract

The initial state creation is a starting point of many quantum algorithms and usually is considered as a separate subroutine not included into the algorithm itself. There are many algorithms aimed on creation of special class of states. Our algorithm allows creating an arbitrary $n$-qubit pure quantum superposition state with precision of $m$-decimals (binary representation) for each probability amplitude. The algorithm uses one-qubit rotations, Hadamard transformations and C-NOT operations with multi-qubit controls. However, the crucial operation is the final controlled measurement of the ancilla state that removes the garbage part of the superposition state and allows to avoid the problem of small success probability in that measurement. We emphasize that rotation angles are predicted in advance by the required precision and therefore there is no classical calculation supplementing quantum algorithm. The depth and space of the algorithm growth with $n$ as, respectively, $O(2^n n)$ and $O(n)$. This algorithm can be a subroutine generating the required input state in various algorithms, in particular, in matrix-manipulation algorithms developed earlier.

Figures (2)

• Figure 1: The circuit for creating an arbitrary quantum state, $R=\prod_{k=1}^m R_{\varphi_k}$.
• Figure 2: The circuit for creating one-qubit state (\ref{['PsiRes']}).