A Topology-Aware Quantum-Classical Framework for Multi-Target Active Debris Removal using Gaussian Boson Sampling

1. Introduction

• Target selection and sequencing are coupled because target utility depends on transfer accessibility and mission architecture, while exhaustive subset exploration becomes impractical as candidate populations grow.The introduction notes that selecting 10 targets from 1,000 candidates already produces an impractically large subset space before sequencing.Source: §1 · pp. 7–8
• Graph weights encode transfer accessibility and target-prioritization information, including consequence, persistence, conjunction risk, centrality, community structure, and source-family information.A random subset-selection procedure serves as an additional experimental baseline.Source: §1 · p. 10
• PyGMO evaluates servicing sequences over candidate launch epochs, jointly minimizing mission Δv and duration subject to modeled propulsion, mass, timing, transfer, and mission constraints.Non-dominated solutions characterize approximate Pareto fronts for candidate mission architectures.Source: §1 · p. 10

1.1. Our Contributions

• The evaluation varies benchmark environment, prioritization strategy, candidate-pool size, and target cardinality to examine sampling behavior and downstream mission performance.The benchmarks are SOCRATES Minefield, Collision Cloud, and Random; pool sizes are N ∈ {15, 18, 22, 25} and target cardinalities are k ∈ {3, 5}.Source: §1 · p. 10, §1.1 · p. 12
• The framework combines weighted orbital-transfer graphs, GBS candidate-subset generation, and PyGMO-based multi-objective mission optimization.The stages separate combinatorial candidate generation from detailed route evaluation, limiting the subsets sent to mission optimization.Source: §1 · p. 9, §1.1 · p. 10

• The preliminaries establish the framework’s foundations in orbital transfer mechanics, graph-theoretic space-environment modeling, GBS probability distributions, and multi-objective evolutionary routing.Source: §2 · p. 13

2.1. Astrodynamics and Orbital Transfer Mechanics

• The framework combines TLE-based orbital-state reconstruction with a Hohmann-type transfer surrogate, inclination-dependent arrival costs, and transfer-time estimates.Pairs exceeding the prescribed inclination-difference limit are treated as inaccessible.Source: §2.1 · p. 13, §2.1.1 · p. 13, §2.1.4 · p. 14

2.1.1. Orbit Reconstruction

• Orbital states reconstructed with SGP4 provide the orbital elements used in pairwise transfer estimation and time-dependent route evaluation.The stored quantities include semimajor axis, eccentricity, inclination, RAAN, and mean anomaly.Source: §2.1.1 · p. 13

2.1.2. Impulsive Velocity Change

• Impulsive velocity change is represented by the difference between pre- and post-maneuver velocity vectors, and the transfer model uses it for the inclination-dependent arrival maneuver.The angle between the velocity vectors is denoted α.Source: §2.1.2 · p. 13

2.1.3. Tsiolkovsky Rocket Equation

• Propellant use follows the Tsiolkovsky rocket equation, with mass propagated across servicing legs and missions rejected below the prescribed dry-mass floor.Source: §2.1.3 · p. 13

2.1.4. Hohmann-Type Transfer Surrogate

• The Hohmann-type surrogate estimates passive transfer cost and time for circular orbits, incorporating orbital-plane separation into the arrival burn.Pairs beyond the prescribed inclination-difference limit are inaccessible in the pairwise transfer matrices.Source: §2.1.4 · p. 14

2.1.5. Secular J2 Evolution and Orbital Phasing

• Secular J2 nodal drift and mean-anomaly evolution determine RAAN and phase waiting times, which are subject to model caps; graph construction combines the two waits with α = 0.7.Ordered mission evaluation also considers an active-transfer alternative using the propagated orbital-plane separation.Source: §2.1.5 · pp. 14–15

• The environment is a directed weighted graph: debris objects are vertices, and directed edges encode transfer accessibility, which can differ by traversal direction.Asymmetry reflects differences in orbital geometry, relative phasing, inclination, and secular evolution.Source: §2.2 · p. 15
• An N × N affinity matrix transforms directional transfer costs through an exponentially decaying kernel, with σ controlling the decay.The resulting matrix is generally asymmetric because transfer costs are directional.Source: §2.2 · p. 15
• Low-cost transfers yield affinities near one, while expensive transfers approach zero; strong directed connections identify accessible collections.In the four-object illustration, three objects in one fragmentation cloud form a strongly connected block, while connections to the dynamically distant fourth object are weak.Source: §2.2 · p. 16

