Unitary Evolution
Closed quantum systems evolve by unitary transformations, which preserve inner products, normalisation, and reversibility.
How closed systems change
Between measurements, an isolated quantum system evolves by a unitary operator: |psi> -> U|psi>, where U-dagger U = I. Unitarity is the quantum version of deterministic, information-preserving dynamics. Every quantum gate is a unitary.
What unitarity preserves
- Norm: |U|psi>| = ||psi>|, so probabilities still sum to one
- Inner products:
- Reversibility: U-dagger undoes U exactly
Because unitaries are invertible, quantum computation is fundamentally reversible. No information is destroyed during gate operations; only measurement is irreversible. This is a sharp contrast with classical logic gates like AND, which discard inputs.
Connection to the Hamiltonian
Unitary evolution comes from the Schrodinger equation. For a time-independent Hamiltonian H, the evolution operator is U = exp(-iHt/hbar). Since H is Hermitian, this U is automatically unitary. Building a specific gate on hardware means engineering a Hamiltonian and turning it on for the right duration.
Gates as unitaries
Every operation in the circuit model is a unitary matrix acting on the register. Single-qubit gates are 2x2 unitaries; two-qubit gates are 4x4. A whole circuit is the product of its gate unitaries, itself one big unitary applied to the input state before the final measurement.
When it fails
Unitarity holds only for closed systems. Real qubits interact with their environment, and that open-system evolution is not unitary — it is decoherence, described by non-unitary quantum channels. Keeping evolution close to unitary is exactly what good hardware and error correction try to achieve.
import numpy as np
X = np.array([[0,1],[1,0]])
print(np.allclose(X.conj().T @ X, np.eye(2))) # True: X is unitary