Noise mitigation of quantum observables via learning from Hamiltonian symmetry decays

We present a new quantum error mitigation technique (QEM), called GUiding Extrapolations from Symmetry decayS (GUESS), which exploits Hamiltonian symmetries to improve accuracy of noisy quantum computations. This method is explicitly designed for quantum algorithms that estimate expectation values of observables and consists in learning the extrapolation coefficients from a symmetry observable of the system to then estimate the value of a target observable. Furthermore, we propose a Hamiltonian impurity technique to enforce symmetries allowing the mitigation of local observables of interest. We employ the IBM Heron r2 quantum processing unit '\texttt{ibm_basquecountry}' to simulate the time evolution of average magnetization and nearest-neighbor correlator observables for transverse field Ising and XZ Heisenberg models in 1D with open boundary conditions. We benchmark the accuracy of our method against baseline Zero Noise Extrapolation (ZNE) and tensor network simulations for systems of 100 qubits. Remarkably, GUESS achieves a relative error around 10\% for circuits containing up to 8000 CZ gates, while showcasing lower variance than ZNE on average across 20 observables and requiring only twice the number of shots per observable compared to baseline ZNE. Furthermore, we demonstrate that GUESS enables statistical post-selection based on the outcomes of the symmetry observable, which provides critical information about the quality of the target qubits by means of its mean and variance. These results indicate that GUESS is a powerful QEM technique capable of mitigating utility-scale circuit outcomes, delivering high accuracy and reduced variance for large-scale circuits with minimal quantum overhead.