A transmon qubit's state is a point on the Bloch sphere: the north/south poles are |0⟩ and |1⟩, and gate pulses rotate the state vector around the sphere. Decoherence — unwanted coupling to the environment — shrinks the vector toward the centre (a fully mixed, useless state).
|ψ⟩ = cos(θ/2)|0⟩ + e^(iφ)sin(θ/2)|1⟩
X, Y, Z: 180° rotations about their axis H: 180° about the (X+Z)/√2 axis
|r(t)| ≈ |r(0)| · e^(−t/T2) (Bloch-vector shrinkage from dephasing)
- X / Y / Z / H gates — real single-qubit rotations, applied instantly as pulses.
- T2 — how long coherence survives before environmental noise randomizes the phase; longer T2 means a more useful qubit.
- Drive noise — imperfections in the control pulses themselves, nudging the rotation axis slightly off target each gate.
Real-world relevance: IBM and Google's superconducting quantum processors are built from exactly this device (the transmon), and T2 — typically tens to hundreds of microseconds — is the hard ceiling on how many gates a real circuit can run before the answer becomes noise.