Iterative Diving Search (IDS) Algorithm

• Iterative Diving Search (IDS) is a depth-first, memory-bounded search method designed to optimize quantum circuit placements in zoned neutral atom architectures.
• It replaces traditional A* search by using a global priority queue and greedy diving strategy to efficiently overcome memory limitations and escape local minima.
• When combined with relaxed routing, IDS significantly reduces atom rearrangement time, achieving up to 45.9% faster placements for circuits with thousands of qubits.

Iterative Diving Search (IDS) is a goal-directed search algorithm introduced to address the scalability and memory limitations of quantum circuit compilation for zoned neutral atom architectures. In these systems, the core task is the placement and routing of atoms to enable the execution of layers of quantum gates—most critically, the efficient scheduling of two-qubit (CZ) operations subject to strict physical constraints. IDS fundamentally departs from conventional A*-based tree search in favor of a depth-prioritized, bounded-memory exploration strategy. When paired with relaxed routing optimizations, IDS achieves circuit placements of superior quality while remaining computationally tractable for circuits involving thousands of qubits and hundreds of parallel gates (Stade et al., 15 Dec 2025).

1. Compilation Challenges in Zoned Neutral Atom Architectures

Zoned neutral atom devices consist of a large, static storage grid; one or more smaller entanglement zones for executing Rydberg-mediated two-qubit gates; and measurement zones. The compiler receives as input both a prescheduled quantum circuit and a machine layout. The mapping challenge is to orchestrate atom movements so that, for each layer of parallel CZ gates, the relevant atoms are efficiently transferred to and from entanglement zones, subject to constraints imposed by acousto-optic deflector (AOD)-based manipulation:

• Non-crossing of active rows/columns,
• Preservation of row/column connectivity (no splitting or merging),
• Avoidance of ghost-spot traps.

The central objective is to minimize the total sequential rearrangement steps—and hence the wall-clock rearrangement time—without violating any hardware-encoded movement restrictions. This directly mitigates decoherence incurred via idle and in-motion qubits (Stade et al., 15 Dec 2025).

2. Conventional Tree Search Approaches and Limitations

Historically, routing-aware placement employs an explicit search tree. Each node reflects a partial placement configuration; child nodes correspond to the placement of an additional atom. An A* algorithm, leveraging carefully designed cost and heuristic functions encoding the three movement constraints, produces high-quality placements by favoring parallelizable pick/drop operations.

However, A* suffers from exponential memory growth: the frontier maintained is of size $O(b^d)$, where $d$ approaches the parallel gate count in the layer, and $b$ is the typical branching factor. Empirical results show that as little as 40–50 parallel CZ gates in a single layer cause memory exhaustion—exceeding tens of gigabytes—thus rendering A* ineffective at modern system scales (Stade et al., 15 Dec 2025).

3. Iterative Diving Search: Mechanism and Pseudocode

IDS substitutes A*'s frontier-driven, breadth-oriented expansion for a memory-constrained, goal-oriented strategy. The essential mechanism is as follows:

1. A global min-priority queue $Q$, capped at $N_{\max}$ nodes, holds “unexpanded” nodes ordered by $f(n) = g(n) + h(n)$.
2. For each of $T$ trials, the search “dives” greedily from the root by repeatedly expanding the child node $c^*$ with minimal $f$; other children are inserted into $Q$ (with eviction if over capacity).
3. Upon reaching a goal node (complete placement), its cost is recorded, $T$ is decremented, and the next trial starts from the current best in $Q$. Dead ends are handled by popping the best available node from $Q$.
4. The process halts when $T$ goals have been found or $Q$ is empty; the lowest-cost goal is returned.

By always advancing along the most promising partial assignment, IDS maintains depth-first focus but is able to escape local minima through the global queue. Its maximum memory consumption is $O(N_{\max} + d)$, several orders of magnitude lower than that of A*.