• GBS uses squeezed light, linear optical interference, and photon measurements to sample distributions associated with matrix Hafnians.The Hafnian aggregates edge-weight products over perfect matchings, and its exact computation is #P-hard.Source: §2.3 · pp. 16–17
• The graph matrix is symmetrized for GBS encoding, while direction-dependent information can be retained separately for later routing or scheduling.Dense regions in the resulting matrix can receive greater sampling weight.Source: §2.3 · pp. 16–17
• For a complex symmetric graph matrix, Autonne–Takagi decomposition supplies singular values for squeezing parameters and a unitary factor for the optical transformation.A scaling factor is chosen so that cσi < 1 for all i, controlling squeezing magnitude and mean photon generation rate.Source: §2.3 · p. 17
• Photon-number-resolving detection yields photon-count patterns with Hafnian-dependent probabilities; threshold detection instead records only whether each mode registers a click.Threshold detection discards exact photon occupation numbers.Source: §2.3 · p. 17
• GBS probabilities can favor candidate subsets with stronger pairwise graph connections, motivating its use for candidate generation before application-specific optimization.In the illustrative two-pair example, the first pair has a sampling-probability ratio of 7225 relative to the second; this is an example, not a benchmark result.Source: §2.3 · p. 18
• The four-node illustration sums perfect-matching contributions to a total Hafnian of 0.0657, with one matching contributing 0.0390.The figure depicts graph-matrix decomposition, optical transformation, and photon detection leading to a possible sampled pattern.Source: §2.3 · p. 18, Figure 5 · p. 19

2.4.1. ADR as a Bi-Objective Optimization Problem

• Mission design minimizes cumulative velocity increment and total duration, with each candidate servicing order evaluated across its k −1 inter-target legs.Passive transfers are evaluated first; active transfers are considered when passive arrival misses a conjunction-assessment deadline, with lateness penalized if the active option is also late.Source: §2.4.1 · pp. 19–20
• The objectives conflict because active plane changes can shorten transfers while increasing fuel use, whereas passive J2 nodal drift can reduce ∆v while lengthening missions.This produces a continuum from rapid, fuel-intensive interventions to slower, fuel-efficient campaigns.Source: §2.4.1 · pp. 19–20

2.4.2. PyGMO and the Archipelago Paradigm

• PyGMO evolves multiple independent populations on islands and exchanges solutions along a directed migration topology.The heterogeneous island design is intended to combine complementary search behaviors while enabling information exchange.Source: §2.4.2 · pp. 20–21

2.4.3. NSGA-II for Pareto Front Exploration

• NSGA-II uses non-dominated sorting, elitist retention, and crowding distance to explore competing mission architectures and maintain a broad Pareto-front approximation.Sparse regions are favored when the marginal front exceeds population capacity.Source: §2.4.3 · pp. 21–22

2.4.4. MOEA/D for Localized Pareto Convergence

• MOEA/D decomposes the objectives into scalar subproblems whose weight vectors encode fuel-focused, duration-focused, or intermediate mission preferences.Subproblems exchange information locally with neighboring weight vectors, refining solutions in different Pareto-front regions.Source: §2.4.4 · pp. 22–23, Figure 7 · p. 24

2.4.5. Complementary Search Dynamics

• The Archipelago combines NSGA-II global exploration and MOEA/D localized refinement, exchanging candidate architectures across heterogeneous islands to build a diverse archive of fuel-time trade-offs.Graph-theoretic and GBS stages first reduce the combinatorial search space to promising debris subsets.Source: §2.4.5 · pp. 24–25

3.1. Benchmark Design Philosophy

• Three suites provide complementary populations: SOCRATES Minefield emphasizes conjunction risk, Collision Cloud inferred provenance, and Random supplies an unfiltered reference.Each is generated independently by candidate-pool size and experimental seed, with the same evaluation procedures applied as appropriate.Source: §3.1 · p. 26, §3.5 · p. 28

3.5. Benchmark Summary

• The benchmark comparison examines how risk concentration, source-family composition, and sampled orbital structure relate to target-selection and mission-planning behavior.The benchmark construction does not predetermine graph topology; topology follows the selected objects' orbital states.Source: §3.1 · p. 26, §3.2.4 · p. 27, §3.3.4 · p. 28

3.2.1. Data Source and Construction

• SOCRATES Minefield ranks catalogue objects by maximum reported conjunction probability, then samples N objects from the highest-risk 200-object tier.This retains a strong risk emphasis while allowing stochastic variation among benchmark instances.Source: §3.2.1 · p. 27

3.2.2. Topological Characteristics

• SOCRATES transfer-network structure follows the orbital states of sampled high-risk objects; any enhanced connectivity is empirical, not imposed by construction.GBS subset distributions are shaped by the induced weighted adjacency matrix and prioritization heuristic, and may vary in concentration and diversity.Source: §3.2.2 · p. 27, §3.2.3 · p. 27

3.3.1. Data Source and Construction

• Collision Cloud samples N objects from the catalogue's largest identifiable source family, with family inferred from recurring object-name patterns.The family label is a catalogue-derived provenance proxy, not an independently reconstructed fragmentation genealogy.Source: §3.3.1 · p. 27

