Abstract
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 1/2+O(n/sqrt{2}^n) by analogously defined classical computations. We experimentally demonstrate computations of 3-, 4-, 5-, and 6-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 quantum scrap space is more valuable than analogous classical space---and calls for an in-depth exploration of space-time tradeoffs in quantum circuits.