The Bloch sphere · Qubits · Learn quantum computing

A globe for every qubit

The circle from the warm-up is a good start, but it can only show amplitudes that are ordinary numbers. Real amplitudes are complex, and they need one more direction to live in. So physicists draw a qubit as an arrow inside a ball.

It's called the Bloch sphere, after the physicist Felix Bloch. Every possible state of one qubit is one point on its surface. Have a look around:

GoalDrag the sphere to look at it from another side.

The arrow's tip can sit anywhere on the surface of this ball. Every point is a different qubit state.

Opposite answers, opposite ends

On the warm-up circle, |0⟩ and |1⟩ were a quarter turn apart. On the sphere they sit at opposite ends: |0⟩ at the North Pole, |1⟩ at the South Pole.

The trick is simple: the sphere doubles every angle. An arrow turned 45° on the circle is turned 90° on the sphere, which lands it on the equator. Slide and compare:

The circle

20°

The sphere, from the side

40° from the top

Turn the arrow on the circle20°

0: 88%1: 12%

GoalTurn the circle's arrow to 45°, then all the way to 90°.

Left: the circle from the warm-up, with |0⟩ right and |1⟩ up. Right: the sphere seen from the side, with |0⟩ at the top and |1⟩ at the bottom. Same qubit, same odds.

Height sets the odds

Here's the best part of the sphere: you can read the odds just by looking at how high the arrow points. At the North Pole, 0 is certain. At the South Pole, 1 is certain. On the equator, it's 50-50.

chance of 0 = (1 + height) ÷ 2

Height is +1 at the North Pole, 0 on the equator and −1 at the South Pole.

Tilt the arrow up or downheight −0.34

chance of 0 = (1 − 0.34) ÷ 2 = 0.33

0: 33%1: 67%

0.57|0⟩ + 0.82|1⟩

GoalTilt the arrow so the qubit gives 0 with a 75% chance.

Height runs from +1 at the North Pole to −1 at the South Pole. The chance of 0 is (1 + height) ÷ 2.

Your turn

A qubit's arrow points at height −0.6, a bit below the equator. What's the chance of measuring 0?

Your qubit on a world map

Think of the sphere as a globe. If the qubit points at a city, the city's latitude (how far north or south it is) sets the odds. A qubit pointing at Delhi leans towards 0. One pointing at Sydney leans towards 1.

Delhi

latitude 28.6° N

0: 74%1: 26%

A qubit pointing at Delhi gives 0 74% of the time.

GoalFind a place where the qubit is (almost exactly) 50-50.

Picture the Bloch sphere as a globe: the North Pole is |0⟩ and the South Pole is |1⟩. Point the qubit at a city and only its latitude decides the odds.

Going around: the phase

So height sets the odds. What about going around, east or west, like longitude? It doesn't change the odds at all. Every point on the equator is a 50-50 qubit.

But they aren't all the same qubit. Remember |+⟩ and |−⟩ from the last lesson, same odds and different phases? On the sphere they sit on opposite sides of the equator. Going around is the phase.

Go around the equator60°

0.71|0⟩ + (0.35 + 0.61i)|1⟩

Measure now

0: 50%1: 50%

Apply H, then measure

0: 75%1: 25%

GoalFind the spot where an H gate would turn the qubit into a certain 1.

Every point on the equator is a 50-50 qubit. Going around changes the phase: invisible now, but an H gate turns it into odds you can see.

Your turn

Which state sits directly opposite |+⟩ on the sphere?

Gates turn the globe

Now for something lovely. Every gate on one qubit is simply a turn of the sphere. X turns it half a turn so the poles swap. Z turns it half a turn around the up-down axis, so the odds stay the same but the phase flips. H turns it about a slanted axis, so a pole moves to the equator.

|0⟩

0: 100%1: 0%

Start at |0⟩, the North Pole. Tap a gate.

|1⟩|+⟩|−⟩|+i⟩

GoalVisit every state above. You can go back to |0⟩ whenever you like.

Every gate on one qubit is a turn of the sphere. Drag the sphere to watch from a better angle.

Why a sphere is enough

A qubit has two complex amplitudes, which is four ordinary numbers. So why does a ball, with just two directions (latitude and longitude), hold every state?

Two numbers drop out. First, the chances must add up to 1, which fixes the arrow's length. Second, turning both amplitudes by the same angle changes nothing you could ever measure. Physicists call that the global phase, and nature simply ignores it. Four numbers, minus two, leaves two: a point on a sphere.

Your turn

A qubit's arrow is somewhere on the equator. What do you know for sure?

What you now know

The Bloch sphere draws every qubit state as an arrow to a point on a ball.
|0⟩ is the North Pole and |1⟩ the South Pole. The sphere doubles the warm-up circle's angles.
Height sets the odds: chance of 0 = (1 + height) ÷ 2.
Going around, like longitude, is the phase: invisible on its own, but gates turn it into odds.
Every one-qubit gate is a turn of the sphere: X swaps the poles, Z flips the phase, H tips a pole onto the equator.

Next: measurement. What exactly happens to the arrow when you look?