3.3.2. Topological Characteristics

• Collision Cloud connectivity is computed from sampled orbital states rather than imposed; orbital dispersion can yield heterogeneous GBS subset distributions.Dense local transfer structures may support subsets associated with connected regions, and sampling behavior is evaluated for each instance.Source: §3.3.2 · pp. 27–28, §3.3.3 · p. 28

3.4.1. Data Source and Construction

• Random uniformly samples N objects from the full catalogue without filtering for conjunction risk, source family, orbital similarity, or other benchmark-specific structure.Its resulting connectivity, dispersion, and community structure can vary by seed.Source: §3.4.1 · p. 28, §3.4.2 · p. 28

3.4.3. Expected Sampling Behavior

• For Random, GBS subset concentration or dispersion depends on the weighted graph induced by each sampled population and the selected prioritization strategy.The suite is an unstructured reference for studying framework behavior without a benchmark-specific selection criterion.Source: §3.4.3 · p. 28, §3.4.4 · p. 28

4. Target Prioritization Framework

• Seven strategies cover physical consequence, persistence, operational threat, transfer-network importance, cluster membership, source-family membership, and random selection; they alter subset generation rather than directly determining servicing sequences.The heuristics are complementary and not independent; all use the same subsequent mission-optimization procedure.Source: §4 · pp. 30–31

4.1. Heuristic-Dependent Graph Weighting

• Heuristic priorities modify the baseline astrodynamic transfer graph supplied to GBS; threshold-detection outcomes retain subsets with exactly k active modes.The empirical subset distribution comes from simulated GBS outcomes, not a simplified product of individual priorities.Source: §4.1 · pp. 31–32

4.2. Physical Consequence Index (SIZE)

• The Physical Consequence Index uses surrogate object mass to prioritize debris with potentially greater fragmentation consequences, without accounting for transfer cost, conjunction risk, or persistence.Mass is estimated from catalog object classification and approximate size; a square-root transform preserves ordering while reducing large-object dominance.Source: §4.2 · pp. 32–33

4.3. Persistence Index (BSTAR)

• The Persistence Index uses TLE BSTAR and mean motion to derive a relative persistence surrogate, not a direct prediction of re-entry time.The measure provides a long-term environmental perspective distinct from immediate conjunction risk.Source: §4.3 · pp. 33–34

4.4. Operational Threat Index (SOCRATES_RISK)

• The Operational Threat Index combines surrogate mass, BSTAR-derived persistence, and conjunction-risk information into one object-level priority score.A square-root transformation moderates the conjunction-risk term's numerical contribution.Source: §4.4 · p. 34

4.5. Transfer-Network Importance (CENTRALITY)

• Transfer-Network Importance prioritizes structurally central objects using PageRank, eigenvector centrality, and betweenness, emphasizing network position rather than direct environmental consequence.The measures represent global influence, ties to important objects, and participation in shortest-path transfer corridors, respectively.Source: §4.5 · pp. 34–35

4.6. Orbital Cluster Membership (COMMUNITY)

• Orbital Cluster Membership applies Louvain modularity maximization to the weighted transfer graph and prioritizes objects in larger communities with greater internal accessibility density.This emphasizes local network structure rather than global centrality.Source: §4.6 · p. 35

4.7. Source-Family Prioritization (SOURCE)

• Source-Family Prioritization uses catalog names, identifiers, and metadata to infer families as a provenance proxy, assigning UNKNOWN when a family cannot be identified.The source-family information is not a reconstructed physical genealogy; the inferred family size informs priority.Source: §4.7 · pp. 35–36
• SOURCE applies an additional multiplicative edge factor to objects assigned to the same known family, while its provenance interpretation is limited by catalog-based family assignments.The strategy emphasizes concentration within inferred debris families as encoded by the adopted classification procedure.Source: §4.7 · p. 36

4.8. Random Subset Selection (RANDOM_SUBSET)

• RANDOM_SUBSET directly samples candidate k-subsets, bypassing GBS while retaining the same downstream trajectory optimization and mission assessment as heuristic-driven strategies.As a baseline, it removes heuristic-specific prioritization from subset generation to isolate the selection mechanism's effect.Source: §4.8 · p. 36

4.9. Summary of Target-Prioritization Strategies

• The strategy comparison holds mission evaluation common while varying subset construction to examine effects on graph structure, GBS sampling, subset composition, and downstream ADR objectives.Table 3 summarizes the seven heuristics and their formulations.Source: §4.9 · p. 36

5.1. Problem Formulation

• The framework splits ADR planning into subset discovery followed by route evaluation and multi-objective optimization, avoiding exhaustive target-space enumeration while retaining astrodynamic and spacecraft feasibility checks.The catalog represents debris objects as vertices and transfer relationships as weighted edges.Source: §5.1 · p. 37

