The paper presents a polynomial-time classical algorithm for simulating noisy quantum circuits. The algorithm can compute the expectation value of any observable for any circuit, with a small average error over input states drawn from an ensemble. The approach leverages the intuition that noise exponentially damps non-local correlations relative to local correlations, allowing the simulation to focus on the dynamics of local quantum information. The algorithm also enables quasi-polynomial-time sampling from the output distribution of a circuit, provided it anti-concentrates. The results have significant implications for near-term quantum experiments, including a fundamental limit on the efficacy of noise mitigation strategies: any quantum circuit for which error mitigation is efficient must be classically simulable. The paper discusses practical applications and provides rigorous proofs for the algorithms, demonstrating their effectiveness in various scenarios, such as uniform noise and gate-based noise models.The paper presents a polynomial-time classical algorithm for simulating noisy quantum circuits. The algorithm can compute the expectation value of any observable for any circuit, with a small average error over input states drawn from an ensemble. The approach leverages the intuition that noise exponentially damps non-local correlations relative to local correlations, allowing the simulation to focus on the dynamics of local quantum information. The algorithm also enables quasi-polynomial-time sampling from the output distribution of a circuit, provided it anti-concentrates. The results have significant implications for near-term quantum experiments, including a fundamental limit on the efficacy of noise mitigation strategies: any quantum circuit for which error mitigation is efficient must be classically simulable. The paper discusses practical applications and provides rigorous proofs for the algorithms, demonstrating their effectiveness in various scenarios, such as uniform noise and gate-based noise models.