Problem:

In quantum information theory, quantum bits, called qubits for short, are used to store information instead of regular bits. A single qubit is typically represented as a vector such that a, B E C with the condition that lo|2 + |3|2 = 1. The quantities |a|2 and | |2 are probabilities that the quantum bit becomes a 0 or 1, respectively, when observed. In the same way that Boolean logic is the basis of many operations in classical computing, the basic operations in quantum computing are in the form of unitary matrices. An important operation is the Hadamard gate, which is a quantum logic gate that acts on a single qubit and is defined by We can represent k qubits as a vector of length 2" and define a k-qubit Hadamard gate Hx recursively: if k = 0, then Ho = [1]; if k > 0, then 1 HK = HK-1 Hk-1 V2 Hk-1 - HK-1 Give an efficient algorithm that when given a quantum state of k qubits as a vector x of length 2", computes the result of the Hadamard transform Hxx. Prove the correctness of your algorithm and analyze its running time