No, the hypothesis you stated is not correct. The number of steps required to find the valid item using Grover's algorithm on a quantum computer is not directly related to the order of the nth root of N.
Grover's algorithm is a quantum search algorithm that provides a quadratic speedup compared to classical search algorithms. It can be used to search an unsorted database of N items and find the desired item in approximately O(√N) steps, where O is the Big O notation representing the order of complexity.
However, it's important to note that this quadratic speedup is achieved when the search problem has an equal probability of finding any given item in the database. In other words, when all N items have the same likelihood of being the desired item. This scenario is commonly referred to as an "unstructured search problem."
In your hypothesis, you specified that there is only one valid item among N items to be searched. In this case, Grover's algorithm is not directly applicable because the search problem is considered "structured" rather than "unstructured." Grover's algorithm assumes an equal probability distribution across the search space, and when applied to a structured search problem, it would not provide the same quadratic speedup.
For structured search problems where there is only one valid item among N items, alternative algorithms or techniques may be more suitable and efficient than Grover's algorithm. The specific approach would depend on the nature of the problem and the available resources.