quantum gate directory

The unitary group is the home of quantum computing: every gate is a unitary matrix, because the evolution of a closed quantum system preserves inner products — and with them, the total probability of measurement outcomes. All other groups on this site are subgroups of this one.

$$\mathsf{U}(n) := \left\{ U \in \mathrm{GL}(n, \mathbb{C}) \mid U^\dag U = I \right\}$$

For $n$ qubits the relevant group is $\mathsf{U}(2^n)$, acting on the $2^n$-dimensional state space.

Equivalent characterizations

A matrix $U$ is unitary if and only if any of the following hold:

• $U^\dag U = U U^\dag = I$, i.e. its inverse is its conjugate transpose.
• It preserves inner products: $\langle U\psi, U\varphi \rangle = \langle \psi, \varphi \rangle$ for all states.
• Its columns (equivalently, rows) form an orthonormal basis.
• All its eigenvalues lie on the unit circle, $\lambda = \mathrm{e}^{i\theta}$.

Properties

• Every unitary is generated by a Hermitian matrix: $U = \mathrm{e}^{-iH}$. This is the gate–Hamiltonian correspondence: circuits are stroboscopic snapshots of continuous physical evolution.
• $\mathsf{U}(n)$ is a compact, connected Lie group of real dimension $n^2$ — so an $n$-qubit unitary takes $4^n$ real parameters to specify, which is why generic circuit synthesis is exponentially hard.
• Global phase is unobservable, so physically distinct gates really form the projective unitary group $\mathsf{PU}(n) = \mathsf{U}(n)/\mathsf{U}(1)$. Fixing $\det U = 1$ instead gives the special unitary group $\mathsf{SU}(n)$.
• Finite gate sets can be dense in $\mathsf{SU}(2^n)$: Clifford + $T$ is the standard example, and the Solovay–Kitaev theorem makes approximating any unitary efficient. This is what "universal gate set" means.

Relations to other groups

• The orthogonal group is its real slice, the diagonal group is a maximal abelian subgroup, and the Pauli and Clifford groups are its most important finite (modulo phase) subgroups.
• Further structured subgroups collect the classical gates (permutation), the free-fermion gates (matchgate), and the excitation-conserving gates (number-preserving).