5.1.1. ADR Subset-Selection Problem

• Candidate subsets contain exactly k targets selected using normalized object priorities and pairwise transfer accessibility encoded in a weighted orbital graph.The attributes include physical size or mass, persistence, conjunction risk, inferred source family, and graph-theoretic importance.Source: §5.1.1 · p. 38
• GBS encodes the weighted orbital graph into a Gaussian photonic state and converts threshold-detection outcomes into candidate subsets, favoring configurations with larger encoded connectivity weights.Cardinality filtering and duplicate removal produce the pool passed to mission optimization.Source: §5.1.1 · p. 38

5.1.2. Multi-Objective Routing Problem

• The routing stage orders each selected target exactly once and jointly minimizes aggregate transfer ∆v and total mission duration using Pareto dominance.The objectives include transfer costs and waiting, time-of-flight, and servicing time; optimization also evaluates spacecraft mass, per-leg maneuver, and conjunction-timing constraints.Source: §5.1.2 · pp. 38–39

5.2. Orbital-State Reconstruction and Transfer Modeling

• Orbital states reconstructed from TLE records supply the orbital parameters used in pairwise transfer and phasing calculations.The stored state includes semimajor axis, inclination, RAAN, eccentricity, velocity, mean anomaly, and secular RAAN-drift rate.Source: §5.2 · p. 39

5.2.1. Pairwise Transfer Model

• The pairwise transfer model excludes pairs outside the inclination bound and estimates admissible transfers with a Hohmann-type surrogate rather than a full Lambert solution.The transfer estimate includes energy-transfer and orbital-plane components and a corresponding time-of-flight.Source: §5.2.1 · pp. 39–40

5.2.2. Secular J2 Phasing Model

• The phasing model combines secular J2-driven nodal drift with in-plane mean-anomaly evolution to estimate passive waiting before a transfer.RAAN waiting is capped at 120 days and in-plane phasing at 30 days; the dynamic evaluator combines both before choosing passive or active transfer.Source: §5.2.2 · p. 40

5.3. Construction of the Orbital Transfer Network

• The network transforms pairwise astrodynamic quantities into a weighted graph for heuristic comparison and GBS encoding, converting transfer cost into affinity with an exponential kernel.The resulting matrix is spectrally normalized to keep its spectral norm below unity before use in the Gaussian photonic model.Source: §5.3 · pp. 40–41, §5.3.1 · p. 41

5.3.1. Heuristic-Weighted Adjacency Matrix

• Heuristic-weighted adjacency combines normalized object priorities with transfer affinity and stochastic perturbation, with SOURCE multiplying same-family non-unknown edges by five.The adjacency matrix is constructed symmetrically before spectral normalization.Source: §5.3.1 · p. 41

5.4.1. GBS Graph Encoding

• The normalized adjacency matrix is supplied to GBS, whose graph encoding uses its eigenvectors as the interferometer transformation and calibrated eigenvalues as squeezing parameters.Threshold-detection sampling follows a distribution governed by graph structure and hafnian-related Gaussian-sampling amplitudes.Source: §5.4 · p. 41, §5.4.1 · p. 41

5.4.2. Photon-Number Calibration

• Photon-number calibration numerically scales the state toward expected photon number k, but does not guarantee exactly k clicks in every measurement.Outcomes are threshold-detected and retained only when they have cardinality k; duplicates are removed.Source: §5.4.2 · p. 42, §5.4.3 · p. 42

5.4.4. Deterministic Fallback

• If GBS yields no valid cardinality-k subset, a deterministic fallback selects high-weighted-degree vertices and appends feasible k-element combinations until the candidate limit.The fallback is activated only when sampling produces no usable candidate.Source: §5.4.4 · p. 42

5.5.1. Route Representation

• The optimizer represents a visitation route with continuous random keys whose sorted order induces a discrete permutation of the selected targets.The implementation assumes the spacecraft begins co-located with the first target, accumulating mission cost from the first inter-target transfer.Source: §5.5.1 · p. 43

5.5.2. Dynamic Route Evaluation

• Dynamic route evaluation prefers passive transfer, switches to an active alternative when passive arrival misses a conjunction TCA, and penalizes remaining deadline violations.The active alternative sets passive waiting time to zero; the passage states that a lateness penalty applies if it also misses the TCA.Source: §5.5.2 · p. 43

5.5.3. Spacecraft Mass Feasibility

• Mass feasibility is checked by propagating spacecraft mass with the Tsiolkovsky rocket equation and enforcing mass and per-leg maneuver bounds.The mass-depletion calculation provides the primary propulsion-feasibility check.Source: §5.5.3 · p. 43

5.5.4. Bi-Objective Evolutionary Optimization

• PyGMO uses NSGA-II and MOEA/D islands to evolve candidate routes, then discards large-penalty solutions and reevaluates the remaining permutations with detailed mission scoring.The islands use a ring topology for candidate migration.Source: §5.5.4 · p. 44

5.5.5. Pareto-Front Construction

• Feasible solutions are compared by Pareto dominance to construct the non-dominated set and its objective-space front.Dominance requires no worse values in both objectives and a strict improvement in at least one.Source: §5.5.5 · p. 44

