Phase Gates: S, T, and Rz

Phase Gates: S, T, and Rz

Phase gates rotate a qubit around the Z axis of the Bloch sphere. They don't change measurement probabilities alone, but they matter enormously in circuits.

Global vs Relative Phase

Multiply an entire quantum state by $e^{i\theta }$ and nothing observable changes: every probability is the squared magnitude of an amplitude, so the factor cancels everywhere. That overall factor is a global phase, and on its own it is physically meaningless.

A relative phase between the |0⟩ and |1⟩ components is completely different. The states (|0⟩ + |1⟩)/√2 and (|0⟩ + $e^{i\theta }$|1⟩)/√2 are genuinely different points on the equator of the Bloch sphere: θ is the azimuthal angle. Phase gates change exactly this angle and nothing else. They leave the |0⟩/|1⟩ balance untouched and spin the state around the Z axis.

Why Phase Matters

Both measure 50% |0⟩ and 50% |1⟩. But they're different quantum states. Applying H to |+⟩ gives |0⟩, while applying H to |−⟩ gives |1⟩. Phase is invisible until you interfere.

You can watch both halves of that statement in the simulator:

# Phase is invisible in the Z basis...
qc = QuantumCircuit(1, 1)
qc.h(0)     # |+>
qc.s(0)     # now (|0> + i|1>)/sqrt(2)
qc.measure(0, 0)
print_counts(simulate(qc, shots=1000))   # still ~50/50

# ...but interference reveals it
qc2 = QuantumCircuit(1, 1)
qc2.h(0)    # |+>
qc2.s(0)
qc2.s(0)    # S twice = Z, so the state is now |->
qc2.h(0)    # H|-> = |1>
qc2.measure(0, 0)
print_counts(simulate(qc2, shots=1000))  # 100% |1>

The S Gate (π/2 phase)

S|0⟩ = |0⟩ S|1⟩ = i|1⟩

The S gate rotates 90° around the Z axis. It takes |+⟩ → |i⟩ = (|0⟩ + i|1⟩)/√2 (pointing in the +Y direction on the Bloch sphere).

qc.s(0)

S² = Z: applying S twice is equivalent to a Z gate.

Need the opposite rotation? qc.sdg(0) applies S† (S dagger), a -90° rotation that exactly undoes S.