S and T Gates

qc/gates qc/math

Two more special cases of the general phase gate $P_{\theta }$, smaller rotations than Z’s full half-turn:

$S=P_{\pi /2}=\left(\begin{array}{cc}1& 0\\ 0& i\end{array}\right),T=P_{\pi /4}=\left(\begin{array}{cc}1& 0\\ 0& e^{i\pi /4}\end{array}\right)$

S (“phase gate,” a quarter-turn about the Z-axis) and T (sometimes called the ”$\pi /8$ gate,” an eighth-turn). $S^2=Z$ and $T^2=S$ — each is literally a finer subdivision of the last, all rotating about the same axis on the Bloch Sphere.

Why T specifically matters

Key insight: T is the standard “magic” gate that shows up throughout fault-tolerant quantum computing. See Universal Gate Sets and the Clifford Group for why — S and H are not enough on their own, but adding T is. This is also the exact T referenced (without being defined) in Error Correction (EC)‘s magic state distillation section: “prepare many noisy magic states… consume one via gate teleportation to apply a fault-tolerant T gate” — this note is what that T actually is.

Related

• Z Gate and Relative Phase
• Bloch Sphere
• Universal Gate Sets and the Clifford Group
• Error Correction (EC) — where T shows up again, in magic state distillation

Self-Check

• Could you write the matrices for S and T and verify $S^2=Z$?
• Why are S and T both described as smaller rotations than Z, geometrically?
• Why does Error Correction (EC) care about T specifically, rather than S or Z?