5.6. Launch-Epoch Sensitivity

• Each selected subset and heuristic is optimized independently at launch epochs sampled every 45 days, making this a timing-sensitivity analysis rather than an operational flight schedule.Solutions from the sampled epochs are pooled for heuristic-level Pareto analysis.Source: §5.6 · p. 44

5.7. Integrated Quantum-Classical Optimization Workflow

• The workflow converts orbital states and pairwise transfer quantities into a weighted accessibility graph, incorporates heuristic priorities, and sends the normalized matrix to GBS, except for RANDOM_SUBSET.GBS outcomes are cardinality-filtered and deduplicated; a weighted-degree fallback supplies candidates when no valid subset is produced.Source: §5.7 · pp. 44–45
• Each candidate subset is evaluated across launch epochs by PyGMO permutation-based multi-objective route optimization, retaining feasible routes for heuristic-level solution pools.The pooled solutions form heuristic-specific Pareto fronts and a global non-dominated set.Source: §5.7 · p. 45
• The framework reduces ADR planning hierarchically from target selection to GBS subset discovery, route optimization, and a Pareto mission set.The solution pools are characterized using sampling, graph, mission, astrodynamic, and multi-objective metrics.Source: §5.7 · p. 45

6. Metrics

• Four metric classes assess quantum sampling, graph structure, mission and astrodynamic properties, and multi-objective optimization.The evaluation quantifies both statistical properties of generated debris subsets and mission-level characteristics and feasibility after evolutionary optimization.Source: §6 · p. 45

6.1. Quantum-Sampling Evaluation

• HCR, diversity, coverage, and ESS describe sampling concentration and breadth across subsets and debris targets.Higher HCR indicates probability concentrated in a few subsets; diversity counts unique subsets, coverage spans the catalog, and ESS reflects effective distribution breadth rather than raw sample count.Source: §6.1.1 · p. 47, §6.1.2 · p. 47, §6.1.3 · p. 47
• Javg compares overlap between sampled subsets, log UAmp compares their internal weighted connectivity with a random-subset baseline, and Hv measures vertex-selection dispersion.Larger Javg indicates greater overlap; low Hv indicates sampling concentrated on fewer debris objects.Source: §6.1.4 · p. 47, §6.1.5 · p. 48, §6.1.6 · p. 48

6.2. Graph-Theoretic Evaluation

• Graph metrics characterize community concentration, subset separation, internal connectivity, and the spectral organization of connectivity.Community Purity reflects whether selected targets share a Louvain community; lower conductance indicates stronger separation from the rest of the graph, while algebraic connectivity and spectral gap provide spectral measures.Source: §6.2.1 · p. 48, §6.2.2 · p. 49, §6.2.3 · p. 49

6.3. Mission-Level and Astrodynamic Evaluation

• HRE measures selected risk addressed per active mission time, while RCR measures the selected risk reached by its target’s TCA deadline.RCR uses modeled target arrival times and the corresponding time of closest approach as the temporal condition.Source: §6.3.1 · p. 49, §6.3.2 · p. 50
• Mission-level metrics also assess perturbed cost variability, transfer accessibility, Δv allocation, orbital diversity, plane-change burden, and J2-driven nodal-drift use.The accessibility index combines normalized transfer cost and waiting time; orbital diversity reflects inclination and RAAN dispersion, and plane-change burden is the fraction of total Δv spent on out-of-plane maneuvers. Differential J2-induced nodal precession can passively modify relative RAAN.Source: §6.3.3 · p. 50, §6.3.4 · p. 50, §6.3.5 · p. 50

6.3.8. J2 Drift Utilization (ηdrift)

• J2 Drift Utilization (ηdrift) measures the fraction of mission duration spent in passive waiting; higher values indicate more passive phasing than active maneuvering.Source: §6.3.8 · p. 51

6.3.9. Family Diversity (FD)

• Family Diversity (FD) is the number of distinct source-family labels in a selected subset normalized by subset size; lower values indicate concentration in fewer families.Higher FD indicates representation from a larger number of distinct families.Source: §6.3.9 · p. 51

6.4. Multi-Objective Optimization Evaluation

• The optimization framework evaluates mission architectures simultaneously across multiple objectives rather than reducing evaluation to a single scalar objective.Source: §6.4 · p. 51

6.4.1. Number of Valid Missions (Nvalid)

• Nvalid counts generated mission architectures satisfying all modeled astrodynamic, temporal, mass, and operational feasibility constraints.A mission is valid only if it satisfies every constraint implemented by the mission-evaluation model.Source: §6.4.1 · p. 51

6.4.2. Hypervolume (HV)

• Hypervolume measures the objective-space volume dominated by a Pareto set relative to a reference point, capturing both convergence and spread.The reference point bounds the objective region used for comparison.Source: §6.4.2 · p. 52

