This paper presents two error mitigation techniques for short-depth quantum circuits, which are essential for near-term quantum devices. The first method, zero-noise extrapolation, uses Richardson's deferred approach to the limit to cancel higher-order noise effects. The second method, probabilistic error cancellation, resamples noisy circuits according to a quasi-probability distribution to cancel errors. Both methods aim to improve the accuracy of quantum simulations and other quantum algorithms in the presence of noise and decoherence. The first method involves estimating the expectation value of an observable in the absence of noise by extrapolating from noisy simulations. This is done by running the quantum circuit at different noise levels and using a linear combination of the results to cancel the noise effects. The second method involves representing an ideal circuit as a quasi-probabilistic mixture of noisy circuits, allowing for the estimation of the ideal circuit's expectation value by sampling from the noisy circuits. The paper discusses the importance of error mitigation for near-term quantum devices, which are limited by the rate of error introduction. It highlights the challenges of quantum error correction and the need for practical error mitigation techniques that do not require additional quantum resources. The two methods presented are simple and efficient, making them suitable for current quantum experiments. The paper also provides examples of how these methods can be applied to different noise models, including depolarizing, amplitude damping, and non-Markovian noise. The results show that these methods can significantly improve the precision of quantum simulations and other quantum algorithms, even in the presence of noise. The paper concludes that these error mitigation techniques are essential for the practical implementation of quantum computing in the near term.This paper presents two error mitigation techniques for short-depth quantum circuits, which are essential for near-term quantum devices. The first method, zero-noise extrapolation, uses Richardson's deferred approach to the limit to cancel higher-order noise effects. The second method, probabilistic error cancellation, resamples noisy circuits according to a quasi-probability distribution to cancel errors. Both methods aim to improve the accuracy of quantum simulations and other quantum algorithms in the presence of noise and decoherence. The first method involves estimating the expectation value of an observable in the absence of noise by extrapolating from noisy simulations. This is done by running the quantum circuit at different noise levels and using a linear combination of the results to cancel the noise effects. The second method involves representing an ideal circuit as a quasi-probabilistic mixture of noisy circuits, allowing for the estimation of the ideal circuit's expectation value by sampling from the noisy circuits. The paper discusses the importance of error mitigation for near-term quantum devices, which are limited by the rate of error introduction. It highlights the challenges of quantum error correction and the need for practical error mitigation techniques that do not require additional quantum resources. The two methods presented are simple and efficient, making them suitable for current quantum experiments. The paper also provides examples of how these methods can be applied to different noise models, including depolarizing, amplitude damping, and non-Markovian noise. The results show that these methods can significantly improve the precision of quantum simulations and other quantum algorithms, even in the presence of noise. The paper concludes that these error mitigation techniques are essential for the practical implementation of quantum computing in the near term.