Grover’s algorithm is one of the primary algorithms offered as evidence that quantum computers can provide an advantage over classical computers. It involves an “oracle” (external quantum subroutine), which must be specified for a given application and whose internal structure is not part of the formal scaling of the quadratic quantum speedup guaranteed by the algorithm. Grover’’s algorithm also requires exponentially many calls to the quantum oracle (approximately 2n calls where n is the number of qubits) to succeed, raising the question of its implementation on both noisy and error-corrected quantum computers. In this work, we construct a quantum-inspired algorithm executable on a classical computer that performs Grover’s task in a linear number of calls to (simulations of) the oracle—an exponentially smaller number than Grover’s algorithm—and demonstrate this algorithm explicitly for Boolean satisfiability problems. The complexity of our algorithm depends on the cost to simulate the oracle once, which may or may not be exponential, depending on its internal structure. Indeed, Grover’s algorithm does not have an quantum speedup as soon as one is given access to the “source code” of the oracle, which may reveal an internal structure of the problem. Our findings illustrate this point explicitly, as our algorithm exploits the structure of the quantum circuit used to program the quantum computer to speed up the search. There are still problems where Grover’s algorithm would provide an asymptotic speedup if it could be run accurately for large enough sizes. Our quantum-inspired algorithm provides lower bounds, in terms of the quantum-circuit complexity, for the quantum hardware to beat classical approaches for these problems. These estimates, combined with the unfavorable scaling of the success probability of Grover’s algorithm, which in the presence of noise decays as the exponential of the exponential of the number of qubits, makes a practical speedup unrealistic even under extremely optimistic assumptions of the evolution of both hardware quality and availability. Published by the American Physical Society 2024

中文翻译：

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打开 Grover 算法中的黑匣子


Grover 算法是量子计算机可以提供优于经典计算机的优势的证据的主要算法之一。它涉及一个 “oracle” （外部量子子例程），必须为给定的应用程序指定，并且其内部结构不是算法保证的二次量子加速的正式缩放的一部分。Grover 的算法还需要对量子预言机进行指数级多次调用（大约 2n 次调用，其中 n 是量子比特数）才能成功，这就提出了它在嘈杂和纠错量子计算机上实现的问题。在这项工作中，我们在经典计算机上构建了一个量子衍生算法可执行文件，该算法在对 oracle 的线性调用（模拟）中执行 Grover 任务——比 Grover 算法的数量小得多——并针对布尔满足性问题明确演示了该算法。我们算法的复杂程度取决于模拟一次预言机的成本，这可能是指数级的，也可能不是指数级的，这取决于它的内部结构。事实上，一旦允许访问预言机的“源代码”，Grover 的算法就不会产生量子加速，这可能会揭示问题的内部结构。我们的研究结果明确地说明了这一点，因为我们的算法利用了用于对量子计算机进行编程的量子电路的结构来加快搜索速度。如果 Grover 的算法可以针对足够大的大小准确运行，则仍然存在一些问题，即 Grover 的算法将提供渐近加速。我们的量子启发算法在量子电路复杂性方面为量子硬件提供了下限，以击败这些问题的经典方法。 这些估计值，再加上 Grover 算法成功概率的不利缩放（在存在噪声的情况下，该算法会衰减为量子比特数指数的指数），这使得实际的加速变得不切实际，即使在对硬件质量和可用性演变的极其乐观的假设下也是如此。美国物理学会 2024 年出版