Simulating T1 Decay

Simulating T1 Decay

T1 decay is the headline noise process in superconducting hardware. A qubit prepared in the excited state |1⟩ does not stay there forever, it spontaneously emits its energy into the environment and relaxes to the ground state |0⟩. The characteristic time for this process is T1. If you measure the excited-state population as a function of waiting time, you find an exponential decay: $P(1,t)=e^{-t/T_1}$. Real transmons have T1 times in the 50-500 microsecond range, and every nanosecond of circuit time nibbles away at that budget.

The Physics in One Sentence

A qubit sitting in |1⟩ is an excited two-level system in a sea of electromagnetic modes. Spontaneous emission into those modes, plus couplings to defect two-level systems in the substrate, flux noise, and stray photons, causes a probabilistic transition down to |0⟩. Because each decay event is independent and happens at a constant rate, the population decays exponentially: the hallmark of a Poisson process.

The Amplitude-Damping Channel

Mathematically, T1 decay is the amplitude-damping channel. In Kraus form it has two operators:

K0 = [[1,        0      ],         K1 = [[0, √γ],
      [0, √(1−γ)]]                         [0,  0]]

Here γ ≈ Δt / T1 is the decay probability over a small time step. After each time step, the state is updated as ρ → K0 ρ K0† + K1 ρ K1†. Importantly, the channel is asymmetric: it drives the qubit toward |0⟩, unlike depolarising noise which drives it toward a uniform mixture. The fixed point is pure |0⟩⟨0|, not the maximally mixed state.

A Practical Simulation Trick

A full Kraus-operator simulation can feel heavy when all you want is a back-of-the-envelope picture. A cheap and surprisingly faithful stand-in is stochastic unraveling: during each small time step, with probability γ apply an X gate (if the qubit was excited, flip it to the ground state), otherwise do nothing. Averaged over many shots, this recovers the correct $P(1,t)=e^{-t/T_1}$ decay curve for a qubit prepared in |1⟩. Keep in mind this X-flip Monte Carlo is a pedagogical approximation, not the real channel: amplitude damping is not a Pauli-X channel. A genuine random X error would also excite |0⟩ to |1⟩, and true T1 decay shrinks superposition coherences by √(1−γ) rather than flipping them. The trick reproduces the excited-state population only.