Decoherence and Error Channels

Five failure locations in one experiment#

Place relaxation, phase loss, imperfect gates, leakage, and readout at distinct locations in a prepare-evolve-measure sequence.

A circuit diagram shows ideal instructions. Hardware executes them with physical systems that interact with control electronics, the surrounding environment, measurement devices, and time itself. Decoherence is the loss of useful quantum coherence through those interactions; an error channel is a simplified mathematical model of what the interactions do to the state.

Open-system evolution is represented with density operators and quantum channels rather than pure-state unitary evolution alone. [nielsen-chuang][watrous-tqi]

Worked mechanism audit: derive before diagnosing#

Put the mechanisms on a timeline. Preparation error changes the initial state. Control error changes the implemented operation. Relaxation and dephasing act during gates and idle intervals. Leakage moves population outside the declared two-level space. Measurement maps the final quantum state to an outcome distribution, after which classical readout confusion changes reported labels. Two mechanisms can produce the same final histogram, so location and intervention are part of the diagnosis.

A minimal record contains the prepared state, delay or gate durations in seconds, the operation schedule, measurement basis, raw outcome alphabet, and readout calibration. Without durations, a coherence time cannot predict attenuation. Without basis, unchanged populations cannot rule out phase loss. Without the raw alphabet, leakage can be silently reassigned to zero or one.

Relaxation is a time-domain process#

Energy relaxation is not the symmetric bit-flip channel. In the zero-temperature amplitude-damping model, excited-state survival after idle time t is $P_1(t)=P_1(0)e^{-t/T_1}$. The loss probability for that interval is $\gamma(t)=1-e^{-t/T_1}$. The ground state remains the ground state, while excited population decays toward it; the process has a preferred direction.

The Kraus operators $K_0=|0\rangle\langle0|+\sqrt{1-\gamma}|1\rangle\langle1|$ and $K_1=\sqrt\gamma|0\rangle\langle1|$ give $\rho(t)=\sum_kK_k\rho(0)K_k^\dagger$. Starting in $\lvert 1 \rangle$, t=60 μs and T1=30 μs produce survival $e^{-2}\approx0.1353$. A Z-basis experiment at several delays can estimate that decay only after state-preparation and readout effects are addressed.

A straight line in $\log P_1$ versus t supports the simple stationary exponential over the measured interval. Curvature or time-ordered residuals can indicate drift, multiple decay modes, thermal repopulation, or calibration error. Fitting still returns a number in those cases; the residual test decides whether calling it T1 under the single-exponential model is warranted.

T1 changes energy populations#

Define amplitude relaxation operationally, state seconds as the unit, and derive a simple exponential survival curve under explicit assumptions.

The bit-flip channel is $\mathcal{E}(\rho)=(1-p)\rho+pX\rho X$. With probability $p$ the qubit flips; otherwise nothing happens. It is the quantum echo of a classical unreliable wire.

Relaxation and dephasing affect populations and coherence differently and require basis-appropriate characterization. [preskill-notes][quantum-computers-review]

Coherence needs a second basis#

Prepare $\lvert + \rangle$, idle for t, and measure X by applying a final Hadamard before Z readout. Under a simple exponential coherence model, $\langle X(t)\rangle=\langle X(0)\rangle e^{-t/T_2}$, possibly multiplied by an oscillation if the frame is detuned. The Z populations can remain near one half while the X contrast decays. That two-basis record separates coherence loss from ordinary population relaxation more clearly than either trace alone.

Relaxation itself contributes to coherence decay: under common Markovian assumptions, $1/T_2=1/(2T_1)+1/T_\phi$, where $T_\phi$ is a pure-dephasing time. Thus $T_2\le2T_1$ for that model. A fitted pair violating the inequality signals a mismatched protocol, drift, uncertainty, or invalid simplifying assumptions; it is not repaired by clipping T2.

Depolarization is different again. The normalized single-qubit convention must be written explicitly because authors attach p to different mixtures. A depolarizing fit can mimic decay in several bases while erasing the mechanism-specific information that T1 and Ramsey experiments were designed to retain. Use it as a declared effective model, not as an explanation of microscopic noise.

T2 changes off-diagonal coherence#

Use Ramsey-style preparation and basis change to distinguish dephasing from computational-basis population loss.

Under the convention used here, the depolarizing channel is $\mathcal{E}(\rho)=(1-p)\rho+\frac{p}{3}(X\rho X+Y\rho Y+Z\rho Z)$. The error is a random Pauli, so the state is nudged toward maximum mixedness from every direction at once.

Fault-tolerance analysis requires an explicit physical error model and cannot infer logical protection from coherence time alone. [fault-tolerant-roads][surface-codes-2012]

Time and probability are linked only by a model#

For amplitude damping over duration Δt, $\gamma=1-e^{-\Delta t/T_1}$; repeating m equal intervals gives survival $(1-\gamma)^m=e^{-m\Delta t/T_1}$. If gates have different durations, each receives a different γ. Assigning the same “T1 error probability” to a 20 ns gate and a 2 μs measurement destroys the time dependence.

