Noise and error

Three places error enters

State-preparation error means the register did not start in the state the circuit assumes. Gate error means the physical pulse was a slightly different map, often slightly non-unitary once the environment is traced out. Readout error means the device reports 0 when the qubit was 1, or the reverse, with some probability that itself depends on the state of neighbors.

A useful report separates those channels. A single “device was noisy” sentence cannot be reproduced. A small model — depolarizing probability per layer, amplitude-damping time, and a 2×2 readout confusion matrix — can be.

Channels, not just extra gates

A noise channel is a completely positive trace-preserving map on density operators. Depolarizing mixes the state toward the maximally mixed state. Amplitude damping pulls |1⟩ toward |0⟩. Dephasing kills off-diagonal terms in a chosen basis and leaves populations alone. These are textbook models used to say which kind of damage dominated, not a claim that the hardware is exactly that channel.

Because a channel acts on operators, the pure state vector is no longer a complete description once noise is on. The honest object is a density matrix, which is much larger, or a stochastic unraveling that averages many pure trajectories. Either way, ideal-state plots and noisy plots belong in different panels.

ρ ↦ Σ_k K_k ρ K_k†,    Σ_k K_k† K_k = I

Mitigation estimates a different quantity

Zero-noise extrapolation runs a family of deliberately worsened circuits and fits the measured observable back toward a hypothetical zero-noise intercept. The intercept is an estimate. It can reduce bias and still be wrong if the noise does not scale the way the fit assumes, or if the fit is unconstrained. Readout mitigation inverts a confusion matrix estimated from calibration circuits; if the calibration drifts, the inverse amplifies the drift.

Kryptur’s fidelity and extrapolation studies treat the mitigated number as a statistical object: show the stretched data, the fit, and the raw device points. Do not replace the raw points with the intercept alone.

What this means for a result

An ideal simulation validates the algorithm. A noisy model explores robustness. A device run measures hardware. Moving a sentence from one of those documents into another changes the claim. When a combinatorial pipeline uses a quantum proposal only as a hint to a classical repair step, say so: the certified object is the repaired solution, not the noisy sample that suggested it.