T2 Time

The coherence time: how long a qubit maintains phase coherence in a superposition state.

T2 (also called the dephasing time or transverse relaxation time) measures how long a qubit can maintain a coherent superposition before the phase relationship is lost.

Definition

For a qubit in superposition $\frac{1}{\sqrt{2}}(|0⟩+|1⟩)$:

The off-diagonal elements of the density matrix decay as: $${\rho }_{01}(t)\propto e^{-t/T_2}$$

T2 characterizes how quickly the qubit “forgets” its phase.

T2 vs T2*

T2* ≤ T2 ≤ 2T1

Physical Picture

A qubit in $|+⟩=\frac{1}{\sqrt{2}}(|0⟩+|1⟩)$:

• Has a definite phase relationship between $|0⟩$ and $|1⟩$
• Noise causes the phase to wander randomly
• After time T2, phase is completely randomized
• Interference effects are lost

Measurement: Ramsey Experiment

|0⟩ ── H ──[wait t]── H ── Measure

1. Create $|+⟩$ with Hadamard
2. Wait time $t$ (phase accumulates)
3. Second Hadamard converts phase to amplitude
4. Oscillations decay with time constant T2*

Measurement: Hahn Echo (T2)

|0⟩ ── H ──[t/2]── X ──[t/2]── H ── Measure

The X gate (π pulse) in the middle refocuses slow phase errors:

• Low-frequency noise is canceled out
• Gives longer T2 compared to T2*

Typical Values

Causes of T2 Decay

Dynamical Decoupling

Extend T2 with more refocusing pulses:

── H ──[τ]── X ──[2τ]── X ──[2τ]── X ──[τ]── H ──

More pulses → better noise cancellation → longer effective T2.

Sequences: CPMG, XY4, XY8, etc.

Why T2 Matters

T2 is often the limiting factor:

• Determines maximum circuit depth
• Sets error rates for idling qubits
• Limits quantum memory duration

The Coherence Budget

If T2 = 100 μs and gate time = 50 ns: $$\text{Max gates}\approx \frac{T_2}{\text{gate time}}=2000$$

This is why faster gates and longer T2 both help.

See also: T1 Time, Decoherence, Circuit Depth