Phase Gate

Gates that add a phase to the |1⟩ state while leaving |0⟩ unchanged.

Phase gates are a family of single-qubit gates that modify the relative phase between $|0⟩$ and $|1⟩$ without changing measurement probabilities in the computational basis.

General Form

$$P(\theta )=\left(\begin{array}{cc}1& 0\\ 0& e^{i\theta }\end{array}\right)$$

Action: $|0⟩\to |0⟩$, $|1⟩\to e^{i\theta }|1⟩$

Common Phase Gates

S Gate (Phase Gate, √Z)

$$S=P(\pi /2)=\left(\begin{array}{cc}1& 0\\ 0& i\end{array}\right)$$

• Quarter turn around Z-axis on Bloch sphere
• $S^2=Z$
• Part of Clifford group

S† (S-dagger)

$$S^{†}=P(-\pi /2)=\left(\begin{array}{cc}1& 0\\ 0& -i\end{array}\right)$$

Inverse of S gate.

T Gate (π/8 Gate, √S)

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

• Eighth turn around Z-axis
• $T^2=S$
• NOT in Clifford group (makes it powerful)
• See T Gate for more

Z Gate

$$Z=P(\pi )=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)$$

Actually a Pauli gate, but also a phase gate with $\theta =\pi$.

Controlled Phase Gates

CZ (Controlled-Z)

$$CZ=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& -1\end{array}\right)$$

Applies Z to target when control is $|1⟩$. Symmetric: control and target are interchangeable!

CP (Controlled-Phase)

$$CP(\theta )=\text{diag}(1,1,1,e^{i\theta })$$

Adds phase only to $|11⟩$ state.

On the Bloch Sphere

Phase gates rotate around the Z-axis:

• $P(\theta )$ rotates by angle $\theta$ around Z
• Only affects superposition states (states on the equator)
• Pure $|0⟩$ or $|1⟩$ states appear unchanged

Why Phase Matters

You might wonder: if phases don’t affect measurement probabilities, why care?

Phases matter because of interference. When paths recombine (through gates like Hadamard), phases determine whether amplitudes add or cancel:

$$H\cdot P(\pi )\cdot H|0⟩=H|-⟩=|1⟩$$

The phase flip changed the final measurement outcome!

See also: T Gate, Pauli Gates, Quantum Gate, Bloch Sphere