The complexity of NISQ.

The recent proliferation of NISQ devices has made it imperative to understand their power. In this work, we define and study the complexity class NISQ, which encapsulates problems that can be efficiently solved by a classical computer with access to noisy quantum circuits. We establish super-polynomial separations in the complexity among classical computation, NISQ, and fault-tolerant quantum computation to solve some problems based on modifications of Simon's problems. We then consider the power of NISQ for three well-studied problems. For unstructured search, we prove that NISQ cannot achieve a Grover-like quadratic speedup over classical computers. For the Bernstein-Vazirani problem, we show that NISQ only needs a number of queries logarithmic in what is required for classical computers. Finally, for a quantum state learning problem, we prove that NISQ is exponentially weaker than classical computers with access to noiseless constant-depth quantum circuits. Our current understanding of the computational abilities of near-intermediate scale quantum (NISQ) computing devices is limited, in part due to the absence of a precise definition for this regime. Here, the authors formally define the NISQ realm and provide rigorous evidence that its capabilities are situated between the complexity classes BPP and BQP. [ABSTRACT FROM AUTHOR]