Single Qubit Gates

Quantum gates

So far, we’ve only talked about quantum gates in an abstract sense, that they are operations that manipulate the quantum state of a qubit. Let’s take a look at how this works formally.

A vector represents a quantum state. The only way we can manipulate this vector is by rotating it in the state space. We cannot change its length because of the normalization constraint.

Matrices play a key role in transformations. We can think of a vector rotation as a transformation in the state space. A matrix transformation can represent each rotation. We can apply this transformation to a vector by performing a matrix multiplication with the matrix.

We want our quantum gates to be reversible. We can achieve this by representing quantum gates as unitary matrices.

A complex-valued square matrix is unitary if its conjugate transpose is also its inverse.

In other words, applying such a transformation twice will revert the state vector to its original position. Think of it as a clockwise rotation followed by an equal anti-clockwise rotation. We would land at the same place in the state space. The Bloch Sphere will play a vital role here to visualize these rotations.