Quantum Gates

Summary: Mathematical operators represented by unitary matrices that transform the state of qubits in a quantum circuit.

Sources: quantum-mechanics-basics.ipynb, build-and-run-your-first-quantum-program.ipynb

1. Mathematical Foundation: Unitary Matrices

In quantum mechanics, all physical state transformations (excluding measurement) must preserve the total probability of 1. Therefore, quantum gates are represented by unitary matrices.

A matrix $U$ is unitary if its conjugate transpose (adjoint) $U^\dagger$ is also its inverse: $U^\dagger U = U U^\dagger = I$

This mathematical property guarantees that all quantum gates are:

Probability-preserving: The length of the state vector remains 1.
Reversible: Any gate operation can be undone by applying its inverse gate $U^\dagger$.

2. Common Single-Qubit Gates

Single-qubit gates act on a 2D state vector.

A. The Hadamard Gate ($H$)

The Hadamard gate creates a superposition by mapping basis states to equal-probability mixtures. It corresponds to a $180^\circ$ rotation around the $X+Z$ diagonal axis of the Bloch Sphere. $H = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}$

Action on basis states: $H \vert 0 \rangle = \frac{1}{\sqrt{2}} (\vert 0 \rangle + \vert 1 \rangle) = \vert + \rangle$ $H \vert 1 \rangle = \frac{1}{\sqrt{2}} (\vert 0 \rangle - \vert 1 \rangle) = \vert - \rangle$

B. The Pauli-X Gate ($X$)

Often called the quantum NOT gate, it swaps the amplitudes of the $\vert 0 \rangle$ and $\vert 1 \rangle$ states (rotation of $180^\circ$ around the $X$-axis). $X = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$

Action on basis states: $X \vert 0 \rangle = \vert 1 \rangle, \qquad X \vert 1 \rangle = \vert 0 \rangle$

C. The Pauli-Y Gate ($Y$)

A rotation of $180^\circ$ around the $Y$-axis. $Y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}$

Action on basis states: $Y \vert 0 \rangle = i \vert 1 \rangle, \qquad Y \vert 1 \rangle = -i \vert 0 \rangle$

D. The Pauli-Z Gate ($Z$)

Also called the Phase Flip gate, it leaves the state $\vert 0 \rangle$ unchanged, but flips the sign of the state $\vert 1 \rangle$. $Z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$

Action on basis states: $Z \vert 0 \rangle = \vert 0 \rangle, \qquad Z \vert 1 \rangle = -\vert 1 \rangle$

3. Common Multi-Qubit Gates

Multi-qubit gates act on multi-qubit state vectors. For two qubits, vectors are 4D (representing coefficients of $\vert 00 \rangle, \vert 01 \rangle, \vert 10 \rangle, \vert 11 \rangle$).

The Controlled-NOT (CNOT / $CX$) Gate

The CNOT gate operates on two qubits: a control qubit (qubit 0) and a target qubit (qubit 1). It flips the target qubit state if and only if the control qubit is in state $\vert 1 \rangle$. $CX = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{pmatrix}$

Action on basis states:

$CX \vert 00 \rangle = \vert 00 \rangle$
$CX \vert 01 \rangle = \vert 01 \rangle$
$CX \vert 10 \rangle = \vert 11 \rangle$ (control is 1, target flips $0 \rightarrow 1$)
$CX \vert 11 \rangle = \vert 10 \rangle$ (control is 1, target flips $1 \rightarrow 0$)

4. Qiskit Implementation Example

Here is how to create a simple circuit in Python using IBM’s Qiskit library to apply these gates:

from qiskit import QuantumCircuit

# 1. Initialize a 2-qubit circuit with 2 classical measurement bits
qc = QuantumCircuit(2, 2)

# 2. Apply Hadamard gate on qubit 0 (puts it in superposition)
qc.h(0)

# 3. Apply CNOT (CX) gate with control=0, target=1 (entangles them)
qc.cx(0, 1)

# 4. Measure both qubits into the classical bits
qc.measure([0, 1], [0, 1])

# 5. Draw the circuit
print(qc.draw(output="text"))

[[qubits]]
[[entanglement]]
[[ibm-quantum-learning]]