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Unitary matrix

Last updated Jun 24, 2022

Unitary matrices are the class of matrices that preserve the length of the input Vector. In other words, if we take any vector $|\psi\rangle$ and compute the length $| U|\psi\rangle |$ it’s always equal to the length $||\psi\rangle|$ of the original vector. In this, they’re much like rotations or reflections in ordinary (real) space, which also don’t change lengths.

Algebraically, this means that $U^\dagger U = I$, that is, the adjoint of $U$, denoted $U^\dagger$, times $U$, is equal to the identity matrix. That adjoint is the complex transpose of $U$:

$$U^\dagger := (U^T)^*$$

So for a $2 \times 2$ matrix, the adjoint operation is just:

$$\left[ \begin{array}{cc} a & b \\ c & d \end{array} \right]^\dagger = \left[ \begin{array}{cc} a^* & c^* \\ b^* & d^* \end{array} \right]$$

Note: $\dagger$ is also sometimes called the dagger operation, or Hermitian conjugation, or just the conjugation operation.

The product of two unitary matrices is also unitary.

All quantum gates are unitary matrices