An Introduction to Quantum Computing
An Introduction to Quantum Computing ____________________________________________________________________________________ Two qubits have four computational basis states: |00), |01), |10) and |11). A two-qubit state is therefore a four-entry column vector: |ψ ) = [a ; b ; c ; d] The order used here is |00), |01), |10), |11). A two-qubit gate is represented by a 4 × 4 matrix. The controlled-NOT (CNOT) gate flips the second qubit only when the first (control) qubit is 1. Its action is: |00) → |00 ); |01) → |01 ); |10) → |11 ); |11) → |10 ) In the stated basis order, its matrix is: CNOT = [1 0 0 0 ; 0 1 0 0 ; 0 0 0 1 ; 0 0 1 0] Worked example: start with |10) = [0;0;1;0]. CNOT maps it to |11) = [0;0;0;1]. CNOT can create entanglement. First apply H to the first qubit of |00), producing (|00) + |10) )/√2. Then apply CNOT: (|00) + |10) )/√2 → (|00 ) + |11) )/√2 This is a Bell state. It cannot be described as two independent qubits with their own separate pure states; the measurement outcomes are correlated. Tensor products: putting gates on separate qubits. If a gate acts on one qubit while another qubit is left alone, the combined operation is written with a tensor product, ⊗ . For example, applying X to the first of two qubits is X ⊗ I, a 4 × 4 matrix: X ⊗ I = [0 0 1 0 ; 0 0 0 1 ; 1 0 0 0 ; 0 1 0 0] The tensor product is the rule that builds a larger system's state space and operation from smaller pieces. It is essential when scaling from one qubit to many. A complete circuit example Consider this circuit, where the top wire is qubit 0 and the bottom wire is qubit 1: q0: |0) )) H )) ● ))) q1: |0) )))))))) X )) Step 3 — Apply CNOT, controlled by q0 and targeting q1: (|00) + |11) )/√2. If both qubits are measured, the results are 00 with probability 1/2 and 11 with probability 1/2. Ideal measurements never produce 01 or 10 for this state. How to work with a matrix: 1. Write the input state as a column vector. 2. Choose the gate matrix. 3. Multiply each row of the matrix by the input column. 4. Check that the resulting amplitudes are normalised: the sum of their squared magnitudes is 1. 5. Convert squared magnitudes into measurement probabilities when measuring. Step 1 — Start with |00). Step 2 — Apply H to q0: (|00) + |10) )/√2.
_______________________________________________________________________________________________ SCTE Technical Committee : Dave Hodges and Dr Anthony Basham. © 2026 Society for Broadband Professionals. All rights reserved Page 26
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