Why Gates Are Unitary

qc/math

A valid quantum operation must preserve the norm of the state vector — probabilities have to sum to 1 both before and after, so $\Vert U|\psi ⟩\Vert =\Vert |\psi ⟩\Vert$ for every valid input state $|\psi ⟩$. Key insight: this requirement is exactly equivalent to unitarity. Since $\Vert U|\psi ⟩{\Vert }^2=⟨\psi |U^{†}U|\psi ⟩$ must equal $⟨\psi |\psi ⟩$ for all $|\psi ⟩$, it forces $U^{†}U=I$ — the definition of a unitary matrix. Norm-preservation isn’t a separate property gates happen to have; it’s the reason the unitary requirement exists in the first place.

Why this rules some matrices out

Not every matrix is a valid gate. A plain scaling matrix breaks norm preservation immediately. More importantly: a Hamiltonian $H$ itself is generally not unitary (see Hamiltonians and Encoding for Quantum Circuits) — which is exactly why you can’t apply $H$ directly as a gate, only $U(H,t)=e^{-iHt}$, which is unitary by construction.

Unitarity means reversibility

Every unitary has an inverse, $U^{-1}=U^{†}$, so every quantum gate (aside from measurement, which is not unitary — see Measurement and Collapse) is reversible. This is a sharp contrast with classical logic: an AND gate has no inverse (you can’t recover both inputs from the output), but every gate in this vault’s gate set can always be undone by applying its adjoint.

Related

• Pauli Operators — states “Hermitian and unitary” as a property; this is why unitary specifically is required
• Hamiltonians and Encoding for Quantum Circuits
• What is a Qubit

Self-Check

• Could you derive $U^{†}U=I$ starting from the requirement that $U$ preserves a state’s norm?
• Why isn’t a Hamiltonian $H$ itself a valid gate, even though it’s a perfectly good Hermitian matrix?
• Why does unitarity imply every quantum gate is reversible, and what’s the classical contrast?
• What would it take to deliberately construct a matrix that is not unitary, and how would you verify computationally (e.g. via an is_unitary()-style check) that it fails $U^{†}U=I$?