An Introduction to Quantum Computing
An Introduction to Quantum Computing ____________________________________________________________________________________ |ψ ) = α|0 ) + β|1 ) = [ α ; β ] Here α and β are probability amplitudes , usually complex numbers. They must satisfy |α|² + |β|² = 1. When measured in the computational basis, the probability of 0 is |α|² and the probability of 1 is |β|². Example: |ψ ) = (1/√2)|0 ) + (1/√2)|1 ) gives a 50% chance of measuring 0 and a 50% chance of measuring 1. It is not simply a classical bit that is secretly either 0 or 1; the amplitudes can interfere when gates are applied. Quantum gates are matrices. A quantum gate changes a state. For a single qubit, a gate is represented by a 2 × 2 unitary matrix U. The calculation is: |ψ′ ) = U|ψ ) “Unitary” means U†U = I, where U† is the conjugate transpose and I is the identity matrix. This condition ensures that total probability remains 1. Ideal quantum gates are reversible. The identity gate does nothing: I = [1 0 ; 0 1] I|ψ ) = |ψ ) The Pauli-X gate is the quantum equivalent of a NOT operation: it swaps |0 ) and |1 ) . X = [0 1 ; 1 0] Worked example: apply X to |0). X|0) = [0 1 ; 1 0][1 ; 0] = [0 ; 1] = |1) Apply X again and the state returns to |0): X² = I. The Hadamard gate creates or removes an equal superposition: H = (1/√2)[1 1 ; 1 -1] Worked example: apply H to |0). H|0) = (1/√2)[1 ; 1] = (|0 ) + |1) )/√2 This state gives equal measurement probabilities. Applying H again returns |0), because H² = I. The minus sign in the matrix is important: it changes phase and can affect interference. A phase gate: changing phase The Z gate leaves |0) alone and changes the sign of |1): Z = [1 0 ; 0 -1] For example, Z(|0) + |1) )/√2 = (|0 ) - |1) )/√2. The measurement probabilities are still 50:50 at this point, but the relative phase has changed. A later gate can make that phase difference visible through interference. A two-qubit matrix
_______________________________________________________________________________________________ SCTE Technical Committee : Dave Hodges and Dr Anthony Basham. © 2026 Society for Broadband Professionals. All rights reserved Page 25
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