T1 and T2 Times

Every qubit is a physical system embedded in a noisy environment. Even if left completely alone, a qubit will lose its quantum information through two distinct mechanisms: energy relaxation (captured by $T_1$) and phase randomization (captured by $T_2$). These two timescales define the window in which a quantum circuit must complete before decoherence overwhelms the computation.

The details

$T_1$: Energy relaxation. The excited state $|1\rangle$ is energetically unstable. Over time, the qubit emits a photon or phonon to the environment and returns to $|0\rangle$. The probability of remaining in $|1\rangle$ decays exponentially:

$P(|1\rangle, t) = e^{-t/T_1}$

This is called amplitude damping. After one $T_1$ period, the excited state population has dropped to $e^{-1} \approx 37\%$ of its original value. $T_1$ is measured by preparing $|1\rangle$, waiting a variable delay $t$, then measuring and fitting the exponential decay.

$T_2$: Dephasing. Even if a qubit stays in the right energy level, its phase can randomize. A qubit in the superposition $\frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)$ gradually loses the phase relationship between its components. The Bloch vector shrinks in the equatorial plane: $\langle X \rangle(t) = e^{-t/T_2}$. Dephasing includes contributions from $T_1$ and from pure dephasing $T_\phi$. The fundamental constraint is $T_2 \leq 2T_1$; in practice, $T_2$ is often much less because low-frequency noise (flux noise, charge noise) dephases without causing energy transitions.

$T_2^*$ vs $T_2$. Free induction decay (FID) measures $T_2^*$, which captures both inhomogeneous broadening and intrinsic dephasing. Spin-echo sequences insert a refocusing pulse to cancel slow, static noise, recovering a longer $T_2 \geq T_2^*$. Dynamical decoupling sequences extend coherence further by applying multiple refocusing pulses.

Platform comparison:

Platform	$T_1$	$T_2$	Gate time (2Q)
Superconducting	100-500 μs	50-300 μs	~100 ns
Trapped ion	1-100 s	1-10 s	~100 μs
Photonic	Very long	Limited by loss	~ns
NV center	~1 ms	~1 ms (RT)	~μs

Simulating T1 decay with a Qiskit noise model:

from qiskit import QuantumCircuit, transpile
from qiskit_aer import AerSimulator
from qiskit_aer.noise import NoiseModel, thermal_relaxation_error

T1, T2, gate_time = 200e-6, 100e-6, 100e-9  # seconds

noise_model = NoiseModel()
noise_model.add_all_qubit_quantum_error(
    thermal_relaxation_error(T1, T2, gate_time), ['id', 'u1', 'u2', 'u3']
)

qc = QuantumCircuit(1, 1)
qc.x(0)       # Prepare |1>
qc.id(0)      # Wait one gate time
qc.measure(0, 0)

sim = AerSimulator(noise_model=noise_model)
counts = sim.run(transpile(qc, sim), shots=4096).result().get_counts()
print(counts)  # Small probability of measuring 0 due to T1 decay

Why the gate time / $T_2$ ratio matters. The error per gate from decoherence is roughly $\Delta t / T_2$. Superconducting qubits achieve $\sim 100\text{ ns}$ gate times against $T_2 \sim 100\text{ μs}$, giving an error fraction around $10^{-3}$ per gate. Trapped ions have longer gate times ($\sim 100\text{ μs}$) but much longer $T_2$ ($\sim 1\text{ s}$), yielding a similar or better ratio. Improving gate speed or coherence time both reduce error rates.

Why it matters for learners

$T_1$ and $T_2$ are the physical root cause of why quantum computing is hard. Every circuit must complete within a small multiple of $T_2$. Error correction extends the effective coherence of logical qubits beyond the physical limit, but at the cost of many physical qubits per logical qubit. Knowing typical $T_1$ and $T_2$ values by platform helps calibrate which algorithms are executable today and which require fault-tolerant hardware.

Common misconceptions

Misconception 1: $T_2$ is always shorter than $T_1$. The inequality $T_2 \leq 2T_1$ is a fundamental bound, but $T_2$ can approach $2T_1$ in systems with low pure dephasing. Trapped-ion qubits encoded in magnetic-field-insensitive transitions can approach this limit; the statement ”$T_2$ is always much shorter” is true only for superconducting qubits.

Misconception 2: Longer $T_1$ and $T_2$ always means better performance. Coherence times matter only relative to gate times. A qubit with $T_2 = 100\text{ ms}$ and 1 ms two-qubit gates is worse than a qubit with $T_2 = 100\text{ μs}$ and 10 ns gates. The useful figure of merit is the number of gates that fit within $T_2$.

Misconception 3: $T_2^*$ and $T_2$ are the same thing. $T_2^*$ includes reversible slow noise that refocusing pulses can cancel; $T_2$ (from spin echo) reflects only irreversible dephasing. Reporting $T_2^*$ without context can make a device appear noisier than its intrinsic dephasing rate implies.

See also

• Decoherence
• Superconducting Qubit
• Trapped Ion
• Quantum Error Correction
• Gate Fidelity