Essential mathematics — complex numbers

Meet the imaginary unit

The number $i$ satisfies $i^2=-1$. A complex number $z=a+bi$ has a real part $a$ and an imaginary part $b$. You can picture it as the point $(a,b)$ on a plane.

Addition works component by component:

$$(1+2i)+(3-i)=4+i.$$

Multiplication uses ordinary expansion and $i^2=-1$:

$$(1+i)(1-i)=1-i+i-i^2=2.$$

Squared magnitude becomes probability

The conjugate of $z=a+bi$ is $z^*=a-bi$. The squared magnitude is

$$|z|^2=z^*z=a^2+b^2.$$

If an amplitude is $\alpha=(1+i)/2$, then $|\alpha|^2=(1/2)^2+(1/2)^2=1/2$. It is the squared magnitude, not the real part and not the amplitude itself, that becomes a basis-measurement probability.

Phase is direction

A unit-magnitude complex number can be written

$$e^{i\phi}=\cos\phi+i\sin\phi.$$

Multiplication by this number rotates an amplitude in the complex plane. For example, multiplying $1$ by $i$ rotates by $\pi/2$ radians, or 90 degrees. Its magnitude stays one.

In a quantum state, relative phase between amplitudes affects interference. Multiplying the whole state by the same unit phase does not change physical predictions.

Work it through

For $\lvert\psi\rangle=(\sqrt3/2)\lvert0\rangle+(i/2)\lvert1\rangle$, the probabilities are $3/4$ and $1/4$. They sum to one. Replacing $i/2$ with $-i/2$ keeps those Z probabilities but changes the relative phase.

Try it: calculate $|3+4i|^2$, then divide $3+4i$ by 5. The results are 25 and a unit-magnitude complex number, respectively.

Reference: IBM Quantum — Single systems.