Parallel Spooky Pebble Games

• Parallel Spooky Pebble Games are computational abstractions that combine mid-circuit measurements (ghost moves) with parallel execution to optimize quantum circuits.
• They employ a blast–unblast paradigm where checkpoint pebbling and ghosting achieve provably optimal depth and logarithmic space scaling for sequential computations.
• These games impact quantum cryptanalysis by reducing circuit depth in modular exponentiation, thereby efficiently managing qubit resources in factoring algorithms.

Parallel Spooky Pebble Games are a computational abstraction that integrates mid-circuit measurements (“spookiness”) and concurrent execution (“parallelism”) into pebble game frameworks. Designed for analyzing space–time tradeoffs in quantum circuits, particularly those simulating inherently sequential classical computations (e.g., modular exponentiation as in integer factorization), these games model the reduction in quantum space via measurement-based uncomputation and leverage parallel scheduling to minimize circuit depth. Their operational regime includes classical reversible pebble games [Blocki, Holman, Lee (TCC 2022)], Gidney-style measurement-based ^^^^1^^^^ing, and, as demonstrated in (Kahanamoku-Meyer et al., 9 Oct 2025), jointly exploiting parallel and spooky moves delivers provably optimal depth and substantial space reductions. The following sections rigorously formalize definitions, strategies, algorithmic analysis, and implications for cryptanalysis and quantum resource management.

1. Formal Model: Parallel Spooky Pebble Game Dynamics

A Parallel Spooky Pebble Game plays out on a directed acyclic graph (most often a line graph for sequential computation), with each vertex $v_i$ bearing a state in $\{ {\rm pebble}, {\rm ghost}, {\rm empty} \}$. The transitions (moves) are:

• Pebble(i): Place a pebble on position $i$ if immediate predecessor constraints are met.
• Unpebble(i): Remove a pebble from %%%%3%%%%, provided the controlling conditions are maintained.
• Ghost(i): Apply a Hadamard basis measurement to $i$—the "spooky" move—which removes the pebble and leaves a phase error ("ghost") to be corrected later.
• Parallelism constraint: At each time step, multiple moves are allowed iff their active sets are non-overlapping (no adjacent operations), ensuring parallel execution does not violate pebble/ghost dependencies.

A configuration at time $t$ consists of a mapping $$\mathcal{C}_t: V \to \{ {\rm pebble}, {\rm ghost}, {\rm empty} \}$$, and a sequence of moves $M_t = \{ m_{t,1}, ..., m_{t,k} \}$ is valid if each $m_{t,j}$ can be performed independently. The execution begins in an initial state encoding the input and aims to arrive at a target output configuration, typically with a single pebble at the rightmost vertex.

2. Blast–Unblast Construction: Achieving Optimal Depth and Space

The recursive blast–unblast paradigm underpins the main constructive result. For a line of length $\ell$, the algorithm proceeds as follows:

• Blast: March a pebble rightwards, leaving marker pebbles at exponentially spaced checkpoints (e.g., $\ell/2$, $3\ell/4$, $7\ell/8$, etc.).
• Unblast: Backtrack by removing intermediate pebbles, correcting ghosts sequentially.
• The scheduling ensures exactly one position advances per parallel time step for the rightmost pebble, enforcing a minimal depth.

Let sequence $A_k$ satisfy $A_1=1$, $A_2=2$, $A_3=2$, and for $k\geq4$, $A_k = A_{k-2} + A_{k-3}$, approximating exponential scaling. Then, for $\ell=A_k$, both blast and unblast require exactly $A_k-1$ steps, so total depth is $2\ell$, which is shown to be optimal.

Space required is tightly bounded:

$$\text{Space} \leq \left\lceil \frac{\log\ell - \log 0.94}{\log\alpha} - 3 \right\rceil \leq 2.47 \log \ell - 2.77,$$

where $\alpha \approx 1.32$ is the unique root of $x^3 - x - 1 = 0$. The log-factor arises from recursion depth in checkpointing.

