What is the Bloch sphere?

Why a sphere?

A qubit in state α|0⟩ + β|1⟩ is described by two complex numbers. That sounds like four separate values, but two of them do not matter: the probabilities must add up to 1, and an overall phase shared by both amplitudes has no observable effect. That leaves two angles, and two angles are exactly what you need to pick a point on the surface of a sphere. The picture is named after the physicist Felix Bloch.

Reading the coordinates

Any single-qubit state can be written as:

|ψ⟩ = cos(θ/2)|0⟩ + e^(iφ) sin(θ/2)|1⟩

• θ (theta) is the angle down from the north pole, like latitude. At θ = 0 you are at |0⟩. At θ = 180° you are at |1⟩. The chance of measuring 0 is cos²(θ/2).
• φ (phi) is the angle around the equator, like longitude. It is the relative phase between the |0⟩ and |1⟩ parts.

Every point on the equator is an equal superposition: measure it and you get 0 or 1 with probability 50% each. Some points have their own names:

• |+⟩ = (|0⟩ + |1⟩)/√2, where the x axis meets the sphere.
• |−⟩ = (|0⟩ − |1⟩)/√2, on the opposite side.
• |+i⟩ and |−i⟩, where the y axis meets the sphere.

Why phase matters

|+⟩ and |−⟩ give exactly the same measurement statistics: 50/50. So does the difference between them matter? Very much. Apply a Hadamard gate and |+⟩ becomes |0⟩, while |−⟩ becomes |1⟩. Two states that look identical when measured directly behave completely differently once you transform them. That is the phase at work, and it is the reason interference is possible.

Gates are rotations

Every single-qubit gate moves the point around the sphere by rotating it:

• X is a half turn (180°) about the x axis, swapping the poles.
• Z is a half turn about the z axis, swapping |+⟩ and |−⟩ and leaving the poles alone.
• H is a half turn about the diagonal axis halfway between x and z, swapping the north pole with |+⟩.
• S is a quarter turn about the z axis, and T an eighth of a turn.

This makes the Bloch sphere a handy tool for working out, without any matrices, what a sequence of gates does to a single qubit.

The quickest way to get a feel for this is to try it. Our free Bloch sphere simulator lets you apply X, Y, Z, H, S and T to a qubit and watch the point turn, with the amplitudes and probabilities shown as it moves.

Why the angle is halved

You may notice that |0⟩ and |1⟩ are opposite each other on the sphere, 180° apart, even though as vectors they are at right angles. That is why the formula uses θ/2: on the Bloch sphere, any two states that can be told apart perfectly by a measurement sit at opposite points.

What the Bloch sphere cannot show

• Only one qubit. There is no simple sphere for two or more qubits.
• No entanglement. A qubit that is entangled with another has no point of its own on the surface. Its state sits inside the sphere.
• Noise shows up as shrinking. Points inside the sphere (the "Bloch ball") represent mixed states, where the qubit has lost some of its quantum information. Decoherence pulls states towards the centre.

Even with those limits, the Bloch sphere is the single most useful picture for building intuition about qubits.