The notion of quantum computers having an exponential growth of speed refers to their potential ability to solve certain problems much faster than classical computers. However, it's important to note that this exponential speedup does not apply to all types of computations but is specific to certain algorithms that can leverage the advantages of quantum computation.
In classical computing, the number of possible states that N classical bits can represent grows exponentially with N. Each bit can be in one of two states (0 or 1), so N bits can represent 2^N distinct states. In contrast, a quantum computer with N qubits can represent 2^N complex probability amplitudes simultaneously, which provides a much larger state space to explore.
Quantum algorithms, such as Shor's algorithm for integer factorization or Grover's algorithm for unstructured search, exploit this larger state space to perform computations more efficiently than their classical counterparts. These algorithms can achieve computational speedups that scale exponentially with the number of qubits involved, allowing certain problems to be solved significantly faster compared to classical algorithms.
Regarding the equivalent of a qubit in bytes, it's not a direct one-to-one correspondence. A qubit represents a unit of quantum information, whereas bytes are units of classical information storage. The number of classical bits required to store the information contained in a single qubit depends on the state of the qubit.
If a qubit is in a general superposition state, it requires an infinite number of classical bits to precisely describe its quantum state. However, if we want to represent a qubit in one of the two basis states (|0⟩ or |1⟩), it can be represented using a single classical bit. So, in terms of classical bits, a qubit can be thought of as being equivalent to one bit of classical information.
It's worth noting that quantum information processing goes beyond classical bits and bytes, and its representation and storage differ fundamentally from classical computing paradigms.