Self-protected quantum simulation and quantum-phase estimation in the presence of classical noise

Decoherence is a major challenge in quantum computing. To execute quantum algorithms, it is crucial to eliminate decoherence and noise, for instance, via dynamic decoupling and quantum error correction protocols based on dynamic zero-noise strategy. We introduced self-protected quantum algorithms as potential alternatives over 15 years ago. Quantum algorithms based on the living-with-noise strategy are now used in the Noisy Intermediate-Scale Quantum regime. Here we introduce self-protected quantum simulations in the presence of weak classical noise. Notably, we prove the equivalence between weak classical noise and noiseless quantum simulations. This equivalence implies that a self-protected quantum simulation does not require any extra overhead in its experimental implementation. Furthermore, we find that the conventional quantum-phase estimation can be upgraded to its corresponding noisy version.