Complex Numbers & Euler's Formula

Every quantum state you will meet on this platform is a list of complex numbers called amplitudes. A single qubit is described by two of them, $\alpha$ and $\beta$, in $\lvert\psi\rangle=\alpha\lvert0\rangle+\beta\lvert1\rangle$; a register of $n$ qubits needs $2^n$ of them. Real numbers are not enough: the phenomenon of interference, in which two computational paths cancel each other, depends on amplitudes that carry a direction (a phase) as well as a size. A phase can rotate a contribution by any angle, not just flip its sign, and the natural language for rotations in a plane is the complex numbers.

This lesson builds the arithmetic of $\mathbb{C}$ from scratch and proves every identity used later in the curriculum. The single most important result is Euler's formula $e^{i\theta}=\cos\theta+i\sin\theta$, which turns multiplication of complex numbers into addition of angles. It is the reason phase gates such as $S=\mathrm{diag}(1,i)$ and $T=\mathrm{diag}(1,e^{i\pi/4})$ are so easy to compose, and the reason the quantum Fourier transform works.

Addition is component-wise, $(a+ib)+(c+id)=(a+c)+i(b+d)$, exactly like adding arrows in the plane. Multiplication is defined by expanding the product with the ordinary distributive law and then replacing $i^2$ by $-1$. The derivation below is the only rule you ever need to memorise; everything else follows from it.

Worked example. Take $z=3+4i$ and $w=1-2i$. With $a=3,b=4,c=1,d=-2$:

$zw=(3\cdot1-4\cdot(-2))+i(3\cdot(-2)+4\cdot1)=(3+8)+i(-6+4)=11-2i.$

Addition gives $z+w=4+2i$ and subtraction $z-w=2+6i$. With the addition and multiplication rules, $\mathbb{C}$ is a field: both operations are associative and commutative, multiplication distributes over addition, $0$ and $1$ are the neutral elements, and (as we prove next) every nonzero number has a multiplicative inverse.

Worked example. For $z=3+4i$ we have $\bar z=3-4i$ and $\lvert z\rvert^2=9+16=25$, so

$\frac{1}{3+4i}=\frac{3-4i}{25}=\frac{3}{25}-\frac{4}{25}i=0.12-0.16i.$

Check: $(3+4i)(0.12-0.16i)=(0.36+0.64)+i(-0.48+0.48)=1$. To divide $w=1-2i$ by $z$ multiply numerator and denominator by $\bar z$:

$\frac{1-2i}{3+4i}=\frac{(1-2i)(3-4i)}{25}=\frac{(3-8)+i(-4-6)}{25}=\frac{-5-10i}{25}=-\frac15-\frac25 i.$

Multiplicativity of the modulus is a useful sanity check: $\lvert 3+4i\rvert=5$, $\lvert1-2i\rvert=\sqrt5$, and indeed $\lvert(3+4i)(1-2i)\rvert=\lvert 11-2i\rvert=\sqrt{121+4}=\sqrt{125}=5\sqrt5$.

A point of the plane can be located either by Cartesian coordinates $(a,b)$ or by its distance $r=\lvert z\rvert$ from the origin together with the angle $\theta$ measured counter-clockwise from the positive real axis. Elementary trigonometry gives $a=r\cos\theta$ and $b=r\sin\theta$, so

$z=r(\cos\theta+i\sin\theta).$

The angle $\theta=\arg z$ is the argument (or phase) of $z$. It is defined only up to adding integer multiples of $2\pi$; the principal value lies in $(-\pi,\pi]$ and is computed as $\theta=\operatorname{atan2}(b,a)$, which respects the quadrant of $(a,b)$. For example $1+i$ has $r=\sqrt2$ and $\theta=\pi/4$, whereas $-1-i$ has $r=\sqrt2$ and $\theta=-3\pi/4$ even though both have $b/a=1$.

The combination $\cos\theta+i\sin\theta$ appears so often that it deserves a shorter name. Euler discovered that it is precisely the exponential function evaluated at the imaginary number $i\theta$. We define the complex exponential by the same power series as the real one, $e^{w}=\sum_{k\ge0}w^k/k!$, which converges for every $w\in\mathbb{C}$, and derive the formula.

Special values follow immediately: $e^{i\pi/2}=i$, $e^{i\pi}=-1$ (so $e^{i\pi}+1=0$), $e^{3i\pi/2}=-i$ and $e^{2\pi i}=1$. The conjugate of a phase is the opposite phase, $\overline{e^{i\theta}}=\cos\theta-i\sin\theta=e^{-i\theta}$, so for $z=re^{i\theta}$ we have $\bar z=re^{-i\theta}$. The number $e^{i\pi/4}=\tfrac{1}{\sqrt2}+\tfrac{i}{\sqrt2}$ is the nontrivial entry of the $T$ gate.

Worked conversions. $-\sqrt3+i$ has $r=\sqrt{3+1}=2$ and lies in the second quadrant with $\cos\theta=-\tfrac{\sqrt3}{2}$, $\sin\theta=\tfrac12$, hence $\theta=5\pi/6$ and $-\sqrt3+i=2e^{5\pi i/6}$. Conversely $4e^{-i\pi/3}=4(\tfrac12-\tfrac{\sqrt3}{2}i)=2-2\sqrt3\,i$.

Worked example. Compute $(1+i)^8$. In polar form $1+i=\sqrt2\,e^{i\pi/4}$, so by De Moivre

$(1+i)^8=(\sqrt2)^8e^{8i\pi/4}=16\,e^{2\pi i}=16.$

Doing this in Cartesian form requires three squarings: $(1+i)^2=1+2i+i^2=2i$, $(2i)^2=-4$, $(-4)^2=16$. The polar route scales to any power in one line.

Worked example (phase gates). The gate $T=\mathrm{diag}(1,e^{i\pi/4})$ applied twice multiplies the $\lvert1\rangle$ amplitude by $e^{i\pi/4}e^{i\pi/4}=e^{i\pi/2}=i$, which is why $T^2=S$; four applications give $e^{i\pi}=-1$, so $T^4=Z$. Multiplying $(1+i)$ by $e^{i\pi/4}$ adds $\pi/4$ to its argument $\pi/4$ and multiplies the modulus by $1$, giving $\sqrt2e^{i\pi/2}=\sqrt2\,i$.

Worked example. For $N=4$ we have $\omega_4=e^{i\pi/2}=i$ and the roots are $1,i,-1,-i$. With $j=1$: $1+i+(-1)+(-i)=0$. With $j=2$: $1+i^2+i^4+i^6=1-1+1-1=0$. With $j=4$: every term is $i^{4k}=1$ and the sum is $4$. Geometrically, for $j\not\equiv0$ the points $\omega_N^{jk}$ are spread symmetrically around the circle and their centre of mass is the origin.

This identity is the engine of the quantum Fourier transform: the QFT matrix has entries $\tfrac{1}{\sqrt N}\omega_N^{jk}$, and the theorem above is exactly the statement that its columns are orthonormal, i.e. that the QFT is unitary. When amplitudes are summed with phases that go all the way around the circle they cancel, and only the frequencies that match survive.