Tight Bounds on the Spooky Pebble Game: Recycling Qubits with Measurements

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.