What is a qubit?

The prevailing explanation of a qubit is odd to me.

It tries to build intuition by separating classical bits from qubits, but they are not separate. Their states inhabit the same space. Both can be described as vectors pointing to a location on a sphere.

The difference is precisely which set of operations can be applied to the vector. A classical bit admits one reversible operation, the flip (the $X$ gate). The classical vector flips between the two poles of the sphere, 0 and 1. A qubit admits every rotation of the sphere, including the flip. The quantum vector can point anywhere on the sphere, not just at the poles.

Physically, the qubit can be any two-level quantum system that we can faithfully prepare, control, and read out. The spin of an electron, the polarization of a photon, and the two lowest levels of a superconducting circuit are a few systems in use today. The modality varies, but the state space does not.

Let's build an intuition through the lens of group theory, one group at a time.

Classical information

S₂ · generator

$$X = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$$

a cyclic group of order 2

Hand-drawn panel, S2 state space: a sphere with only its two poles marked 0 and 1, and an arrow between them.

The symmetry group of a classical bit is $S_2 = \langle x \mid x^2 = I \rangle$, a cyclic group of order 2. It acts on the set $\{0, 1\}$. The action is free and transitive, so the orbit of 0 is $\{0, 1\}$. Choosing the outcomes as a basis of $\mathbb{C}^2$ turns this action into a faithful representation $\rho: S_2 \to U(2)$, which is how the classical group sits inside the quantum one.

Quantum information

The Pauli group

P₁ · generators

$$X = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$$

$$Y = \begin{bmatrix} 0 & -i \\ i & 0 \end{bmatrix}$$

$$Z = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}$$

Hand-drawn panel, P1 state space: an octahedron inscribed in the sphere, its six vertices labelled 0, 1, plus, minus, plus i, minus i.

The Pauli group is $P_1 = \langle X, Y, Z \rangle \subset U(2)$, of order 16, with $X^2 = Y^2 = Z^2 = I$. The six axis states $0, 1, +, -, +i, -i$ are the eigenstates of $Z$, $X$, and $Y$, and the vertices of an octahedron inscribed in the sphere. Under the Pauli group they form three orbits, the antipodal pairs $\{0, 1\}$, $\{+, -\}$, and $\{+i, -i\}$. Each pair is the eigenbasis of one generator, which fixes it, and the other two generators swap it. The orbit of 0 is still $\{0, 1\}$; the Pauli group moves quantum states, but it maps classical states to classical states.

The Clifford group

C₁ · generators

$$H = \frac{1}{\sqrt{2}} \begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix}$$

$$S = \begin{bmatrix} 1 & 0 \\ 0 & i \end{bmatrix}$$

Hand-drawn panel, C1 state space: the sphere with its six axis states and arrows carrying one state to the next around a single orbit.

The Clifford group is $C_1 = \langle H, S \rangle \subset U(2)$, 24 elements up to phase. It contains the Pauli group ($S^2 = Z$, $HZH = X$). $H$ exchanges the $x$ and $z$ axes and $S$ turns $x$ into $y$, so the three Pauli pairs merge into one orbit of six, with stabilizer $\{I, S, S^2, S^3\}$. It is the first layer to carry a classical state off the poles, and the last finite one.

U(2), a Lie group

U(2) · generators

$$I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$$

$$X = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$$

$$Y = \begin{bmatrix} 0 & -i \\ i & 0 \end{bmatrix}$$

$$Z = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}$$

exponentiated

$$\mathrm{Ph}(\varphi) = e^{i\varphi I} = \begin{bmatrix} e^{i\varphi} & 0 \\[4pt] 0 & e^{i\varphi} \end{bmatrix}$$

$$R_x(\theta) = e^{-i\theta X/2} = \begin{bmatrix} \cos\tfrac{\theta}{2} & -i\sin\tfrac{\theta}{2} \\[4pt] -i\sin\tfrac{\theta}{2} & \cos\tfrac{\theta}{2} \end{bmatrix}$$

$$R_y(\theta) = e^{-i\theta Y/2} = \begin{bmatrix} \cos\tfrac{\theta}{2} & -\sin\tfrac{\theta}{2} \\[4pt] \sin\tfrac{\theta}{2} & \cos\tfrac{\theta}{2} \end{bmatrix}$$

$$R_z(\theta) = e^{-i\theta Z/2} = \begin{bmatrix} e^{-i\theta/2} & 0 \\[4pt] 0 & e^{i\theta/2} \end{bmatrix}$$

Hand-drawn panel, SU(2) state space: a fully shaded sphere with a short arc of rotation drawn on its surface.

$U(2)$ is a Lie group: a group that is also a smooth manifold. It contains every group above exactly as written. Its generators are the identity and the Pauli matrices themselves. Exponentiating $X$, $Y$, and $Z$ gives the rotations $R_x(\theta)$, $R_y(\theta)$, $R_z(\theta)$, which reach every point of the sphere.

Clifford

Clifford + T

U(2)

In practice a quantum computer is programmed with a finite set of gates, and every operation is composed from that set. We have seen that the Clifford gates reach six points on the sphere. But if we add an operation called the $T$ gate, an eighth-turn about $z$, we can approximate the reach of $U(2)$. The $T$ gate breaks the octahedron's symmetry, and sequences of $H$, $S$, and $T$ then come arbitrarily close to every point on the sphere. Any rotation can be approximated by a long enough gate sequence. This is what makes the gate set universal. Clifford circuits on their own can be simulated efficiently on a classical computer. The $T$ gate is what takes a quantum computer beyond them.