S and T Phase Gates

S and T Phase Gates

The S gate (also called the phase gate or $\sqrt{Z}$) and T gate ($\sqrt{S} = \sqrt[4]{Z}$) are fundamental single-qubit phase gates.

S Gate

The S gate applies a $\frac{\pi}{2}$ phase to $|1\rangle$:

$S = \begin{pmatrix} 1 & 0 \\ 0 & i \end{pmatrix}$

$S|0\rangle = |0\rangle \qquad S|1\rangle = i|1\rangle$

T Gate

The T gate applies a $\frac{\pi}{4}$ phase to $|1\rangle$:

$T = \begin{pmatrix} 1 & 0 \\ 0 & e^{i\pi/4} \end{pmatrix}$

$T|1\rangle = e^{i\pi/4}|1\rangle = \frac{1+i}{\sqrt{2}}|1\rangle$

Why These Gates Matter

Phase gates do not change measurement probabilities on their own — $|\alpha|^2$ and $|\beta|^2$ are unchanged. Their power lies in interference: applying S or T before a Hadamard creates a different superposition.

The T gate, combined with H and CNOT, forms a universal gate set for quantum computation.

The S² = Z Relationship

Applying S twice gives the Pauli-Z gate:

$S^2 = \begin{pmatrix} 1 & 0 \\ 0 & i^2 \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} = Z$

Implementation

To apply S to a state $[\alpha, \beta]$:

$S[\alpha, \beta] = [\alpha,\ i\beta]$

To apply T to a state $[\alpha, \beta]$:

$T[\alpha, \beta] = [\alpha,\ e^{i\pi/4}\beta]$

import cmath, math

def s_gate(state):
    return [complex(state[0]), state[1] * 1j]

def t_gate(state):
    return [complex(state[0]), state[1] * cmath.exp(1j * math.pi / 4)]

Your Task

Implement s_gate(state) and t_gate(state) that apply the S and T gates to a 2-element complex state vector.