Qubit Decoherence Time Calculator

Introduction

Qubits store information in amplitudes and phase, so their useful lifetime is not described by a single classical failure clock. Energy relaxation, commonly reported as T₁, measures how quickly an excited qubit falls back toward the ground state. Pure dephasing, commonly written as T_φ, measures phase scrambling that can occur even when the energy state has not changed. Together they set the transverse coherence time T₂, the timescale that matters for preserving superposition phase during idle periods and coherent gates.

This calculator uses the standard exponential-rate approximation for a single qubit with independent relaxation and pure dephasing channels. It estimates T₂, compares it with the relaxation-limited ceiling of 2T₁, and converts the result into a simple gate-duration budget. It is meant for quick hardware comparisons, lab-note sanity checks, and back-of-the-envelope circuit-depth estimates.

How to use this calculator

Enter the measured relaxation time T₁ in microseconds, a pure dephasing time T_φ in microseconds, and an approximate one-qubit or idle gate duration in nanoseconds. The dephasing field should be pure dephasing, not a measured Ramsey or echo T₂ value. If your device report already gives T₂, treat that number as the coherence result and use the formula below only to infer what pure dephasing would have to be.

The output shows the transverse decay rate, estimated T₂, the fraction of the ideal 2T₁ limit achieved, the number of gate-duration intervals in one e-folding time, and the estimated phase-coherence amplitude remaining after 100 such intervals. Those gate outputs are not full process fidelities; they are a compact way to see whether a proposed gate schedule is comfortably faster than the coherence clock.

Formula and method

The page models relaxation and pure dephasing as independent exponential decay rates. Under that assumption, the transverse coherence rate is

The first term says that energy relaxation also erases phase information, but at half the population-relaxation rate. The second term adds phase-only noise such as magnetic-field drift, charge noise, flux noise, oscillator phase jitter, or slow environmental fluctuations. The calculator adds those two rates and then inverts the sum to estimate T₂.

For a gate or idle interval τ, the simple phase-coherence amplitude scale is

That exponential is a useful timescale indicator, but it is not a calibrated gate-error model. Real quantum processors have pulse-dependent control errors, leakage, crosstalk, readout errors, and noise spectra that a one-number lifetime cannot capture.

Example calculation

Suppose a superconducting qubit has T₁ = 120 µs, pure dephasing time T_φ = 160 µs, and a 40 ns gate interval. The half-relaxation contribution is 1 / 240 µs, while the pure dephasing contribution is 1 / 160 µs. Adding those rates gives 1 / T₂ = 0.0104167 µs^-1, so T₂ is about 96 µs.

A 40 ns interval is 0.04 µs, so this simplified model places about 2,400 such intervals inside one e-folding time. After 100 idle-size intervals, or 4 µs total, the exponential coherence amplitude is roughly 95.9%. That does not mean a 100-gate circuit has 95.9% fidelity; it only says that decoherence alone is not yet the dominant clock in this idealized scenario.

Interpreting the result

If the result is close to 2T₁, relaxation is the main ceiling and making the device quieter in phase will produce diminishing returns until the relaxation channel improves. If the result is far below 2T₁, pure dephasing is the stronger bottleneck, so shielding, cleaner bias control, better oscillator phase noise, materials improvements, or dynamical decoupling may help.

Comparing technologies requires care. Trapped ions, spin qubits, superconducting circuits, neutral atoms, and photonic qubits expose different measurement conventions, gate speeds, and noise environments. A longer lifetime is valuable, but the relevant engineering question is how many reliable operations fit inside that lifetime after calibration, measurement, and error-correction overhead are included.

Limitations

• Independent exponential channels. The calculator assumes relaxation and pure dephasing add as simple rates. Non-Markovian noise, 1/f spectra, revivals, leakage, and strongly driven dynamics can violate this approximation.
• No pulse-sequence distinction. Ramsey, Hahn echo, Carr-Purcell, and dynamically decoupled measurements can report different apparent coherence times. Enter a pure dephasing timescale consistent with the experiment you are modeling.
• No full error budget. The gate-depth output ignores coherent control error, leakage, crosstalk, state preparation, measurement, thermal population, and correlated faults.
• No multi-qubit coupling model. Entangling gates and idle spectators can introduce extra channels that are not represented by a single-qubit T₁ and T_φ.

FAQ

Can I enter a measured T₂ value as the pure dephasing time?

No. This calculator expects T₁ and pure dephasing time T_φ, then estimates T₂ from the rate equation. If you already have a measured T₂, use it directly as the measured coherence result or rearrange the equation to infer T_φ.

Why can T₂ be longer than T₁?

Transverse coherence is limited by half of the energy relaxation rate plus pure dephasing. If pure dephasing is very small, T₂ can approach 2T₁, but this model should not produce a value above that ceiling.

Does this estimate quantum gate error?

No. The gate-depth output is only an exponential idle-coherence scale. Real gate error also depends on calibration, pulse shape, leakage, crosstalk, readout, thermal population, and the frequency content of the noise.