3. A* Search and State-Space Reduction

To explore space-depth tradeoffs beyond the theoretical construct, an optimized A* search implemented in Julia traverses the configuration space:

• Each state encodes pebble positions; ghosts are derived deterministically from pebble placements, dramatically reducing the search space.
• Only transitions following the natural blast and immediate ghosting pattern are considered.
• Admissible heuristics, notably the position of the rightmost pebble, furnish lower bounds on remaining steps, allowing pruning of non-optimal branches.
• The search verifies that, for large $\ell$, the blast–unblast schedule remains nearly optimal, while for smaller $s$, tailored schedules achieve sub-logarithmic depth-space product in return for marginal space expansion.

This approach supports weighted cost metrics, e.g., penalizing moves that trigger expensive ancilla usage during modular multiplication, enabling fine-tuned space/time tradeoff evaluations.

4. Application: Modular Exponentiation in Regev's Factoring Algorithm

Regev's factoring algorithm, like Shor's, relies on modular exponentiation implemented as a sequence of squaring and multiplication steps—an archetypal instance of sequential computation in quantum circuits. Mapping this to a line graph of length $D$ (exponent length), the blast–unblast methodology enables:

• Multiplication depth reduction to $4 \log D$ for exponentiation (scheduling squaring and multiplication with pebble/ghost orchestration).
• For 4096-bit $N$, minimum depth empirically achieved is 193 multiplications; this is substantially lower than the 680 for prior Regev implementation and the 444 reported for Shor (Kahanamoku-Meyer et al., 9 Oct 2025).
• Space overhead is $2.47\log D$ additional qubits, governed by checkpoint recursion.

Weighted pebbling further adapts to product tree steps and ancilla usage, so the schedule coordinates both ghosting and parallel step assignment, minimizing qubit utilization without inflating depth.

5. Algorithmic and Quantum Complexity Implications

Parallel spooky pebble games establish new lower bounds in quantum arithmetic:

• Optimal depth-space tradeoff: Parallelization and ghosting together enforce optimal depth $2\ell$ with logarithmic space scaling for line graphs.
• Quantum space reduction: Measurement-based ghosting substitutes classical bits for costly quantum registers during reversible simulation; tradeoffs are systematically explored for arbitrary DAGs via SAT-based solvers and heuristic post-optimizers (Quist et al., 19 Jan 2024).
• Hardness: PSPACE-completeness for finding optimal strategies is established for general DAGs (Kornerup et al., 2021, Quist et al., 19 Jan 2024); parallel pebbling schedules in multiprocessor models are NP-hard and inapproximable within any finite factor (Böhnlein et al., 5 Sep 2024).
• Cryptanalytic impact: Resource determinations for cryptanalytic attacks—e.g., memory-hard functions and integer factorization—are now governed by blast–unblast parallel spooky pebbling bounds.

6. Broader Applications and Prospects

Parallel spooky pebbling informs several computational paradigms:

• Quantum circuit engineering: By interleaving parallel pebbling with ghost moves, circuit designers can optimize resource utilization in sequential computations, time-lock puzzles, and proofs-of-work.
• Cryptanalysis and post-quantum security: Lower resource bounds directly influence the feasibility of quantum attacks against cryptosystems (Kahanamoku-Meyer et al., 9 Oct 2025).
• Algorithmic design: The principle of checkpoint-based recursion and concurrent scheduling is generalizable to classical and quantum algorithms where space–time tradeoffs are paramount.
• Experimental outlook: As mid-circuit measurements improve in reliability, these methodologies are poised to drive practical efficiency in quantum hardware.

7. Summary Table: Core Properties

Property	Parallel Spooky Pebbling	Classical/Reversible Pebbling
Depth on line length $\ell$	$2\ell$ (optimal)	$\Omega(\ell \log \ell)$
Space usage	$\leq 2.47 \log \ell$ pebbles	$O(\log \ell)$–$O(\ell)$
Quantum space optimization	Measurement enables classical	No ghosting; all quantum
Computational hardness	PSPACE-complete, NP-hard	PSPACE-complete
Application points	Factoring, cryptanalysis	Simulation, complexity
Parallelism supported	Yes	Limited (sequential or block)

Parallel spooky pebble games unify parallel scheduling and space reduction via ghosting in the quantum simulation of sequential functions. Their analysis yields optimal depth and exponential space improvements, both theoretically via recursive blast–unblast construction and practically via A* search. These techniques define new frontiers in quantum circuit optimization and the cryptanalytic evaluation of post-quantum schemes.