6. Complex Numbers: Arithmetic

Complex Arithmetic

That is the essential next step. If linear algebra is the structure of quantum mechanics, complex numbers are the very substance that makes it quantum. They are non-negotiable.

In quantum computing, the numbers $\alpha$ and $\beta$ in your state vector are not just any scalars; they are complex numbers. They are called probability amplitudes.

1. The Imaginary Unit and Complex Number Definition

A complex number $z$ is composed of a real part ($a$) and an imaginary part ($b$):

$$z = a + bi$$

where $i$ is the imaginary unit, defined such that $i^2 = -1$.

The crucial concept for QC is the phase. The complex number $z$ can be visualized on the complex plane (the Argand diagram), where its position defines both its magnitude (length from the origin) and its phase (the angle $\phi$ it makes with the real axis). It is this phase that quantum gates manipulate to create interference.

2. The Complex Conjugate ($z^*$)

The complex conjugate $z^*$ is found by flipping the sign of the imaginary part:

$$z^* = a - bi$$

We need this conjugate for the Inner Product because the probability of measuring a state is the squared magnitude ($|z|^2$) of the amplitude $z$.

$$P = |z|^2 = z^* z$$

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Your Task: Calculating Probability

Suppose we have an amplitude $\alpha$ for the $|0\rangle$ state, defined as:

$$\alpha = \frac{1}{\sqrt{2}} + \frac{i}{\sqrt{2}}$$

Calculate the probability $P$ of measuring the state $|0\rangle$. Remember, $P = \alpha^* \alpha$.