Noise Models and Error Mitigation: Zero Noise Extrapolation and Probabilistic Error Cancellation

Updated 2026-10-11 | 4 min read | QUANTUM (QNT) community

Noise is the default state

Today's machines make mistakes in almost every operation. Qubits drift, see T1 and T2 times, gates are imperfect and measurement sometimes reports the wrong answer. Full fault tolerance, described in the error correction guide, needs many extra qubits. Until then, researchers use cheaper tricks that reduce the impact of noise on the final answer. These tricks are called error suppression and error mitigation, and they belong to the era described in the NISQ guide.

What is a noise model?

A noise model is a recipe describing how errors happen. Software such as Qiskit Aer lets you build one from parts. Its tutorial lists building blocks like depolarizing error (a chance a gate scrambles the state), thermal relaxation error (set by T1, T2 and gate duration) and readout error (for example, a 5 percent chance of reading 1 when the true state is 0). You attach errors to gates, then simulate a circuit as if it ran on a noisy chip. Noise models serve two jobs: testing algorithms on a laptop before using scarce hardware time, and feeding mitigation methods that need to know the noise.

Suppression versus mitigation

IBM's documentation describes dynamical decoupling (pulse sequences on idle qubits) as error suppression, and Pauli twirling (random gates that turn messy noise into simpler noise) as a technique often combined with mitigation. Methods that fix up the measured numbers afterward, such as TREX, zero noise extrapolation and probabilistic error cancellation, are mitigation. Both aim to improve result quality as workloads grow, and neither makes a qubit permanently error free.

Zero noise extrapolation (ZNE)

The idea is almost cheeky: if you cannot turn the noise off, turn it up and see the trend. ZNE measures the same quantity at several deliberately raised noise levels, then extrapolates back to the zero noise point. The Mitiq documentation describes three ways to raise noise: stretching pulses (only on devices that allow pulse-level access), unitary folding (replacing a gate G with G followed by its inverse and G again, which deepens the circuit) and inserting identity gates that add waiting time. Then you fit a curve, for example a line, a polynomial or an exponential, and read off its value at zero.

Strengths: it needs no detailed noise map and works with any circuit. Weaknesses: the answer depends on the curve you choose, each noise level needs its own runs, and it is not guaranteed to be unbiased. IBM notes that its default settings sample at three noise factors, a roughly 3x overhead.

Probabilistic error cancellation (PEC)

PEC takes the opposite approach. It first learns the noise, then builds an inverse. According to Mitiq, each ideal gate is written as a weighted sum of noisy operations the hardware can really run. The weights add to 1 but some are negative, so they are not true probabilities. Add up their absolute values and you get a number called gamma, which measures how costly the cleanup will be. To avoid running a huge number of circuits, PEC randomly samples noisy circuits and attaches a sign and a scale to each, then averages. The result is an unbiased estimate of the ideal answer.

The catch is the price. Samples needed scale roughly as gamma squared divided by the square of the target error, and gamma multiplies across gates. IBM states the sampling overhead grows exponentially with circuit depth. Short circuits on good hardware are fine. Deep circuits on noisy chips quickly become impractical. PEC also needs an accurate noise description, which is why noise models and calibration are linked, see calibration.

Where these ideas came from

Both methods appear together in a 2016 paper by Kristan Temme, Sergey Bravyi and Jay Gambetta, "Error mitigation for short-depth quantum circuits," published in Physical Review Letters in 2017. The paper proposed extrapolation using Richardson's method and error cancellation by resampling from a quasi-probability distribution. The authors aimed them at near-term devices, with no extra qubits needed.

A fair summary

• Mitigation is a bridge, not a destination. Costs rise quickly with circuit size.
• For large experiments IBM's page says for utility-scale experiments a noise-amplification method called PEA is often the best choice.
• Always ask which mitigation was used when you read a result, see reading company claims.

This is education, not financial advice. The QNT memecoin is independent of Quantinuum Ltd and every company mentioned.

Sources and further reading

• IBM Quantum docs: error mitigation and suppression techniques
• Mitiq: zero noise extrapolation theory
• Mitiq: probabilistic error cancellation theory
• arXiv 1612.02058: Error mitigation for short-depth quantum circuits (Temme, Bravyi, Gambetta)
• Qiskit Aer: building noise models

Reported as of 2026-10-09. Research moves fast, so check the original papers and company pages.

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