The paper investigates the impact of uncorrected, possibly non-unital noise on quantum circuits, focusing on the task of estimating Pauli expectation values. Key findings include: 1. **Effective Logarithmic Depth**: Any noisy quantum circuit, under non-unital noise, effectively has a logarithmic depth. This means that only the last few layers of the circuit significantly influence the Pauli expectation values. 2. **Lack of Barren Plateaus**: Quantum circuits under non-unital noise exhibit a lack of barren plateaus for cost functions composed of local observables. This implies that the cost landscape is never flat, and the gradient never vanishes at any depth. 3. **Classical Simulation**: A classical algorithm is designed to estimate Pauli expectation values within inverse-polynomial additive error with high probability over the ensemble. The runtime is independent of circuit depth and polynomial in the number of qubits for one-dimensional architectures, and quasi-polynomial for higher-dimensional ones. 4. **Non-Exponential Concentration**: Local expectation values of random quantum circuits with non-unital noise are not exponentially concentrated around a fixed value, unlike in the depolarizing noise scenario. 5. **Trainability**: Only the last few layers of the circuit are trainable, meaning they have significantly large partial derivatives for local cost functions. This implies that the norm of the gradient does not exponentially vanish in the number of qubits at any depth. 6. **Improved Upper Bounds**: The paper also provides improved upper bounds on barren plateaus for unital noise, which are more precise and applicable at deeper circuit depths. These results highlight that, unless circuits are carefully engineered to take advantage of the noise, noisy quantum circuits may not offer computational advantages over shallow quantum circuits for tasks like estimating Pauli expectation values.The paper investigates the impact of uncorrected, possibly non-unital noise on quantum circuits, focusing on the task of estimating Pauli expectation values. Key findings include: 1. **Effective Logarithmic Depth**: Any noisy quantum circuit, under non-unital noise, effectively has a logarithmic depth. This means that only the last few layers of the circuit significantly influence the Pauli expectation values. 2. **Lack of Barren Plateaus**: Quantum circuits under non-unital noise exhibit a lack of barren plateaus for cost functions composed of local observables. This implies that the cost landscape is never flat, and the gradient never vanishes at any depth. 3. **Classical Simulation**: A classical algorithm is designed to estimate Pauli expectation values within inverse-polynomial additive error with high probability over the ensemble. The runtime is independent of circuit depth and polynomial in the number of qubits for one-dimensional architectures, and quasi-polynomial for higher-dimensional ones. 4. **Non-Exponential Concentration**: Local expectation values of random quantum circuits with non-unital noise are not exponentially concentrated around a fixed value, unlike in the depolarizing noise scenario. 5. **Trainability**: Only the last few layers of the circuit are trainable, meaning they have significantly large partial derivatives for local cost functions. This implies that the norm of the gradient does not exponentially vanish in the number of qubits at any depth. 6. **Improved Upper Bounds**: The paper also provides improved upper bounds on barren plateaus for unital noise, which are more precise and applicable at deeper circuit depths. These results highlight that, unless circuits are carefully engineered to take advantage of the noise, noisy quantum circuits may not offer computational advantages over shallow quantum circuits for tasks like estimating Pauli expectation values.