Trade-offs between classical and quantum space using spooky pebbling

TL;DR

The paper advances the study of memory-time trade-offs by applying the spooky pebble game to general DAGs, proving a quantum-space upper bound and PSPACE-completeness for the problem. It introduces a SAT-based solver augmented with optimization heuristics to find memory-efficient strategies that trade quantum space for classical space and time. Empirical results on benchmark circuits demonstrate that allowing classical memory (ghosts) can significantly reduce quantum space, yielding practical Pareto fronts for quantum/classical/time trade-offs. The work also provides an open-source solver and lays groundwork for broader applications of measurement-based uncomputation in quantum memory management.

Abstract

Pebble games are used to study space/time trade-offs. Recently, spooky pebble games were introduced to study classical space / quantum space / time trade-offs for simulation of classical circuits on quantum computers. In this paper, the spooky pebble game framework is applied for the first time to general circuits. Using this framework we prove an upper bound for quantum space in the spooky pebble game. We also prove that solving the spooky pebble game is PSPACE-complete. Moreover, we present a solver for the spooky pebble game based on satisfiability solvers combined with heuristic optimizers. This spooky pebble game solver was empirically evaluated by calculating optimal classical space / quantum space / time trade-offs. Within limited runtime, the solver could find a strategy reducing quantum space when classical space is taken into account, showing that the spooky pebble model is useful to reduce quantum space.

Figures (10)

• Figure 1: This circuit describes measurement-based uncomputation. The three wires depict the input and output of the function $f$ and the ghost wire to classically store the measurement outcome. The labels (1), (2), etc correspond to the corresponding spooky pebbling configurations as shown in Figure \ref{['fig:spooky_pebble_configs-mmbased-uncomp']}.
• Figure 2: Three examples of steps for different types of pebble games. A white node represents an empty vertex, a black vertex is a pebbled vertex and a grey vertex is a ghosted vertex.
• Figure 3: Spooky pebbling configurations for the labels in the circuit of Figure \ref{['fig:measurement_based_uncomputation_explanation']}. In Definition \ref{['def:spooky_pebbling']}, the spooky pebble game is introduced. The function inputs in vertices (A) and (B) are used to compute a function $f$ and store the output in vertex (C). In other words: $f\left((A),(B)\right)=(C)$. A white vertex is unpebbled, a black one is pebbled and a grey vertex is ghosted. In this example not the entire input is changed: vertex (B) is pebbled (i.e. hold in memory) all over the computation and only vertex (A) is unpebbled (i.e. removed from memory).
• Figure 4: Classical circuit on input bits $x_1,x_2,x_3$. The gates A,B,C,D,E are classical gates, e.g. AND, OR.
• Figure 5: Quantum circuit to compute the circuit from Figure \ref{['fig:example-irreversible-circuit']} on a superposition on a quantum computer without measurement-based uncomputation. The reversible pebbling strategy that leads to this circuit consists of the following consecutive moves: pebble(A), pebble(B), pebble(C), pebble(D), unpebble(A), pebble(E), unpebble(D), unpebble(B), pebble(A), unpebble(C), unpebble(A). This strategy has a pebbling cost of 4.

Theorems & Definitions (16)

• Definition 2.1: (Ir)reversible pebbling
• Definition 2.2: Spooky pebbling
• Theorem 3.1
• Theorem 3.2
• proof
• Lemma 3.3
• proof
• Lemma 3.4
• proof
• Theorem 3.5: gilbert1979pebbling