The math behind a qubit

Here's the most important thing to know about a qubit: when you measure it, you only ever get one of two answers. That's it. The measurement device always picks one of two outcomes.

We give those two outcomes names. By convention, we call them:

The vertical bar and angle bracket are called ket notation. They're just a label saying “this is a quantum state.” You can think of |0⟩ and |1⟩ as the quantum equivalents of the classical bits 0 and 1.

Try measuring a qubit a few times below. Each measurement gives you exactly one of two outcomes.

Here's the key move. We're going to picture the two possible outcomes as two perpendicular axes on a graph, and the qubit's state as an arrow on that graph.

The arrow above represents a qubit state, which we call |ψ⟩ (“ket psi”). Its projection onto the |0⟩ axis has length α, and its projection onto the |1⟩ axis has length β. We write this as:

That equation is doing exactly what it looks like. We've broken the arrow |ψ⟩ into two pieces: a piece in the |0⟩ direction (with size α) and a piece in the |1⟩ direction (with size β). This is called a linear combination of |0⟩ and |1⟩.

The numbers α and β are called amplitudes. They tell us how much of each outcome is mixed into the state.

When a qubit is in superposition and you measure it, you still only get one of the two outcomes. But which one? The amplitude tells you.

The rule is named after the physicist Max Born. It says:

The probability of each outcome is the square of its amplitude. That's the Born rule, one of the most important formulas in all of quantum mechanics.

For example, if α = 0.6 and β = 0.8, then measuring the qubit will give you:

• |0⟩ with probability 0.62 = 0.36 (36% of the time).
• |1⟩ with probability 0.82 = 0.64 (64% of the time).

Notice something about the example above: 0.36 + 0.64 = 1. That isn't a coincidence. The probabilities of all possible outcomes have to add up to 100%, because something definitely happens when you measure. Plugging in the Born rule:

That equation has a beautiful geometric meaning. Look back at the axes chart: the state |ψ⟩ is an arrow with horizontal component α and vertical component β. By the Pythagorean theorem, the length of that arrow is:

So a valid qubit state is an arrow of length 1. It can point in any direction, but it always has length 1. In math language, the state is a unit vector.

Here's a strange and important fact: we chose |0⟩ and |1⟩ as our two outcomes by convention, but that's not the only choice. We could just as well pick any two perpendicular directions on the plane and call those our outcomes.

Two of the most common alternative outcomes are |+⟩ (“ket plus”) and |−⟩ (“ket minus”). They're defined as 45° rotations of |0⟩ and |1⟩:

When a quantum computer “measures in the X basis,” it's measuring along the |+⟩ / |−⟩ axes instead of the |0⟩ / |1⟩ axes. The same physical qubit gives different probabilities depending on which axes you measure along.

This is one of the most counter-intuitive parts of quantum mechanics. The qubit doesn't have a hidden “true” answer waiting to be revealed. What it tells you depends on the question you ask.

To find the probability of measuring |+⟩ instead of |0⟩, we just rewrite the state in the new basis. If we write

then the probabilities are γ2 and δ2, and they still add up to 1. The Born rule is the same; only the axes change.

Everything we've done so far works on a 2D plane with real-number amplitudes. But there's one more piece of the puzzle: amplitudes can actually be complex numbers, not just real numbers.

(If you haven't seen complex numbers before: they're numbers of the form a + bi, where i = √−1. They show up all the time in physics and engineering.)

When the amplitudes are complex, the Born rule needs a small adjustment. Instead of squaring the amplitude, we take its squared magnitude:

For complex a + bi, the squared magnitude is a2 + b2. If the amplitudes happen to be real, then |α|2 is just α2 and everything from before still works.

The complex part of the amplitude has a name: phase. It doesn't affect the probability of any single measurement, but it absolutely matters when qubits interact, when you apply gates, or when you measure in a different basis. Phase is what makes interference patterns possible, and it's what makes quantum computers more powerful than classical ones.

We started with a 2D plane: |0⟩ on one axis, |1⟩ on the other. Then we added |+⟩ and |−⟩, which are 45° rotations in that same plane. So far we're still in 2D.

The phase changes everything. Because the amplitudes are complex, the “plane” we've been drawing is actually a slice through a higher-dimensional space. When you fold in the phase, the qubit's state lives on the surface of a 3D sphere: the Bloch sphere.

Each pair is a valid pair of measurement outcomes. Whichever pair you pick is the “axes” you measure along. The qubit's state is one specific point on the surface of the sphere, and the probabilities you get depend on the angle between that point and the axis you measured along.

That's the math. Everything else you'll see (gates, entanglement, error correction) is built on top of these ideas.

Three good next steps:

• What is the Bloch sphere? takes the 3D picture seriously.
• The Hadamard gate is the gate that turns |0⟩ into |+⟩. It's the first thing you do in nearly every quantum algorithm.
• What is quantum entanglement? shows what happens when two qubits become connected.