6.4.3. Global Pareto Front Hits (PFhits)

• PFhits counts a heuristic’s solutions that belong to the pooled global non-dominated set, indicating its participation in the global Pareto solution space.The global set is formed by considering candidate solutions from all evaluated strategies jointly.Source: §6.4.3 · p. 52

7.1. Physical Constants and Mission Parameters

• The conceptual servicer uses 9200 kg wet mass, a 500 kg dry-mass floor, and Isp = 300 s; feasibility also applies a 50-day servicing allocation and transfer and mass constraints.Cumulative propulsive expenditure is bounded through propagated spacecraft mass rather than a separate cumulative ∆v limit.Source: §7.1 · pp. 53–54

7.2. Experimental Benchmarks

• 840 seeded experimental instances span three benchmark environments, eight (N, k) configurations, and seven target-selection strategies.The environments use random catalogue samples, SOCRATES high-risk objects, or objects from the most populated identifiable inferred source family; RANDOM_SUBSET bypasses GBS.Source: §7.2 · p. 55

7.3. Gaussian Boson Sampling Configuration

• For each graph-informed strategy, the weighted adjacency matrix is spectrally normalized for GBS; threshold detections yield unique candidate subsets with exactly k active modes, with a graph-based fallback if sampling fails.Collection stops at 20 unique valid subsets or when requested samples are exhausted.Source: §7.3 · pp. 55–56

7.4. Implementation of the Orbital Transfer Graph

• The transfer graph represents estimated accessibility between debris objects and uses centrality and community descriptors; heuristic-specific edges can also receive a boost for shared inferred source-family labels.The source-family boost is not applied to the separate aggregate GBS summary graph.Source: §7.4 · pp. 56–57

7.5. Dynamic Orbital Transfer and Phasing Model

• Mission evaluation propagates orbital states with J2 nodal drift and allows passive phasing or active maneuvers when waiting would miss a TCA deadline, while tracking spacecraft mass and operational feasibility.SGP4 initializes orbital propagation from TLE data; passive phasing can reduce propellant for a subsequent rendezvous maneuver.Source: §7.5 · p. 57, §7.9 · pp. 59–60

7.6. Evolutionary Optimization Configuration

• A two-island PyGMO archipelago runs NSGA-II and MOEA/D under identical mission constraints, optimizing total mission ∆v and duration while retaining feasible solutions for Pareto fronts.Each island uses 20 individuals and evolves for 30 generations; scoring propagates mass and checks available SOCRATES arrival-time conditions.Source: §7.6 · p. 58

7.7. Launch Epoch Sampling

• Each candidate subset is optimized at 18 launch epochs sampled every 45 days across an 800-day horizon beginning 1 January 2026 UTC.A common epoch grid supports comparisons across benchmark configurations.Source: §7.7 · pp. 58–59

7.8. Stochastic Repetition and Statistical Treatment

• Five predetermined seeds vary catalogue sampling, graph perturbation, GBS sampling, Louvain detection, and evolutionary optimization; reported statistics aggregate runs, with averages representing computational performance within the benchmark-generation procedure.GBS metrics use valid raw detection samples, while mission and Pareto metrics use feasible optimizer solutions.Source: §7.8 · p. 59

8.1.1. The SOCRATES Minefield Environment Topology

• SOCRATES sampling broadens as scale increases: HCR falls from 0.99 to 0.54 and ESS rises from 1.7 to 10.9, while mean mission Δv increases from 3.53 to 6.67 km/s.Across the bounding configurations, OD, AI, and PCB remain relatively stable, but duration grows from 136.96 to 326.56 days; the longer sequence exhausts favorable local transfers and requires less favorable links.Source: §8.1.1 · pp. 60–62

8.1.2. The Collision Cloud Environment Topology

• Collision Cloud’s fragmentation-derived coherence persists at larger scale, although HCR falls from 0.94 to 0.51 and mean mission Δv rises from 3.30 to 5.58 km/s.Purity remains 0.89 at the larger scale; mission duration increases from 134.3 to 233.6 days, while the average waiting fraction falls from 18.0% to 11.7%, consistent with more direct transfers between nearby fragments.Source: §8.1.2 · pp. 62–64

8.1.3. The Random Environment Topology

• In the Random suite, HCR decreases from 0.85 to 0.48 as orbital dispersion rises, while mean mission Δv increases from 4.24 to 6.67 km/s and duration from 153.7 to 326.6 days.Waiting fraction rises from 28.1% to 35.4% alongside drift efficiency from 0.43 to 0.51, consistent with greater use of passive J2-assisted phasing; kinematic robustness remains approximately 0.96–0.97.Source: §8.1.3 · p. 65

