Structured Scaling of AI Discovery Across Diverse Scientific Domains — part 9/12

Cleaned-text code points [125360, 140742). Bibliography omitted here; source.md includes it.

Our scaffold exposes the compiler as executable code inside a fixed framework that defines the interfaces, solver, and evaluator. The compilation pipeline has four components: a scheduler that partitions a logical ZNAACircuit into one-qubit and two-qubit stages, a reuse analyzer that marks qubits reused across adjacent two-qubit stages, a placer that outputs full placement snapshots, and a router that converts consecutive placements into legal Open , Move , and Close operations. A fixed solver invokes these components in order, and the fixed evaluator replays the emitted operations on a ZNAAMachine , and rejects invalid compilations. In our setting, the editable evolve block modifies only the placer ; the scheduler, reuse analyzer, and router remain fixed.

Initial program. The initial program exposes the editable evolve block while keeping the surrounding pipeline fixed. In Round 1, that block contained a trivial placeholder placer. Each later round initialized the editable block from the strongest placer discovered in the previous round.

Evaluation. We evaluate candidate placers on a 36-circuit suite built from QASMBench [ 75 ] circuits and MQT Bench [ 110 ] generated circuits, spanning textbook quantum algorithms, state preparation, variational circuits, quantum simulation, quantum machine learning, and arithmetic circuits, with sizes from 6 to 500 qubits. The fixed evaluator executes each compiled plan, records correctness, total execution time, and fidelity-related losses from gate execution, transport, transfers, and decoherence, and scores each circuit against a cached baseline from the fixed scaffold. Our main comparison baseline is the reuse-aware ZAC-style compiler [ 80 ] .

#### H.3 Discovered programs

##### Discovered superconducting algorithm.

The discovered algorithm can be summarized as follows. First, the discovered algorithm invests heavily in initial layout : it seeds high-degree logical qubits onto central, high-degree physical qubits, and then refines the mapping through an aggressive stack of hill-climbing and restart-based local search. Second, it strengthens online SWAP selection : it broadens the candidate neighborhood beyond front-layer incident edges to include look-ahead and shortest-path edges, changes the look-ahead term into dynamic horizon with exponential decay, and reshapes the swap objective to explicitly reward immediate gate executability. Together, these changes preserve the overall LightSABRE-style structure while materially improving robustness against long-range interactions and stagnation.

##### Discovered zoned neutral-atom algorithm.

The discovered placer replaces stage-local travel minimization with whole-trajectory routing-aware search, so it explicitly optimizes the parallelism that survives under the fixed router rather than only local travel distance. It begins by constructing a moderate set of heuristic initial storage layouts, including interaction-graph-based qubit orderings, their reversed counterparts, and a few shuffled orders. From each initial layout, it runs a full forward placement pass to build a complete placement trajectory, and then compares these candidates using router-aware evaluation.

Within a single forward pass, active qubits are placed stage by stage. When a stage loads qubits from storage into the entangling zone, a qubit whose partner is already reused is assigned to the complementary column of the same entangling site, whereas the remaining two-qubit pairs are assigned by Hungarian matching, with physical Euclidean distance as the placement cost. When active non-reused qubits return from the entangling zone to storage, they are grouped by source entangling row and ordered by source column. A monotone dynamic program then selects destination storage cells that preserve this order while minimizing column deviation, after which an intra-row swap refinement is applied whenever it shortens travel distance and remains compatible with the router’s non-crossing constraint.

After a complete trajectory has been constructed, the algorithm evaluates the full placement sequence with the fixed router. Candidates are ranked first by the number of router-emitted Move operations and then, for ties, by total Euclidean travel distance. Starting from the best candidate, the algorithm performs several rounds of reverse-through-time refinement, in which a reverse pass seeded by the current final layout is fed back into a new forward pass, and then applies hill-climbing swaps on the initial storage layout. An update is accepted only when it reduces the number of router-emitted Move operations or preserves that count while improving travel distance.

#### H.4 Discovered superconducting routing policy using Gemini

The superconducting qubit routing experiment was additionally run with gemini-3-pro-preview . This run produced a policy family in which both the initial placement and the online routing rule are explicitly shaped by circuit structure and hardware geometry. For initial layout, the discovered program classifies logical interaction components by morphology, such as path-like, ring-like, or dense structure. It then selects physical regions with matching shape and connectivity, prioritizes central and well-connected physical qubits for high-traffic logical qubits, and refines the mapping through large-scale randomized restarts and local improvement.

