Quantum Gates: Hadamard, Pauli, and CNOT Gates

Learning Objectives Define a qubit and represent its state using a column vector. Represent the Hadamard, Pauli (X, Y, Z), and CNOT gates as matrices. Apply single-qubit gates to a qubit's state vector using matrix multiplication to determine the resulting state. Explain how the Hadamard gate creates a state of superposition. Describe the function of the two-qubit CNOT gate and its role in creating entanglement. Trace the state of a simple two-qubit system through a circuit involving H and CNOT gates to produce a Bell state. What if a computer bit could be both 0 and 1 at the same time? 🤯 Let's explore the fundamental logic gates that unlock this quantum power. This tutorial introduces the core building blocks of quantum circuits: the Hadamard, Pauli, and CNOT ga...

TermDefinitionExample Qubit (Quantum Bit)The fundamental unit of quantum information. Unlike a classical bit which is either 0 or 1, a qubit can exist in a state of superposition, being a combination of both |0⟩ and |1⟩ simultaneously.A qubit's state |ψ⟩ is represented as a vector: |ψ⟩ = α|0⟩ + β|1⟩, where α and β are complex numbers called probability amplitudes and |α|² + |β|² = 1. For example, |+⟩ = (1/√2)|0⟩ + (1/√2)|1⟩ is a qubit in a perfect superposition. State VectorA column vector in a 2-dimensional complex vector space that represents the state of a qubit. The basis states |0⟩ and |1⟩ are represented by orthogonal vectors.The state |0⟩ is represented by the vector [1, 0]ᵀ, and the state |1⟩ is represented by the vector [0, 1]ᵀ. Quantum GateA fundamental quantum circuit oper...

Single-Qubit Gate Application |ψ'⟩ = U |ψ⟩ To find the state of a qubit |ψ'⟩ after a gate U is applied, you perform matrix multiplication. The gate's matrix U is multiplied by the qubit's initial state vector |ψ⟩. The order of operations matters. Hadamard and Pauli Gate Matrices H = (1/√2)[[1, 1], [1, -1]] X = [[0, 1], [1, 0]] Y = [[0, -i], [i, 0]] Z = [[1, 0], [0, -1]] These are the matrix representations for the most common single-qubit gates. H creates superposition. X is a bit-flip (NOT). Z is a phase-flip. Y is both a bit-flip and a phase-flip. Controlled-NOT (CNOT) Gate Matrix CNOT = [[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 0, 1], [0, 0, 1, 0]] This is a two-qubit gate. It flips the second qubit (target) if and only if the first qubit (control...