Unitary Operators

Hermitian operators tell you what you can measure; unitary operators tell you how states evolve. Between measurements, a closed quantum system changes by a unitary operator. Every quantum gate is unitary, and the Schrödinger time-evolution operator $U(t) = e^{-iHt/\hbar}$ is unitary. The defining feature is that they preserve probability.

Definition

An operator $U$ is unitary if its adjoint is its inverse:

U^\dagger U = U U^\dagger = \mathbb{1}.

Equivalently, $U^{-1} = U^\dagger$ . Because the dagger is the conjugate transpose, unitarity is the complex generalization of an orthogonal (rotation/reflection) matrix from real linear algebra.

Unitaries preserve inner products and norms

The single most useful property: a unitary operator leaves every inner product unchanged. For any states $|\psi\rangle, |\phi\rangle$ ,

\langle U\phi \,|\, U\psi\rangle = \langle \phi \,|\, U^\dagger U \,|\, \psi\rangle = \langle \phi \,|\, \psi\rangle.

Setting $|\phi\rangle = |\psi\rangle$ shows the norm is preserved:

\| U\psi \|^2 = \langle U\psi | U\psi\rangle = \langle\psi|\psi\rangle = \|\psi\|^2.

This is exactly why unitary evolution is physical: a normalized state $\|\psi\| = 1$ stays normalized, so total probability stays equal to $1$ at all times. Applying the Hadamard gate to $|0\rangle$ , for instance, gives $\tfrac{1}{\sqrt2}(|0\rangle + |1\rangle)$ , whose squared amplitudes $\tfrac12 + \tfrac12 = 1$ still sum to one.

Orthonormal columns

Reading $U^\dagger U = \mathbb{1}$ in components gives a concrete test. The $(i,j)$ entry of $U^\dagger U$ is $\sum_k \overline{U_{ki}}\,U_{kj}$ , which is the inner product of column $i$ and column $j$ of $U$ . Setting this equal to $\delta_{ij}$ says:

the columns of a unitary matrix form an orthonormal set.

The same holds for the rows. So you can verify unitarity by checking that each column has norm $1$ and is orthogonal to the others. For the Hadamard matrix $H = \tfrac{1}{\sqrt2}\begin{pmatrix} 1 & 1 \\ 1 & -1\end{pmatrix}$ , each column has norm $\sqrt{\tfrac12 + \tfrac12} = 1$ and the two columns have inner product $\tfrac12(1\cdot 1 + 1\cdot(-1)) = 0$ — unitary.

Eigenvalues lie on the unit circle

Just as Hermitian operators have real eigenvalues, unitary operators have eigenvalues of modulus one. If $U|\psi\rangle = \lambda|\psi\rangle$ , norm preservation gives $\|\psi\| = \|U\psi\| = |\lambda|\,\|\psi\|$ , so $|\lambda| = 1$ . Every eigenvalue can therefore be written $\lambda = e^{i\theta}$ for some real phase $\theta$ . Unitary operators rotate phases; they never shrink or grow amplitudes.

Connection to Hermitian operators

The two families are linked by the exponential map. If $H$ is Hermitian, then

U = e^{-iHt}

is unitary for every real $t$ , because $U^\dagger = e^{+iH^\dagger t} = e^{+iHt} = U^{-1}$ . This is the content of Stone's theorem and the reason the Hamiltonian (a Hermitian observable) generates unitary time evolution. We will make sense of "the exponential of an operator" precisely when we study functions of operators later in this module.

Try it

Build a one-qubit circuit that applies the Hadamard gate to $|0\rangle$ . The simulator will confirm the output $\tfrac{1}{\sqrt2}(|0\rangle + |1\rangle)$ remains a unit-norm state — the signature of a unitary operation.

Run your code to see the quantum state.