For online routing, the program strengthens SWAP selection with criticality-aware scoring. Front-layer and look-ahead gates are weighted by reverse depth and critical-path relevance, while stronger decay and centrality-sensitive preferences steer active qubits toward more useful regions of the device and help avoid stagnation. Quantitatively, the discovered program reduces added SWAPs by 24.7 % 24.7\% relative to SABRE and 18.3 % 18.3\% relative to LightSABRE. The largest gains occur on Q20 , where the reductions in added SWAPs reach 40.7 % 40.7\% and 32.9 % 32.9\% , respectively. Table 5 gives the topology-level breakdown. These results suggest that effective qubit routing benefits from a more explicitly graph-theoretic treatment, in which circuit morphology, interaction criticality, and hardware connectivity are exploited jointly.

### I Astrodynamics trajectory design

This section provides the information needed to interpret and reproduce the astrodynamics experiments in the main text. We define the executable-program interface and independent trajectory evaluator, document the benchmark and historical-reference construction, and examine how evolution changes the allocation of search effort. Throughout, “cost” denotes the propulsive cost computed by the benchmark evaluator, not the reported Δ ​ v \Delta v budget of a flown mission.

#### I.1 Task formulation and program interface

We consider impulsive multi-gravity-assist trajectory design under a patched-conic model [ 148 , 121 ] . Each candidate produced by SimpleTES is an executable optimizer that maps a numerical mission specification to a complete trajectory rather than to a single scalar design vector.

The feasible domain Ω s \Omega_{s} depends on the selected discrete design. Changing the encounter sequence or DSM placement can change the dimension and meaning of the continuous variables, whereas the transfer branch selects among distinct transfer solutions for the same variables. These choices remain coupled through the transfer-time and flyby constraints.

The reported runs expose the numerical mission specification to the executable program through a structured problem object. Mission names, descriptions, historical trajectory-family labels, reference encounter sequences and event epochs, and the optimized reference trajectories and costs are withheld. The program is instead given the departure body and target endpoint, the broad mission windows, the admissible flyby bodies, the endpoint models, and the feasibility limits listed above. This removes direct historical encounter-sequence information from the task interface, but does not preclude a pretrained language model from inferring dynamically plausible sequences, or recognizing a familiar mission class, from the numerical specification itself.

#### I.2 Independent trajectory evaluator

Candidate programs are executed in isolated processes and return only the node list in Eq. ( 24 ); they do not return their own validity or score. The evaluator first checks the structural schema and mission limits, and reconstructs planetary states from DE430 ephemerides [ 38 ] . To verify each heliocentric leg, it propagates the declared outgoing state ( 𝐫 i , 𝐯 i + ) (\mathbf{r}_{i},\mathbf{v}_{i}^{+}) from epoch t i t_{i} to the next node’s declared epoch t i + 1 t_{i+1} under solar two-body dynamics, then compares the propagated position and velocity with the next node’s declared incoming state ( 𝐫 i + 1 , 𝐯 i + 1 − ) (\mathbf{r}_{i+1},\mathbf{v}_{i+1}^{-}) . This verification requires neither a Lambert-branch label nor a Lambert solve. The evolved programs may use Lambert solvers internally to construct candidate arcs [ 62 ] ; the evaluator instead tests the returned states directly for propagation consistency.

At every GA node, the evaluator recomputes the planetary velocity and applies a powered-flyby model to the incoming and outgoing hyperbolic-excess velocity vectors. A flyby is rejected if the required periapsis radius lies below the configured planetary radius plus minimum altitude. A feasible powered flyby contributes its required periapsis impulse to the cost [ 148 , 121 ] . DSM cost is the norm of the instantaneous velocity discontinuity, Δ ​ v DSM , i = ‖ 𝐯 i + − 𝐯 i − ‖ 2 . \Delta v_{\mathrm{DSM},i}=\left\|\mathbf{v}_{i}^{+}-\mathbf{v}_{i}^{-}\right\|_{2}. (27)

Departure and arrival costs are specified independently for each benchmark. For a piecewise-linear boundary, the evaluator computes the velocity discontinuity relative to the departure planet or target state and maps its magnitude through prescribed breakpoints. At launch, the configured quantity C 3 , 0 C_{3,0} defines a zero-cost threshold v ∞ , 0 = C 3 , 0 v_{\infty,0}=\sqrt{C_{3,0}} rather than a hard feasibility boundary. Within the configured linear range, excess departure speed is charged one for one, such that Δ ​ v start = max ⁡ ( 0 , v ∞ − v ∞ , 0 ) . \Delta v_{\mathrm{start}}=\max\!\left(0,v_{\infty}-v_{\infty,0}\right). (28) The same piecewise-linear form is used for flyby or rendezvous endpoints with a prescribed relative-speed allowance. For a planetary-capture endpoint, the evaluator models a tangential periapsis insertion into an elliptic orbit with period T T . If the configured altitude is h factor ​ R p h_{\mathrm{factor}}R_{p} above the planetary reference radius R p R_{p} , the insertion radius is r p = R p ​ ( 1 + h factor ) r_{p}=R_{p}(1+h_{\mathrm{factor}}) . With a = ( μ p ​ T 2 4 ​ π 2 ) 1 / 3 , a=\left(\frac{\mu_{p}T^{2}}{4\pi^{2}}\right)^{1/3}, (29) the capture cost is Δ ​ v cap = v ∞ 2 + 2 ​ μ p r p − 2 ​ μ p r p − μ p a . \Delta v_{\mathrm{cap}}=\sqrt{v_{\infty}^{2}+\frac{2\mu_{p}}{r_{p}}}-\sqrt{\frac{2\mu_{p}}{r_{p}}-\frac{\mu_{p}}{a}}. (30)

