Error Mitigation Without Error Correction

Error Mitigation Without Error Correction

Full quantum error correction requires many physical qubits per logical qubit and is not available on NISQ hardware. Error mitigation is a weaker but more practical alternative: post-processing techniques that reduce the effect of noise on measurement results without protecting the quantum state during computation.

Key Distinction

Error mitigation cannot reduce errors to zero, but it can dramatically improve the accuracy of expectation value estimation.

Technique 1: Zero-Noise Extrapolation (ZNE)

ZNE runs circuits at intentionally higher noise levels and extrapolates back to zero noise.

If noise affects expectation values as $E(\lambda )=E_0+a\lambda +b{\lambda }^2+\dots$, then ZNE recovers $E_0=E(0)$ by fitting and extrapolating.

Practical noise amplification: Inserting pairs of X gates (X then X = I) adds gates without changing the circuit logic, but amplifies noise at each added gate location.

Technique 2: Probabilistic Error Cancellation (PEC)

PEC represents the ideal (noise-free) channel as a linear combination of noisy channels:

By sampling from this decomposition and weighting results by $c_i$, PEC estimates noiseless expectation values. The cost: a factor of ${\gamma }^2$ more measurements, where $\gamma$ depends on the noise level.

For typical error rates of 1%, $\gamma \approx 1.01$ per gate. For 100-gate circuits, the sampling overhead is $1.01^{200}\approx 7\times$ more shots. Manageable for shallow circuits.