Error Mitigation in Chemistry

VQE returns an upper bound on the ground-state energy, so on noisy hardware the estimate drifts above the true value as decoherence corrupts the trial state. Error mitigation is a set of techniques that recover accurate expectation values from noisy data without the full overhead of error correction — making chemical accuracy reachable on today's devices.

Mitigation versus correction

Quantum error correction encodes a logical qubit into many physical qubits and actively detects and fixes errors, but demands thousands of physical qubits per logical one. Error mitigation instead runs the bare circuit, accepts that the result is noisy, and uses extra measurements plus classical post-processing to estimate and subtract the noise-induced bias. It adds sampling overhead, not qubits — which is why it dominates NISQ-era chemistry.

Zero-noise extrapolation

The flagship technique is zero-noise extrapolation (ZNE). The idea: deliberately amplify the noise by a known factor, measure the energy at several noise levels, and extrapolate back to the zero-noise limit.

Run the circuit at the native noise level $\lambda = 1$ and measure $E(1)$ .
Stretch the noise to $\lambda = 1.5, 2, 3, \dots$ — for example by unitary folding, replacing each gate $G$ with $G G^\dagger G$ (which equals $G$ ideally but triples its exposure to noise).
Fit $E(\lambda)$ with a polynomial or exponential model and evaluate the fit at $\lambda = 0$ .

If the noise scales smoothly, the extrapolated $E(0)$ is much closer to the noiseless energy than the raw $E(1)$ . ZNE needs no knowledge of the noise model — only the ability to scale it.

Other practical techniques

Readout-error mitigation. Measurement is one of the noisiest operations. Calibrate a confusion matrix $M$ (the probability of reading $j$ when the true outcome is $i$ ) on known input states, then apply $M^{-1}$ to correct measured distributions.
Symmetry verification. The exact molecular state has fixed particle number and spin. Post-select on shots that satisfy these symmetries (e.g. discard outcomes with the wrong electron count), discarding runs corrupted by symmetry-breaking errors.
Purification / virtual distillation. Prepare several copies of the noisy state and combine them so the estimate approaches the dominant (ground) component, suppressing incoherent error.

Putting it in the VQE loop

In a mitigated VQE run, every energy evaluation $E(\boldsymbol\theta)$ inside the optimization loop is itself a mitigated estimate: measure at several noise scales, extrapolate, optionally post-select on symmetry, and hand the corrected energy to the optimizer. For $\text{H}_2$ this combination has demonstrated chemical accuracy on real superconducting and trapped-ion hardware, even though the raw unmitigated energies miss by far more than $1.6 \times 10^{-3}$ Hartree. The next lesson brings the full pipeline together to estimate a ground-state energy end to end.