The four terms in the verified propulsive cost J ⁡ ( 𝒯 ) J(\mathcal{T}) in Eq. ( 26 ) account for departure, DSMs, powered flybys, and arrival, respectively. The number of DSMs is therefore not itself the objective: a DSM is useful only when its added impulse is outweighed by a reduction elsewhere in the verified cost. Valid trajectories are ranked by a monotone transformation of J J , whereas any schema, dynamics, flyby, boundary, timeout, or memory-limit failure receives zero score. We report J J in km ​ s − 1 \mathrm{km}\,\mathrm{s}^{-1} because the transformation does not change the ranking.

The evaluator is deliberately lower fidelity than an operational mission design model. It does not attempt to reproduce navigation margins, finite-burn dynamics, spacecraft mass evolution, detailed launch-vehicle performance, or full N N -body perturbations. It nevertheless enforces a common dynamical and cost model across every candidate and reference, including ephemeris consistency, heliocentric propagation, minimum flyby altitude, launch allowance, powered-flyby cost, and target-specific arrival conditions. The reported comparisons should therefore be interpreted as controlled within-evaluator comparisons, not as claims to outperform the flown missions in their operational models.

#### I.3 Benchmark instances

The five historical mission windows were selected to span distinct trajectory-design structures. Mariner 10 represents an early inner-planet gravity-assist transfer to a flyby target [ 44 ] ; Galileo represents a classical Venus–Earth–Earth-assisted Jupiter transfer [ 32 ] ; Cassini–Huygens combines repeated inner-planet encounters, a Jupiter assist, and Saturn capture [ 106 ] ; Voyager 2 traverses the four giant planets in one continuous tour [ 71 ] ; and Rosetta uses repeated Earth and Mars encounters to rendezvous with a small body [ 46 ] .

Tables 7 and 8 summarize the boundary conditions and admissible topology of each benchmark. Planet abbreviations are Me (Mercury), V (Venus), E (Earth), M (Mars), J (Jupiter), S (Saturn), U (Uranus), and N (Neptune). The quantity C 3 , 0 C_{3,0} is the zero-cost launch threshold defined in Eq. ( 28 ).

For a planetary boundary, the evaluator obtains the body’s ephemeris state at the candidate’s reported boundary epoch. Thus, the listed arrival windows constrain the epoch rather than specify a stored Cartesian state. The relatively permissive terminal-speed ranges in the Mariner 10 and Voyager 2 benchmarks represent flyby rather than rendezvous conditions. Their endpoint costs remain zero up to 10 and 30 ​ km ​ s − 1 30\,\mathrm{km}\,\mathrm{s}^{-1} , respectively, and increase with unit slope up to the configured admissible limits of 20 and 40 ​ km ​ s − 1 40\,\mathrm{km}\,\mathrm{s}^{-1} . The Rosetta endpoint is the fixed heliocentric state 𝐫 67 ​ P \displaystyle\mathbf{r}_{67\mathrm{P}} = ( 185144048.790 , − 438016487.910 , − 252361615.057 ) ​ km , \displaystyle=(185144048.790,\;-438016487.910,\;-252361615.057)\,\mathrm{km}, 𝐯 67 ​ P \displaystyle\mathbf{v}_{67\mathrm{P}} = ( 7.655449311 , 12.114354862 , 5.612458233 ) ​ km ​ s − 1 , \displaystyle=(7.655449311,\;12.114354862,\;5.612458233)\,\mathrm{km}\,\mathrm{s}^{-1}, at MJD 56875.3791667. This state was obtained from the NASA/JPL Horizons on-line ephemeris system [ 45 ] , using 67P record 90000701 (target solution JPL#K154/5), the Sun as the centre (“500@10”; DE441), and a geometric heliocentric ICRF vector at 6 August 2014, 09:06 TDB.

#### I.4 Construction of historical-sequence references

The historical comparison does not use a reported mission Δ ​ v \Delta v budget. Reported mission budgets also reflect operational decisions and navigation margins that do not match the benchmark model. Instead, the cited mission chronologies supply the historical encounter orders and nominal event epochs. We reconstruct each reference through fine local optimization around those epochs, then save the resulting node sequence and replay it through the same full evaluator used for SimpleTES . This estimates how the historical trajectory structure performs under the benchmark model; it is not intended as a competing broad mission-design optimizer.