Pebble games are commonly used to study space-time trade-offs in computation. We present a pebble game that explores this trade-off in quantum simulation of arbitrary classical circuits on non-classical inputs. Many quantum algorithms, such as Shor's algorithm and Grover's search, include subroutines that require simulation of classical functions on inputs in superposition. The current state of the art uses generic reversible simulation through Bennett's pebble game and universal reversible gate sets such as the Toffoli. Using measurement-based uncomputation, we replace many of the ancilla qubits used by existing constructions with classical control bits, which are cheaper. Our pebble game forms a natural framework for reasoning about measurement-based uncomputation, and we prove tight bounds on the time complexity of all algorithms that use our pebble game. For any $\epsilon \in (0,1)$, we present an algorithm that can simulate irreversible classical computation that uses $\mathcal{T}$ time and $\mathcal{S}$ space in $O(\frac{1}{\epsilon}\frac{\mathcal{T}^{1+\epsilon}}{\mathcal{S}^\epsilon})$ time with $O(\frac{1}{\epsilon}\mathcal{S})$ qubits. With access to more qubits we present algorithms that run in $O(\frac{1}{\epsilon}\mathcal{T})$ time with $O(\mathcal{S}^{1-\epsilon}\mathcal{T}^\epsilon)$ qubits. Both of these results show an improvement over Bennett's construction, which requires $O(\frac{\mathcal{T}^{1+\epsilon}}{\mathcal{S}^\epsilon})$ time when using $O(\epsilon2^{1/\epsilon} \mathcal{S} (1+\log \frac{\mathcal{T}}{\mathcal{S}}))$ qubits. Additionally the results in our paper combine with Barrington's theorem to provide a general method to efficiently compute any log-depth circuit on quantum inputs using a constant number of ancilla qubits. We also explore a connection between the optimal structure of our pebbling algorithms and backtracking from dynamic programming.

翻译：在计算中通常使用 Pebble 游戏来研究空间- 时间取舍 。 我们展示了一个可以用来在非古典输入的任意古典电路的量模拟中探索这种交易的卵子游戏 。 许多量子算法, 如Shor的算法和 Grover 的搜索, 包括需要模拟用于超位输入的经典函数的子例程 。 目前艺术状态通过 Bennett 的振荡游戏和通用可逆门套件, 如 Toffoli 。 使用基于测量的非comput, 我们用经典控制点取代了现有构造中的许多 amcilla qubits, 更便宜。 我们的比目游戏形成了一个自然框架, 用于基于测量的算术的推理, 并且我们在使用这些振荡游戏的所有算法的复杂时间里, 任何以 $@ mocial (0, 1美元), 我们用一种可以模拟不可逆转的计算结果, 使用 $mathcal; 时间, 和 美元=cal_ masial1 max 。