Decoherence: T₁ and T₂ Times

Overview

Decoherence is the main reason real quantum computers cannot run arbitrarily long circuits. Even if every gate were perfect, a qubit sitting idle still interacts with its environment.

Two standard timescales summarize how a qubit loses quantum information:

• $T_1$ (energy relaxation): how quickly population decays toward the ground state.
• $T_2$ (dephasing): how quickly phase relationships become scrambled.

These quantities matter because:

• algorithms rely on coherent phase relationships across many gates
• decoherence accumulates with time and circuit depth

Intuition

$T_1$: energy relaxation

Think of a qubit as a small system with an excited state $|1\rangle$ and a ground state $|0\rangle$. If the qubit is in $|1\rangle$, it can “leak” energy into the environment and relax to $|0\rangle$.

This is like a ball rolling down to a valley: once it falls, you cannot recover the original energy without actively pumping it back.

$T_2$: dephasing

Dephasing is loss of phase coherence. Even if the qubit stays in the same energy level population-wise, the relative phase between $|0\rangle$ and $|1\rangle$ can drift unpredictably.

A helpful picture:

• Each run of the experiment accumulates a slightly different phase.
• When you average over many runs (or when the phase becomes effectively random), interference disappears.

Relationship between $T_1$ and $T_2$

$T_2$ is never longer than a bound set by $T_1$. Intuitively:

• if the qubit frequently relaxes, it certainly cannot keep stable phase for long

But $T_2$ can be much shorter than $T_1$ if there is strong dephasing noise.

Formal Description

We keep it descriptive with just the standard exponential idea.

$T_1$ as exponential decay of excited population

If you prepare a qubit in $|1\rangle$ and wait time $t$, the probability of still being in $|1\rangle$ typically decays approximately like:

$$P(1\text{ at time }t) \approx e^{-t/T_1}.$$

Interpretation:

• after time $T_1$, the excited population has dropped to about $1/e$ of its initial value

$T_2$ as decay of coherence

If you prepare a superposition like $|+\rangle$, the coherence (the “off-diagonal” structure in the state) decays approximately like:

$$\text{coherence}(t) \approx e^{-t/T_2}.$$

Interpretation:

• after time $T_2$, interference visibility has dropped to about $1/e$ of its initial value

What these times do to quantum states

• $T_1$ tends to push states toward $|0\rangle$.
• $T_2$ tends to remove phase information, turning superpositions into mixtures.

Both reduce the effectiveness of deep circuits.

Worked Example

Suppose a device has:

• $T_1 = 100\,\mu\text{s}$
• $T_2 = 40\,\mu\text{s}$

If you prepare $|+\rangle$ and run a circuit that takes $t=40\,\mu\text{s}$ total time, then coherence drops to roughly $e^{-1}\approx 0.37$.

That means:

• interference effects are much weaker
• algorithm outcomes can look noisy even if gates were otherwise ideal

This illustrates why $T_2$ is often the most immediately limiting number for algorithms that depend on phase.

Turtle Tip

Common Pitfalls

Quick Check

What’s Next

Decoherence limits how long quantum information survives. Next we look at what happens even during active computation: gate errors, where the intended operation is slightly wrong and errors accumulate with circuit depth.