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.
翻译:最近 NISQ 设备的扩散使得有必要理解它们的计算力。 在这项工作中, 我们定义并研究复杂等级 $\ textsf{ NIS{ $, 目的是通过使用 NISQ 设备的古典计算机来包罗能够有效解决的问题。 模型化现有设备, 我们假设设备可以(1) 隐性地初始化所有qubits, (2) 应用许多吵闹的量子门, (3) 对所有qubits进行噪音测量。 我们首先提供证据证明 $\ textsf{ BP ⁇ subsetneq\ textsf{ NISsubsetneq\ textsf{ BQP} $, 通过显示三个类别之间的超极极极或极分分离, 根据西蒙问题的修改, 我们然后考虑 $\ textf{ 量的量子门的功率。 对于未结构化的搜索, 我们证明, $ textf{ { { { $ subsrecitral_ ral 无法在 $\ ral_ rial_ ride rial ride 中取得类似的加速速度。