Probability vs Amplitude

Two kinds of number

A probability is a nonnegative number that directly says how likely an outcome is. An amplitude is a number — often complex — that contributes to a probability only after you take its squared magnitude. Amplitudes can be positive, negative, or complex, which means they can cancel each other or reinforce each other before measurement enters the story.

That cancellation and reinforcement is the heart of every quantum algorithm. A randomized classical algorithm can spread probability mass across many outcomes, but probabilities only ever add. A quantum algorithm can arrange amplitudes so that paths leading to wrong answers cancel and paths leading to right answers survive. Measurement then samples from a distribution that interference has already shaped.

One sentence is worth memorizing: amplitudes are the quantities quantum operations transform; probabilities are what measurement produces from them. Most confused claims about quantum computing violate this sentence somewhere.

The rules, written down

A one-qubit state is written $|\psi ⟩=\alpha |0⟩+\beta |1⟩$, where $\alpha$ and $\beta$ are complex amplitudes. The state is normalized when $|\alpha {|}^2+|\beta {|}^2=1$ — total probability must come to one, because measurement always returns something.

Measurement in the computational basis follows the Born rule: the outcome 0 appears with probability $|\alpha {|}^2$, and 1 with probability $|\beta {|}^2$. Square first, then interpret — never the other order.

Here is the subtlety the rest of the book exploits. Take $\alpha =1/\sqrt{2}$ with $\beta =1/\sqrt{2}$, and both outcomes have probability $\frac{1}{2}$. Now flip the sign so $\beta =-1/\sqrt{2}$: the probabilities are still $\frac{1}{2}$ each. The relative sign is invisible to an immediate measurement — but it is real, it is stored in the state, and later gates respond to it.

Worked example: one gate separates two states

Compare the two states from the figure:

For both, $P(0)=|1/\sqrt{2}{|}^2=\frac{1}{2}$ and $P(1)=\frac{1}{2}$. No single computational-basis measurement, and no number of repeated measurements on identically prepared copies analyzed in that basis alone, distinguishes them reliably.

Now apply a Hadamard gate — the workhorse single-qubit transformation this book uses constantly — to each:

Identical measurement statistics went in; opposite, perfectly predictable outcomes came out. The Hadamard converts relative phase into amplitude difference, and the amplitudes that used to cancel now reinforce. This is why a quantum state is more than a probability distribution. If you compress a state to its probabilities too early, you throw away exactly the information quantum algorithms spend their whole budget manipulating.

Where the intuition fails

The first trap is linguistic: saying "the probability of 0 has a negative sign" or "the probability is complex." Probabilities are never negative and never complex. Amplitudes carry signs and phases; probabilities are squared magnitudes. Keeping the two vocabularies separate is not pedantry — it is the difference between describing interference and describing noise.

The second trap is grandeur: assuming that because a state has amplitudes on many basis states, measurement reveals all of them. It does not. The amplitudes shape a distribution; measurement returns one classical sample drawn from it. A register with amplitudes on $2^n$ basis states still yields $n$ classical bits per measurement, and the algorithm's job is to make those bits useful.

The engineering view

For a computer scientist, amplitudes behave like internal representation, not output. They are hidden state that affects later behavior but is never returned by the API — the API is measurement, and it returns samples. You would not judge a service by dumping its heap; do not judge a quantum algorithm by staring at its statevector.

This is also why simulators mislead. A statevector simulator will happily print every amplitude, because your classical machine is expensively simulating the quantum state. Real hardware hands you no such thing: it returns measurement outcomes from the bases you chose, at the shot count you paid for. Designs that depend on "reading the statevector" are designs that only work on simulators, and the distinction decides what scales.

What this buys you in diligence

When a pitch says the machine explores $2^n$ states and therefore solves the problem, ask one question: how does the algorithm make the desired answer more likely to be measured? If the answer is only that many amplitudes exist, the claim is incomplete. Amplitudes on every basis state are the default condition of a quantum register, not an achievement.

The serious version of any such claim names the interference structure, the measurement strategy, the error budget, and the classical baseline. Those four items are where the business value either exists or doesn't, and all four are checkable without taking anyone's word for anything.

Exercise

Normalize and interpret. Take the state $|\psi ⟩=3|0⟩+4i|1⟩$ and work it end to end.

• Submit: the normalized state, both measurement probabilities, and one sentence explaining why the phase on $|1⟩$ could matter after a later gate even though it does not change either probability.
• Check: confirm the probabilities sum to 1, and re-read your write-up to make sure you never called an amplitude a probability.
• Repair: if you found yourself taking the square root of a probability to get an amplitude, you crossed the line in the wrong direction — redo the worked example and note where the squaring happens.

Check your understanding

Without notes: normalize the state $|\psi ⟩=1|0⟩+i|1⟩$, compute both measurement probabilities, and state what information stays invisible if you only look at those probabilities.

A passing answer divides by $\sqrt{2}$, reports $P(0)=P(1)=\frac{1}{2}$, and names the casualty: the relative phase $i$, which a later gate could convert into a measurable difference. An answer that calls the state "50 percent zero and 50 percent one, full stop" has described the measurement and missed the state.

Oral defense: explain to a colleague why "the qubit is 0 and 1 at the same time" is a worse description than "the qubit is a specific weighted combination whose weights can cancel."

If you get stuck

If you catch yourself treating amplitudes as stored classical answers, return to Chapter 1 (Computation Is Physical) and re-read what a state is allowed to be; then redo this chapter's worked example slowly, tracking the sign through the Hadamard. The plus/minus pair is small enough to hold in your head, and once it clicks, the rest of the book's interference arguments follow the same shape.