In classical computing, logic gates like AND, OR, and NOT manipulate bits. Quantum computing has its own set of fundamental operations, called quantum gates. Instead of flipping bits from 0 to 1, these gates perform precise rotations on a qubit's state vector.

The most useful way to visualize the state of a single qubit is with the Bloch Sphere It's a 3D sphere where any point on the surface represents a possible pure state for a qubit. The north pole is typically designated as the state $|0\rangle$, and the south pole as $|1\rangle$. All other points on the sphere represent a superposition of these two states. A quantum gate's job is to move the state vector to a new point on the sphere.

Every quantum gate can be represented by a matrix. When a gate acts on a qubit, its matrix is multiplied by the qubit's state vector to find the new state. For a quantum operation to be valid, it must be reversible. This physical constraint means that all quantum gate matrices must be unitary

A matrix $U$ is unitary if its conjugate transpose, $U^†$, is also its inverse. This ensures that the total probability of the system remains 1 after the operation. In other words, the length of the state vector is always preserved.

The most fundamental gates for a single qubit are rotations around the x, y, and z axes of the Bloch sphere. For data scientists, these are analogous to applying a transformation or activation function to a single feature.

More general rotation gates, $R_x(\theta)$, $R_y(\theta)$, and $R_z(\theta)$, allow for rotations by any angle $\theta$. These are the bread and butter of many quantum algorithms, providing the fine-tuned control needed to navigate the qubit's state space.

Single-qubit gates are powerful, but they can't create entanglement. To do that, we need multi-qubit gates that make qubits interact. The most important of these is the Controlled-NOT, or CNOT gate.

A CNOT gate has two inputs: a control qubit and a target qubit. It performs a Pauli-X (or NOT) operation on the target qubit only if the control qubit is in the state $|1\rangle$. Otherwise, it does nothing to the target.

This conditional logic is how we create entanglement. Consider two qubits, both starting in the $|0\rangle$ state. If we apply a Hadamard gate to the first (control) qubit to put it in superposition, and then apply a CNOT gate, the result is the famous Bell state—a maximally entangled pair.

Putting these gates together in a sequence creates a quantum circuit. This is the quantum equivalent of an algorithm or, for a data scientist, a model architecture. Two key metrics describe a circuit:

Just as a deep neural network has many layers, a deep quantum circuit has many sequential gates. Both width and depth are critical resources. Today's quantum computers are limited in both, making the design of shallow, narrow circuits a major challenge in the field.

By arranging these fundamental gates, we can construct algorithms that solve problems intractable for classical computers.