Pebble games are popular models for analyzing time-space trade-offs. In particular, the reversible pebble game is often applied in quantum algorithms like Grover's search to efficiently simulate classical computation on inputs in superposition. However, the reversible pebble game cannot harness the additional computational power granted by irreversible intermediate measurements. The spooky pebble game, which models interleaved measurements and adaptive phase corrections, reduces the number of qubits beyond what reversible approaches can achieve. While the spooky pebble game does not reduce the total space (bits plus qubits) complexity of the simulation, it reduces the amount of space that must be stored in qubits. We prove asymptotically tight trade-offs for the spooky pebble game on a line with any pebble bound, giving a tight time-qubit tradeoff for simulating arbitrary classical sequential computation with the spooky pebble game. For example, for all $\epsilon \in (0,1]$, any classical computation requiring time $T$ and space $S$ can be implemented on a quantum computer using only $O(T/ \epsilon)$ gates and $O(T^{\epsilon}S^{1-\epsilon})$ qubits. This improves on the best known bound for the reversible pebble game with that number of qubits, which uses $O(2^{1/\epsilon} T)$ gates. We also consider the spooky pebble game on more general directed acyclic graphs (DAGs), capturing fine-grained data dependency in computation and show that this game can outperform the reversible pebble game on trees. Additionally any DAG can be pebbled with at most one more pebble than is needed in the irreversible pebble game, implying that finding the minimum number of pebbles necessary to play the spooky pebble game on a DAG with maximum in-degree two is PSPACE-hard to approximate.

翻译：卵石游戏是分析时间-空间权衡的经典模型。特别地，可逆卵石游戏常应用于量子算法（如Grover搜索），以有效模拟叠加态输入上的经典计算。然而，可逆卵石游戏无法利用不可逆中间测量带来的额外计算能力。幽灵卵石游戏通过建模交错测量与自适应相位修正，能在量子比特数上超越可逆方法。虽然幽灵卵石游戏不降低模拟的总空间复杂度（比特加量子比特），但它减少了需存储在量子比特中的空间量。我们证明了在线性结构上任意卵石界约束下幽灵卵石游戏的渐近紧致权衡，给出了利用幽灵卵石游戏模拟任意经典串行计算的时间-量子比特紧致折衷。例如，对所有$\epsilon \in (0,1]$，任何需要时间$T$和空间$S$的经典计算，可在量子计算机上仅用$O(T/ \epsilon)$门和$O(T^{\epsilon}S^{1-\epsilon})$量子比特实现。这一结果改进了已知最优的可逆卵石游戏在相同量子比特数下的界——后者需要$O(2^{1/\epsilon} T)$门。我们还考虑了更一般的有向无环图（DAG）上的幽灵卵石游戏，以捕获计算中细粒度的数据依赖关系，并证明该游戏在树结构上的表现优于可逆卵石游戏。此外，任何DAG的卵石数至多比不可逆卵石游戏所需多一个，这意味着在最大入度为二的DAG上寻找幽灵卵石游戏所需的最小卵石数是PSPACE-难近似问题。