In simple terms, a qubit can be thought of as a mathematical object that represents the state of a quantum system. This state is described using a unit vector in a two-dimensional mathematical space called a Hilbert space.
In classical computing, a bit can be in one of two states: 0 or 1. It can represent information as either true or false, on or off, etc. Similarly, a qubit can also be in one of two states, conventionally represented as |0⟩ and |1⟩. However, what makes qubits special is that they can exist in a superposition of both states simultaneously.
When we say that a qubit corresponds to a unit vector in a two-dimensional Hilbert space, we mean that the state of the qubit can be described by a combination of these two basis vectors, |0⟩ and |1⟩, with complex coefficients. These coefficients give us information about the probability amplitudes associated with each basis state.
For example, a qubit can be in a state |ψ⟩ = α|0⟩ + β|1⟩, where α and β are complex numbers known as probability amplitudes. The coefficients α and β determine the probability of measuring the qubit in the states |0⟩ and |1⟩, respectively. The probabilities are given by the squared magnitudes of the probability amplitudes: |α|^2 and |β|^2.
The two-dimensional Hilbert space is a mathematical framework that allows us to describe and manipulate the quantum states of qubits. By representing qubits as unit vectors in this space, we can perform calculations and transformations on the qubit states using mathematical operations such as rotations, superpositions, and entanglement, which form the basis of quantum computation and information processing.