Quantum Cookbook

Part II · Chapter 12

Amplitude & Phase Damping

Read ~22 min Prerequisites Ch. 9, Ch. 11 Tools Bloch-slice damping widget, code cell Live
Basic

Ch. 11 gave you the general framework for noise. This chapter covers the two noise processes that actually show up on every piece of real quantum hardware in existence: a qubit losing energy to its surroundings, and a qubit losing phase coherence without losing any energy at all. They look superficially similar in their math, but they do genuinely different things — and the difference matters enormously for anyone actually building a quantum computer.

Amplitude damping — losing energy

Picture a qubit as a two-level atom: |0⟩ the ground state, |1⟩ an excited state. Left alone long enough, an excited atom spontaneously decays to the ground state, emitting a photon. That's amplitude damping — γ is the probability of decaying from |1⟩ to |0⟩:

$$K_0=\begin{bmatrix}1&0\\0&\sqrt{1-\gamma}\end{bmatrix} \qquad K_1=\begin{bmatrix}0&\sqrt{\gamma}\\0&0\end{bmatrix}$$

Phase damping — losing coherence, not energy

Phase damping describes a different kind of disturbance: the environment "peeks" at the qubit's phase relationship without exchanging any energy. Populations (the |0⟩ vs. |1⟩ probabilities) stay exactly where they were; only the off-diagonal coherence terms shrink.

$$K_0=\begin{bmatrix}1&0\\0&\sqrt{1-\lambda}\end{bmatrix} \qquad K_1=\begin{bmatrix}0&0\\0&\sqrt{\lambda}\end{bmatrix}$$
Intermediate
Fig. 12.1 — set a starting state, dial in noise, pick a channelx-z Bloch slice
Blue ring = input state, amber dot = output after the channel. With amplitude damping, the point is pulled up toward |0⟩ — try θ=90° and watch it. With phase damping, the point only moves horizontally, toward the vertical axis — the z-coordinate never changes at all, no matter how large γ gets.

Reading the difference directly off the matrices

Both channels share the exact same K₀ — that's not a coincidence, and it's worth noticing. Where they differ is K₁: amplitude damping's K₁ has its nonzero entry in the corner that moves population from |1⟩ to |0⟩ (an off-diagonal placement); phase damping's K₁ is purely diagonal, so it can never move population between basis states — it can only attack the coherence between them.

Advanced

Connecting γ, λ to real hardware: T₁ and T₂

If you've ever seen a real quantum computing paper, you've seen T₁ and T₂ times. T₁ ("relaxation time") is the characteristic decay time for amplitude damping: γ(t) = 1−e^(−t/T₁). T₂ ("dephasing time") plays the same role for phase damping. These aren't abstract — they're literally the numbers superconducting and trapped-ion qubits are benchmarked on, and Fig. 12.1's γ slider is sweeping through exactly the same physical process those papers are timing.

Fig. 12.2 — code, runs in your browser
Compare the diagonal entries between the two outputs directly — that's the entire conceptual difference between these two channels, in two printed matrices.
Check your understanding

What physical process does amplitude damping model?

Energy loss — spontaneous decay from the excited state |1⟩ to the ground state |0⟩, with probability γ.

Check your understanding

Does phase damping change the probability of measuring |0⟩ vs. |1⟩?

No — the diagonal (population) entries of ρ stay exactly the same. Only the off-diagonal coherence terms shrink.

Check your understanding

In Fig. 12.1, why does amplitude damping pull the point toward |0⟩ specifically, rather than toward the center?

Because amplitude damping is asymmetric by design — it only describes decay from |1⟩ to |0⟩, never the reverse, so every application nudges population toward the |0⟩ pole rather than uniformly toward the maximally mixed center.

Check your understanding

What's the single matrix-structure difference between the K₁ of amplitude damping and the K₁ of phase damping?

Amplitude damping's K₁ is off-diagonal (it moves population between |0⟩ and |1⟩); phase damping's K₁ is purely diagonal (it can only attack coherence, never move population).

Check your understanding

What do T₁ and T₂ measure on real quantum hardware?

T₁ is the characteristic relaxation time for amplitude damping (energy loss); T₂ is the characteristic dephasing time for phase damping (coherence loss). Both are standard benchmarking numbers for real qubits.

Review queue

Scroll back to a chapter's quiz cards to review them once they're due — a full cross-chapter queue view is a planned upgrade.