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QUANTUM COMPUTING AND CIRCUIT ARCHITECTURES
QUANTUM LOGIC GATES
Mathematical Transformations in Multi-Qubit Circuits
An axiomatic guide to unitary operations, matrix representations, and universal gate sets.
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1. MATHEMATICAL FOUNDATIONS OF QUANTUM GATES
Quantum gates are mathematical operators that transform a quantum state. Because physical systems must conserve
total probability (Postulate 4), every closed-system quantum gate is modeled as a Unitary Matrix U.
Unitary Condition: U †U = UU † = I ←→ Consequence: Reversible computation is natively mandatory.
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2. SINGLE-QUBIT OPERATIONS & THE PAULI GROUP
Single-qubit gates manipulate state vectors on the complex Bloch Sphere, mapping transformations on the basis states
{|0⟩, |1⟩}.
The Pauli Matrices & Hadamard Transformation:
0 1 0 −i 1 0 1 1 1
X = Y = Z = H = √
1 0 i 0 0 −1 2 1 −1
Superposition Generation: H |0⟩= |0⟩+|1⟩√ = |+⟩ and H |1⟩= |0⟩−|1⟩√ = |−⟩
2 2
✦Phase Gates: The S gate diag(1, i) executes a π/2 rotation, while the T gate diag(1, eiπ/4) executes a critical π/4
rotation around the Z-axis.
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3. MULTI-QUBIT ENTANGLING GATES
Multi-qubit systems scale via tensor products (⊗). Controlled gates apply a target transformation conditioned strictly
on the state of control qubits.
Controlled-NOT (CNOT) Operator: Flips the target qubit if and only if the control qubit is |1⟩.
1 0 0 0
0 1 0 0 |00⟩→|00⟩ |10⟩→|11⟩
CNOT = Mapping: and
0 0 0 1 |01⟩→|01⟩ |11⟩→|10⟩
0 0 1 0
Controlled-Phase (CZ) Operator: Injects a negative phase factor (−1) if and only if both system qubits occupy the
|11⟩state. CZ = diag(1, 1, 1, −1).
GENERATING THE 3-QUBIT GHZ STATE
Global coherence is built sequentially using a mix of single-qubit and multi-qubit entangling operations:
|000⟩+ |111⟩
Circuit Operations: (CNOT2→3)(CNOT1→2)(H ⊗I ⊗I) |000⟩= √
2
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4. COMPUTATIONAL UNIVERSALITY
A finite set of quantum gates is classified as Universal if any arbitrary unitary operation can be approximated to any
desired degree of accuracy ε.
1. The Clifford Set: Consisting of {H, S, CNOT}, this set can efficiently be simulated classically (Gottesman-Knill
Theorem).
2. Clifford + T Set: Adding the non-Clifford T gate breaks classical simulatability, yielding absolute universal
quantum computational power.