8.1.4. Concluding Quantitative Synthesis of Environmental Topologies

• Collision Cloud combines the lowest dispersion, zero family diversity, and high purity with mean Δv of 1.29–6.81 km/s; Random is more heterogeneous and diffuse, while SOCRATES retains concentrated sampling around hazard structure.Across the full design space, Collision Cloud has OD = 0.10–0.20, FD = 0, and Purity = 0.89–1.00; Random has OD = 0.60–2.60, FD = 0.63–0.95, and HCR = 0.48–0.85. SOCRATES has HCR = 0.53–0.99 and Purity = 0.86–1.00.Source: §8.1.4 · p. 66, Table 11 · p. 66
• Across suites, organized transfer structure concentrates GBS probability on coherent candidate subgraphs, whereas heterogeneous topology broadens sampling and exploration, with differences carrying through to transfer cost, waiting, maneuver diversity, and duration.The reported relationship reflects interactions between object-level properties and the orbital-transfer network rather than any single environmental metric.Source: §8.1.4 · pp. 66–67, Figure 9 · p. 67

8.2. Synthesis of Combinatorial Scaling Dynamics: The Interplay between N and k

• Increasing N adds graph nodes and potential transfer corridors, broadening GBS candidate exploration; increasing k extends the servicing sequence and raises transfer complexity and physical mission burden.The scaling behavior therefore reflects the interaction of graph enrichment with route extension, including greater need for favorable phasing as target cardinality grows.Source: §8.2 · p. 68

8.2.1. Parametric Sensitivity Analysis

• Sensitivity analysis separates exploration, driven primarily by N, from mission complexity, more sensitive to k; these are dominant observed trends, not universal monotonic laws.Graph exploration, coverage, and sampling diversity respond more strongly to catalog expansion, while propellant expenditure, mission duration, and drift utilization respond more strongly to target cardinality.Source: §8.2.1 · p. 68

8.2.3. Operational Scaling Laws

• Increasing N mainly enriches the graph and broadens candidate exploration, whereas increasing k raises propellant, plane-change, passive-drift, and duration burdens as coherent transfers are exhausted.Benchmark topology modulates these effects: Collision Cloud retains fragmentation structure, SOCRATES Minefield concentrates probability around hazard corridors, and Random is more dispersed.Source: §8.2.3 · pp. 69–71

8.3.1. Objective-Space Performance and Robustness

• SOURCE achieves the highest mean Hypervolume (0.982), while CENTRALITY and COMMUNITY each reach 0.977; CENTRALITY and COMMUNITY also show lower relative variation and Random-suite losses than SIZE and BSTAR.Their CV values are below 2%, and their Randomization Losses are 3.84% and 3.44%, versus 12.21% for SIZE and 15.19% for BSTAR. SOCRATES_RISK’s low sensitivity and loss exclude some infeasible high-complexity configurations.Source: §8.3.1 · pp. 72–73, Table 15 · p. 72

8.3.2. Global Kinematic Performance and Algorithmic Reliability

• COMMUNITY and CENTRALITY have mean mission costs of 3.21 km/s and 3.32 km/s, versus 4.95 km/s for RANDOM_SUBSET; SOURCE and SOCRATES_RISK converge in 95.8% and 91.7% of configurations.The remaining heuristics achieve 100% convergence; COMMUNITY and CENTRALITY’s lower mean costs correspond to reductions of 35.2% and 32.9% versus RANDOM_SUBSET.Source: §8.3.2 · pp. 73–74, Table 16 · p. 73

8.3.3. Temporal Dynamics and Phasing Efficiency

• In the Random suite, COMMUNITY and CENTRALITY missions last 230.5 and 232.7 days, versus 287.1 days for BSTAR, while their Wait Fractions are 29.1% and 30.2%, versus BSTAR’s 44.9%.Temporal Degradation is 27.0% for COMMUNITY and 30.4% for CENTRALITY, compared with 57.9% for SIZE and 50.4% for BSTAR; SOCRATES_RISK’s 1.7% excludes infeasible configurations.Source: §8.3.3 · p. 74, Table 17 · p. 74

8.3.4. Mission Feasibility and Pareto Dominance

• CENTRALITY and COMMUNITY account for 178 of 289 global Pareto-front hits and about 68% of Random-suite hits; both yield about 1.1% Pareto hits per feasible mission versus 0.14% for RANDOM_SUBSET.SOCRATES_RISK has the lowest global valid-mission count, 4,598, showing that objective-space quality need not imply broad mission-generation coverage.Source: §8.3.4 · p. 75, Figure 10 · p. 75

8.3.5. Inter-Heuristic Diversity and Search-Space Exploration

• In the Random suite, COMMUNITY overlaps with BSTAR and SIZE by only 6.0% and 8.0%, while CENTRALITY–COMMUNITY overlap remains 30.2%, 41.0%, and 35.6% across suites.This distinct, topology-dependent selection behavior accompanies suite-specific heuristic strengths; no heuristic dominates every metric and condition simultaneously.Source: §8.3.5 · p. 76, §8.3.7 · p. 77, Table 19 · p. 78