Pseudocode for IDS is as follows (with blue lines indicating IDS-specific modifications atop A*):

function IDS_Search(root, h, N_max, T): Initialize open-queue Q ← {} best_goal ← ∞; trials ← T repeat n ← root; g(root) ← 0 while true do if n.is_goal() then if g(n) < best_goal.g then best_goal ← n trials ← trials – 1 break // restart dive end children ← expand(n) if children.empty() then // hit dead end if Q.empty() then return best_goal n ← Q.pop_min_f() continue end // pick best child to dive into c* ← argmin_{c ∈ children}(g(c) + h(c)) for each c ∈ children \ {c*} do push Q, c // may evict worst if |Q| > N_max end n ← c* end until trials = 0 or Q.empty() return best_goal

4. Cost and Heuristic Functions

The IDS framework retains the cost and heuristic strategies of routing-aware placement. Let

• $S$ denote atoms that must enter/leave the entanglement zone in this layer,
• $d_a(\pi)$ denote the Manhattan distance for atom $a \in S$ under partial placement $\pi$.

The accelerated-norm heuristic is employed:

$$h(\pi)\;=\;\delta\!\sum_{a\in S}d_a(\pi)\;+\;\beta\,|S|\;+\;\alpha\,\max_{a\in S}d_a(\pi),$$

with tunable parameters $\alpha, \beta, \delta \geq 0$; recommended values are $\delta=0.01$, $\beta=0$, $\alpha=0.4$. The cost-to-come $g(\pi)$ accumulates an estimate of rearrangement steps incurred so far, scaled by per-step time overhead. The $\ell_\infty$ term steers the search to reduce maximal outstanding displacement early, breaking potential row/column conflicts.

5. Complexity, Memory, and Empirical Performance

The dominant limitation of A* search is its exponential frontier size. In contrast, IDS’s bounded queue and linear stack yield overall memory consumption $O(N_{\max}+d)$, suitable for deployments under stringent RAM quotas. In practice, IDS achieves placement for circuits with up to 5,000 qubits and 306 parallel CZ gates within a few minutes, never exceeding 20 GB of RAM.

Benchmarking against prior A* methods, IDS plus relaxed routing consistently outperforms both in both quality and efficiency:

• A* routing-aware placement: mean rearrangement time 2022.5 ms; fails above 200 qubits or 50 parallel CZs.
• IDS: mean time 1768.5 ms ($-27.1$\%).
• IDS + relaxed routing: mean time 1745.2 ms ($-28.1$\%); up to $-45.9$\% on highly parallel “graph-state” circuits.

IDS typically explores far fewer nodes, yet reliably finds placements superior to A*. The method’s effectiveness is attributed to rapidly “locking in” low-cost solutions via deep promising dives, with the global queue used for escaping suboptimal local branches (Stade et al., 15 Dec 2025).

6. Relaxed Routing: Routing Optimization after Placement

After atom placement, minimizing the number of physical AOD rearrangement steps is crucial. Strict routing strategies enforce monotonic row-by-row or column-by-column pick/drop ordering, which limits parallelism. Relaxed routing, by contrast, allows for temporary reordering of already-active rows or columns during multi-stage pick/drop phases provided the non-crossing constraint holds.

A simple decision metric compares the time cost of these “offset shifts” ($T_{\rm extra\_offset}$) against the savings achieved by collapsing steps ($T_{\rm saved\_step}$), and selects relaxed routing only when it yields a net time reduction. This allows what would be 4–5 serial steps under strict routing to become 1–2 consolidated steps, amortizing minor offset penalties over overall fewer rearrangement steps.

7. Trade-Offs, Limitations, and Future Directions

IDS introduces a principled depth-over-breadth bias, trading the completeness of exhaustive search for scalability and memory efficiency. While this implies that, in rare cases, the absolute global optimum may be missed, empirical evidence indicates superior results and robust scalability when compared to classical A*. Adjusting the queue size ($N_{\max}$) and trial count ($T$) tunes the search exploration/exploitation balance; these parameters can be adapted dynamically for workload-dependent strategies.

Relaxed routing may be suboptimal for very short moves due to accumulated offset penalties—a distance-based threshold is used to mitigate this. Potential extensions include global multi-layer placement, parallelization of IDS trials, co-optimization with scheduling, refinement of AOD-movement models (including time-dependent and 2D constraints), and integration of learning-based heuristics to anticipate routing difficulty (Stade et al., 15 Dec 2025).

IDS and relaxed routing have been implemented in open-source form within the Munich Quantum Toolkit (MQT), enabling high-quality compilation for large-scale neutral atom quantum devices using accessible hardware resources.