🌀 Phase & T Gates
Master phase rotation gates essential for universal quantum computing
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🌀 What Are Phase Gates?
Phase gates add phase shifts to quantum states without changing amplitudes. The S gate (Phase gate) adds π/2 phase, while the T gate adds π/4 phase. These gates are crucial for universal quantum computation and fault-tolerant quantum computing.
The combination of Hadamard, T gate, and CNOT forms a universal gate set, meaning any quantum computation can be built from these gates.
📊 Phase Gate Family
π/2 phase rotation (90°)
π/4 phase rotation (45°)
🎯 Key Properties
Phase Addition
Only affects |1⟩ state, leaving |0⟩ unchanged
Diagonal Matrices
Both gates are represented by diagonal matrices
Fault-Tolerant
T gate is crucial for fault-tolerant quantum computing
Gate Hierarchy
S² = Z and T² = S (nested square roots of Z)
🎓 Historical Context
Phase gates emerged from the study of universal quantum gate sets. The T gate gained special importance with the discovery of quantum error correction and fault-tolerant computing. The Clifford+T gate set (using H, S, CNOT, and T) is now the standard for implementing fault-tolerant quantum algorithms, as T gates are the only non-Clifford gates needed for universality.