T Gate (π/8 Gate) | Non-Clifford Fault-Tolerant Primitive
Quick Technical Answer:
The T gate applies a π/4 (45°) phase rotation to state |1⟩: T|0⟩ = |0⟩ and T|1⟩ = e^{iπ/4}|1⟩. Because it is non-Clifford (T⁴ = Z), adding the T gate to Clifford operations enables universal fault-tolerant quantum computing.
Formula / Unitary:
T = \begin{pmatrix} 1 & 0 \\ 0 & e^{i\pi/4} \end{pmatrix}, \quad T^2 = S, \quad T^4 = Z
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Non-Clifford Universality and Fault Tolerance
By the Eastin-Knill theorem, no quantum error-correcting code can implement a universal set of transversal logic gates. While Clifford gates are readily implemented transversally on surface codes, the T gate must be synthesized via fault-tolerant magic state distillation.
Frequently Asked Questions
Why is the T gate called the π/8 gate?
Up to an overall global phase of e^{iπ/8}, the diagonal elements of T are e^{-iπ/8} and e^{iπ/8}, which historically led to the name π/8 gate.