 This work defines and studies the complexity class NISQ, which captures problems that can be efficiently solved by a classical computer with access to noisy quantum circuits, and establishes super-pollinomial separations among classical computation, NISQ, and fault-tolerant quantum computation for some problems based on modifications of Simon's problems. The work also considers the power of NISQ for three well-studded problems, showing that NISQ cannot achieve a grover-like quadratic speed-up over classical computers for unstructured search, only needs a number of queries logarithmic in what is required for classical computers for the Bernstein-Vazirani problem, and is exponentially weaker than classical computers with access to noiseless constant-depth quantum circuits for a quantum state, learning problem. This article was authored by Seitan Chen, Jordan Kotler, Hinyuan Huang, and others.