Complex Numbers for Quantum Computing

Why amplitudes are complex

Quantum amplitudes are complex numbers: quantities of the form $z=a+bi$, where $a$ and $b$ are ordinary real numbers and $i$ is defined by $i^2=-1$. The magnitude $|z|=\sqrt{a^2+b^2}$ is the point's distance from the origin, and the squared magnitude $|z{|}^2=a^2+b^2$ is what measurement turns into probability.

The reason for dragging in a second dimension of number is interference. A probability table can only add; it has no way for two live possibilities to erase each other. Complex amplitudes carry both a size and an angle, and the angle lets two contributions to the same outcome cancel without either contribution ceasing to exist. Strip the theory down to real numbers and you lose exactly the phenomenon Part I identified as the engine of every algorithm.

So the deal is simple: amplitudes live on the complex plane, gates move them around that plane, and measurement reads out squared lengths. Every calculation in this part of the book is some variation on those three moves.

Conjugate, magnitude, phase

The complex conjugate of $z=a+bi$ is $z^{\ast }=a-bi$ — the same point, reflected across the real axis. Its job is to make magnitudes computable: $z^{\ast }z=(a-bi)(a+bi)=a^2+b^2$, a plain nonnegative real number. For an amplitude $\alpha$, the probability contribution is always $|\alpha {|}^2={\alpha }^{\ast }\alpha$. Never ${\alpha }^2$ — that is a different, generally complex, object, and using it is the most common arithmetic error in early quantum calculation.

Phase comes in two grades, and confusing them causes real bugs. Global phase — multiplying an entire state by the same factor like $i$ — changes nothing measurable: every amplitude picks up the same angle, every squared magnitude is untouched, and no experiment can detect the change. Relative phase — one branch's angle shifting against another's — is physically real. Later interference can convert it into different outcome probabilities, which is the whole trick behind the algorithms in Part IV.

Worked example: normalizing a state by hand

Take the unnormalized state $|\psi ⟩=1|0⟩+i|1⟩$. First the norm squared: $|1{|}^2+|i{|}^2=1+1=2$. Normalizing means dividing each amplitude by $\sqrt{2}$:

Now the measurement probabilities. $P(0)=|1/\sqrt{2}{|}^2=\frac{1}{2}$. For the second, use the conjugate: $(i/\sqrt{2}{)}^{\ast }=-i/\sqrt{2}$, so $P(1)=(-i/\sqrt{2})(i/\sqrt{2})=\frac{1}{2}$. The probabilities sum to one, as they must.

Notice the quiet event in that last step: the $i$ had no effect on either immediate probability. It is not, however, absent — it is the relative phase between the two branches, and a later gate can rotate it into a measurable difference. Chapter 6's three-gate circuit is precisely that conversion, run on a sign instead of an $i$.

Where the intuition fails

The first trap is treating $i$ as decoration — an exotic flourish on an otherwise real theory. In a probability table a phase would indeed be meaningless, which is why probability tables cannot describe quantum states. The phase is how the state remembers which way each amplitude is pointing, and interference is what happens when those pointing directions meet.

The second trap is arithmetical: squaring a complex number directly instead of multiplying by its conjugate. $(a+bi{)}^2=a^2-b^2+2abi$ is complex, sign-dependent, and useless as a probability; $|a+bi{|}^2=a^2+b^2$ is real, nonnegative, and correct. When a computed "probability" comes out negative or imaginary, this is the first bug to look for.

The engineering view

If you ever write a statevector simulator, this chapter is your data model. The state is an array of complex numbers; a gate is a matrix of complex numbers; applying a gate is matrix-vector multiplication over that arithmetic. There is no separate probability array to update — probabilities are derived on demand by squaring magnitudes, and only then.

This shapes debugging. A state can carry phases that no single measurement step reveals but that corrupt every later gate, and if your test harness inspects only probabilities at each step it will wave the bug through. Compare amplitudes, or at least compare in a second measurement basis; probability-only testing is blind to exactly the errors quantum code is most prone to.

What this buys you in diligence

Phase talk is cheap, and this chapter makes it inspectable. When a paper or product claims interference, phase estimation, amplitude amplification, or coherent control, ask where the complex phase is introduced, where it is preserved, and where it is converted into output probability. A real claim points to a calculation or a circuit; a rhetorical one points to adjectives.

The same reflex applies to error discussions. Decoherence is, mathematically, the uncontrolled scrambling of relative phase — so a hardware team that can speak precisely about phase preservation is telling you, in this chapter's language, that it understands its own enemy.

Exercise

Normalize with a complex amplitude. Take $2|0⟩+(1+i)|1⟩$ and carry it from raw state to measurement probabilities.

• Submit: the norm-squared calculation, the normalized state, and $P(0)$ and $P(1)$, showing the conjugate multiplication for $(1+i)$ explicitly.
• Check: the probabilities must sum to 1, and both must be real and nonnegative. If either fails, find the step where a square replaced a conjugate.
• Repair: if you cannot say what information would be lost by keeping only the two probabilities, redo the worked example and watch where the relative phase goes.

Check your understanding

Without notes: for a complex-amplitude state of your choosing, compute the conjugates, the squared magnitudes, the normalized state, and both measurement probabilities.

A passing answer shows ${\alpha }^{\ast }\alpha$ at least once, never writes ${\alpha }^2$ where a probability belongs, and includes one sentence on why global phase is invisible while relative phase is not. That sentence is the chapter; the arithmetic is scaffolding for it.

Oral defense: explain to a colleague why the Born rule needs $z^{\ast }z$ rather than $z^2$ — and what goes wrong, numerically, if you confuse them.

If you get stuck

If the amplitude-versus-probability boundary keeps blurring, return to Chapter 3 (Probability vs Amplitude) and redo its worked example with this chapter's conjugate arithmetic in hand. The two chapters teach one idea from two sides: Chapter 3 builds the intuition, this one makes the calculation mechanical.