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

2021-10-18

AI Summary

This work addresses the time–space trade-off in classical simulation of quantum computation, introducing the “spooky pebble game” model to characterize qubit reuse under intermediate measurement and adaptive phase correction. Methodologically, it integrates quantum circuit modeling, DAG analysis, and measurement-driven resource recycling. Key contributions include: (i) the first asymptotically tight time–space trade-off bound for spooky pebbling on linear graphs; (ii) a proof that approximating the pebbling number of an arbitrary DAG is PSPACE-hard; and (iii) an optimal pebbling strategy for binary trees. Results show that classical computation with time T and space S can be simulated using only O(T/ε) quantum gates and O(T^ε S^{1−ε}) qubits—exponentially improving upon reversible pebbling’s gate overhead—while the binary-tree scheme achieves the theoretical minimum qubit count.

Abstract

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. We show that for an arbitrary DAG even approximating the number of required pebbles in the spooky pebble game is PSPACE-hard. Despite this, we are able to construct a time-efficient strategy for pebbling binary trees that uses the minimum number of pebbles.

Problem

• Optimizing qubit usage in quantum computation
• Modeling interleaved measurements in pebble games
• Reducing space complexity in quantum simulations

Innovation

• Spooky pebble game reduces qubits
• Tight time-qubit tradeoff proven
• Efficient strategy for binary trees