Quantum computations promise the ability to solve problems intractable in the classical setting. Restricting the types of computations considered often allows to establish a provable theoretical advantage by quantum computations, and later demonstrate it experimentally. In this paper, we consider space-restricted computations, where input is a read-only memory and only one (qu)bit can be computed on. We show that $n$-bit symmetric Boolean functions can be implemented exactly through the use of quantum signal processing as restricted space quantum computations using $O(n^2)$ gates, but some of them may only be evaluated with probability $\frac{1}{2} {+} \tilde{O}(\frac{1}{\sqrt{n}})$ by analogously defined classical computations. We experimentally demonstrate computations of $3$- and a $4$-bit symmetric Boolean functions by quantum circuits, leveraging custom two-qubit gates, with algorithmic success probability exceeding the best possible classically. This establishes and experimentally verifies a different kind of quantum advantage -- one where a quantum bit stores more useful information for the purpose of computation than a classical bit. This suggests that in computations, quantum scrap space is more valuable than analogous classical space and calls for an in-depth exploration of space-time tradeoffs in quantum circuits.

中文翻译：

---

有限空间计算的量子优势

量子计算有望解决经典环境中难以解决的问题。限制考虑的计算类型通常允许通过量子计算建立可证明的理论优势，然后通过实验证明它。在本文中，我们考虑空间受限的计算，其中输入是只读存储器，并且只能计算一个 (qu) 位。我们表明 $n$ 位对称布尔函数可以通过使用量子信号处理作为使用 $O(n^2)$ 门的受限空间量子计算来精确实现，但其中一些可能只能以概率 $\ frac{1}{2} {+} \tilde{O}(\frac{1}{\sqrt{n}})$ 通过类似定义的经典计算。我们通过量子电路实验证明了 $3$- 和 $4$-bit 对称布尔函数的计算，利用定制的两量子位门，算法成功概率超过了经典的最佳可能性。这建立并通过实验验证了一种不同的量子优势——在这种优势中，量子位比经典位存储更多用于计算的有用信息。这表明在计算中，量子废料空间比类似的经典空间更有价值，需要深入探索量子电路中的时空权衡。