The Complexity of NISQ

• URL: http://arxiv.org/abs/2210.07234v1
• Date: Thu, 13 Oct 2022 17:58:32 GMT
• Title: The Complexity of NISQ
• Authors: Sitan Chen, Jordan Cotler, Hsin-Yuan Huang, Jerry Li
• Abstract summary: NISQ devices have made it imperative to understand their computational power. In this work, we define and study the complexity class $textsfNISQ $. We consider the power of $textsfNISQ$ for three well-studied problems.
• Score: 15.842125145638693
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: The recent proliferation of NISQ devices has made it imperative to understand their computational power. In this work, we define and study the complexity class $\textsf{NISQ} $, which is intended to encapsulate problems that can be efficiently solved by a classical computer with access to a NISQ device. To model existing devices, we assume the device can (1) noisily initialize all qubits, (2) apply many noisy quantum gates, and (3) perform a noisy measurement on all qubits. We first give evidence that $\textsf{BPP}\subsetneq \textsf{NISQ}\subsetneq \textsf{BQP}$, by demonstrating super-polynomial oracle separations among the three classes, based on modifications of Simon's problem. We then consider the power of $\textsf{NISQ}$ for three well-studied problems. For unstructured search, we prove that $\textsf{NISQ}$ cannot achieve a Grover-like quadratic speedup over $\textsf{BPP}$. For the Bernstein-Vazirani problem, we show that $\textsf{NISQ}$ only needs a number of queries logarithmic in what is required for $\textsf{BPP}$. Finally, for a quantum state learning problem, we prove that $\textsf{NISQ}$ is exponentially weaker than classical computation with access to noiseless constant-depth quantum circuits.

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