1. Linear Algebra: Vectors

Linear Algebra: Vectors

Let's get to work. In classical mechanics, we might use a vector to describe the position or velocity of a ball. In Quantum Computing, we use a vector to describe the state of the system—literally everything we know about it.

These aren't just arrows in space; they are specific mathematical objects called State Vectors, and they live in a complex vector space called a Hilbert Space.

1. The Standard Basis (The "Axes")

In a classical computer, a bit is either 0 or 1. In a quantum computer, we treat these as column vectors. We call this the "Computational Basis."

• The state "Zero" is denoted as $|0\rangle$: $$|0\rangle = \begin{pmatrix} 1 \\ 0 \end{pmatrix}$$
• The state "One" is denoted as $|1\rangle$: $$|1\rangle = \begin{pmatrix} 0 \\ 1 \end{pmatrix}$$

(Note: The symbol $|\psi\rangle$ is called a "ket". This is Dirac Notation, and it's the standard language of quantum mechanics.)

2. Superposition (Linear Combination)

The power of quantum computing comes from Linearity. We can add these vectors together. A qubit can be in a state that is a linear combination (superposition) of $|0\rangle$ and $|1\rangle$:

Here, $\alpha$ (alpha) and $\beta$ (beta) are numbers that describe "how much" of 0 and 1 are in the state.