# Unitary Operator

- [[quantum-computing]]
- a matrix $`\mathcal{U}`$ operating on a [[quantum-state]] to produce another [[quantum-state]]
  - maps unit vectors to unit vectors
  - [[norm]]
- other equivalent definitions:
  - columns of $`\mathcal{U}`$ are [[orthonormal]]
  - rows of $`\mathcal{U}`$ are [[orthonormal]]
  - $`\mathcal{U}^{-1} = \mathcal{U}^\dagger = \overline{\mathcal{U}^\intercal} = \overline{\mathcal{U}}^\intercal`$
  - preserves [[norm]]
  - preserves [[inner-product]]