Arbitrary State Creation via Controlled Measurement

We propose an algorithm to create 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. The depth of the circuit is O(2ⁿn), its space is O(n). We emphasize that the parameters of the utilized unitary transformations are predicted in advance by the required precision and therefore there is no classical calculation supplementing this quantum algorithm. Finalizing the state-creation we perform the measurement of the ancilla state with a certain desired output with the purpose of removing all the garbage from the created superposition state. If n and m are independent and n ≫ m, the probability of access to the desired ancilla-state (success probability) is ~ 2⁻ⁿ, which requires O(2ⁿ) runs of the algorithm and leads to the overall depth O(4ⁿn). However, the situation can be significantly improved by replacing the usual measurement with the controlled measurement of the ancilla state that, first of all, removes the garbage part of the superposition state and, second (and most important), allows to avoid both the problem of small success probability to the desired ancilla state and multiple runs of the algorithm. As a consequence, the depth of the algorithm is O(2ⁿn) in this case. If m serves to provide the required fidelity of state approximation, then the above parameters increase. This algorithm can be a subroutine generating the required input state in various algorithms, in particular, in matrix-manipulation algorithms developed earlier.