The Amplitude-Damping Channel

Amplitude damping models energy loss, the decay of an excited qubit toward its ground state, capturing T1 relaxation.

Energy relaxation

The amplitude-damping channel describes a qubit losing energy to its environment, for example an excited atom emitting a photon. It has two Kraus operators: K0 = diag(1, sqrt(1 - gamma)) and K1 with a single sqrt(gamma) entry in the upper-right corner. The parameter gamma is the probability of decay from |1> to |0> during the interval. It captures the T1 relaxation time of real qubits.

Asymmetry

Unlike depolarizing or dephasing noise, amplitude damping is not symmetric between |0> and |1>. It drives every state toward the ground state |0>, which is a fixed point. Populations relax and coherences shrink, but the ground state is preserved perfectly. This directionality reflects a genuine energy asymmetry: spontaneous emission goes downhill, not up, at low temperature.

import numpy as np
def amp_damp(rho, g):
    K0=np.array([[1,0],[0,np.sqrt(1-g)]])
    K1=np.array([[0,np.sqrt(g)],[0,0]])
    return K0@rho@K0.conj().T + K1@rho@K1.conj().T
rho=np.array([[0,0],[0,1]])   # excited
print(np.round(amp_damp(rho,0.3),3))  # some pop moved to |0>

Effect on the Bloch sphere

Amplitude damping moves the sphere's center toward the north pole (ground state) and contracts it anisotropically: the z-component relaxes toward +1 while the x and y components shrink by sqrt(1 - gamma). The transformation is an affine map that both scales and translates the Bloch vector, distinguishing it from the purely contracting depolarizing channel.

Relevance

Amplitude damping is one of the two dominant error channels in superconducting and atomic qubits, the other being dephasing. Together they set the coherence budget that limits circuit depth. Error-correcting codes must handle the bit-flip-like and phase components damping produces; because damping is asymmetric and non-Pauli, accurate modeling requires the full channel rather than a Pauli approximation, especially for leakage-sensitive and biased-noise codes.