The ratio T1 divided by layer duration is at best a scale estimate. It ignores state dependence, dephasing, gate errors, parallel scheduling, idle imbalance, measurement time, and the success criterion. A circuit does not suddenly fail at one coherence time; its observable degrades continuously under a process model. Publish the computed attenuation or channel composition, not a mythical maximum depth.

Channel probabilities are not coherence times#

Show how a duration-dependent channel parameter can be derived only after choosing a model and interval.

The coherence budget is a scale estimate, $N_{\mathrm{layers}}\approx T_{\mathrm{coherence}}/t_{\mathrm{layer}}$. This has engineering teeth: it bounds how deep a circuit can run before the state decays, and it turns "is this device good enough?" into arithmetic. All three are teaching models — they do not replace calibrated device physics — but they force you to name the error type, its probability, and its timescale.

NISQ-era device behavior includes control, measurement, and noise constraints beyond ideal channel examples. [nisq-preskill]

Leakage, readout, and control require distinct tests#

Leakage introduces at least one state outside $\{\lvert0\rangle,\lvert1\rangle\}$. A trace-preserving channel on a $2\times2$ density matrix cannot represent population leaving that space unless the missing population is disguised as loss. Extend the Hilbert space or keep an explicit leakage probability and return path. State whether measurement distinguishes the leaked level, discards the shot, or assigns it to a computational outcome.

Readout confusion acts after quantum measurement. With false-one probability $e_{0\to1}$ and false-zero probability $e_{1\to0}$, the observed distribution is a $2\times2$ stochastic matrix times the true distribution. Preparation of known zero and one states estimates its columns. Applying that matrix before a basis rotation would be physically wrong: it would turn a classical label error into basis-dependent quantum evolution.

Control error has another signature. A systematic over-rotation accumulates coherently and can oscillate with repeated gates, whereas a stochastic Pauli approximation often predicts monotone decay. Vary the number and sign of repeated gates, and insert echo sequences, to distinguish coherent calibration error from stationary decoherence. The same average error rate can imply very different long-circuit behavior.

Leakage and readout need different state spaces#

Explain why a two-level Pauli model cannot represent leaving the subspace and why classical confusion belongs after measurement.

An ideal Bell circuit outputs only 00 and 11. Your noisy run shows plenty of 01 and 10. Before blaming decoherence, enumerate the suspects:

A full-stack account assigns device, control, measurement, and feedback errors to different system layers. [full-stack-review]

Close the diagnosis three ways#

Build each row from an observation, at least two candidate mechanisms, and a controlled discriminator. Falling Z excited-state population suggests relaxation or readout drift; repeat at zero delay and swap prepared basis states. Stable Z populations with falling Ramsey contrast suggests dephasing or detuning; inspect phase versus delay and add an echo. Missing outcomes suggest leakage or filtering; retain the full discriminator record.

The synthetic pair is internally consistent: excited-state survival at 60 μs gives T1=30 μs, and X coherence at 20 μs gives T2=20 μs. Symmetric readout confusion would introduce an approximately delay-independent affine offset in both traces rather than those two exponentials. A basis-swapped readout calibration is the decisive follow-up, not a larger shot count alone.

Record residuals and rejected alternatives. A diagnosis is stronger when it states what observation would reverse it. Parameter agreement across Z and X fixtures, limiting behavior at zero delay, and trace-preserving executable channels form a three-way check: time-domain derivation, basis response, and implementation must tell the same story.

Shot allocation follows the discriminating question. More repetitions at one delay reduce sampling error there, but additional delay points test the exponential shape and additional bases test mechanism. Spend measurements where competing models make different predictions. A design that collects a million Z shots at t=0 cannot identify T2, no matter how narrow its error bar.

When fitting both traces, keep seconds and probabilities in separate columns. T1 and T2 are time constants with uncertainty in seconds; γ(t), outcome probability, and readout confusion are dimensionless quantities tied to a duration or measurement. Converting a time constant into “percent error” without a named interval is a unit failure, not a convenient summary.

The companion fixture is synthetic and stationary. Its purpose is to catch algebra, basis, and record-schema mistakes. It provides no estimate for a named device, and the correct conclusion is not that one mechanism is universally dominant. The conclusion is that these particular generated observations are consistent with the declared pair of exponential channels and inconsistent with the specified symmetric readout-only alternative.

Check limiting cases before accepting the fit. At t=0, amplitude damping and dephasing reduce to the identity map. At long time, the amplitude-damping fixture approaches ground-state population one, while the simple coherence term approaches zero. Trace and Hermiticity remain invariant throughout. A model that matches the two quoted points but fails those limits is not the declared channel.

A diagnosis matrix with falsifying experiments#

Map each observed signature to alternative mechanisms and the next controlled measurement.

The next run should isolate one explanation: raise the shot count, remove the noise model, change the channel probability, or inspect the exact statevector before sampling. One controlled change per run — anything more is astrology, not debugging.

Open-system evolution is represented with density operators and quantum channels rather than pure-state unitary evolution alone. [nielsen-chuang][watrous-tqi]