8.3.6. Quantum Sampling Topology and Graph-Theoretic Profile

• SOCRATES has the highest HCR (0.77) and lowest ESS (6.18), COLLISION has the highest ESS (7.26) and entropy (Hv = 0.68), and Random has the lowest HCR (0.71) and highest Conductance (0.47).Pur remains between 0.92 and 0.96 across suites, so greater Random-suite dispersion does not mean uniform sampling; the distribution retains preference for structurally favorable regions.Source: §8.3.6 · p. 77, Table 18 · p. 77

8.4.1. Case Study I: Deep-Drift Dominance in a Dispersed Environment

• A Random-suite CENTRALITY solution at N = 25, k = 5 achieved 0.629 km/s transfer cost by using repeated waits and differential J2 nodal precession to resolve a near-300° RAAN separation.The mission lasted 538.4 days and consumed 1,771.6 kg of propellant; the case studies show distinct routing regimes shaped by orbital clustering, passive alignment, and conjunction urgency.Source: §8.4 · p. 78, §8.4.1 · p. 79, Table 20 · p. 79

8.4.2. Case Study II: Risk-Driven Interception under Conjunction Constraints

• In the SOCRATES Minefield case, waiting times collapse to zero, and conjunction urgency can make a large maneuver part of a Pareto-efficient interception despite its energy cost.The representative N = 15, k = 5 solution, selected using CENTRALITY, has total ∆v = 3.250 km/s and total duration = 200.1 days.Source: Table 21 · p. 79, §8.4.2 · p. 79

8.4.3. Case Study III: Kinematic Degeneracy within Fragmentation Structures

• With total ∆v = 0.008 km/s, the Collision Cloud case shows that selecting highly correlated, topologically clustered objects can make downstream transfers intrinsically compatible.The N = 15, k = 3 case was selected using SIZE and had a total duration of 193.1 days.Source: Table 22 · p. 80, §8.4.3 · p. 80

8.4.4. Case Study IV: Adaptive Hybrid Control

• In the SOCRATES Minefield case, an initially passive phasing interval gives way to immediate transfers in a hybrid strategy that emerges from competing objectives and orbital conditions.The N = 18, k = 5 case was selected using COMMUNITY and has total ∆v = 1.071 km/s and total duration = 285.2 days.Source: Table 23 · p. 80, §8.4.4 · p. 80

8.4.5. Case Study V: Secular Rephasing in Dispersed Fragmentation Families

• For dispersed fragmentation-derived populations, differential nodal regression enables secular rephasing across RAAN separations exceeding 300°, alternating extended phasing with opportunistic near-zero-cost transfers.The Collision Cloud N = 22, k = 5 case, selected using SIZE, has total ∆v = 0.400 km/s and total duration = 452.6 days.Source: §8.4.5 · p. 80, Table 24 · p. 81

8.4.6. Origins of the Emergent Trajectory Regimes

• GBS-guided subset sampling and trajectory optimization jointly shape the regimes: dense communities can yield kinematic degeneracy, sparse accessible subsets deep drift, and conjunction-critical subsets interception.GBS biases sampling toward subsets with strong transfer accessibility and heuristic utility, so the optimizer works on topologically filtered collections rather than arbitrary targets.Source: §8.4.6 · p. 81

8.4.7. Synthesis of Emergent Operational Regimes

• Pareto solutions balance passive dynamics, impulsive maneuvers, urgency, and orbital accessibility, yielding deep-drift, risk-driven, and hybrid strategies according to subset structure and mission constraints.The results frame multi-target ADR as a coupled graph-theoretic and astrodynamic optimization problem.Source: §8.4.7 · p. 81

9. Discussion

• CENTRALITY and COMMUNITY select targets relationally through transfer accessibility and orbital structure, complementing object-centric heuristics that rank targets individually.Their strong Hypervolume and Pareto-front performance is consistent with network-level information helping account for collective servicing costs.Source: §9 · p. 82
• GBS serves as topology-aware candidate-subset generation, while evolutionary optimization evaluates whether sampled subsets yield feasible, competitive mission architectures.As scale increases, sampling probability spreads across more structurally distinct subsets; the sampler is not limited to one dominant transfer corridor.Source: §9 · pp. 82–83
• Increasing N broadens topological flexibility, whereas increasing k raises servicing depth and physical burden through greater Δv, longer durations, and lower route reliability.A larger candidate pool can partly offset the burden of larger k by adding stepping-stone objects and alternative transfer pathways.Source: §9 · p. 83
• The framework combines a weighted transfer graph, GBS candidate generation, and evolutionary mission optimization, evaluated across conjunction-driven, fragmentation-driven, and unstructured environments.The experiments support this decomposition's feasibility but do not establish general computational speedup or end-to-end scalability for operational catalogues.Source: §9 · pp. 81, 83, 84
• Reduced graph-connectivity metrics did not produce proportional degradation in mission quality, as competitive servicing sequences continued to be found across structurally distinct candidate regions.Increasing mission scale had topology-dependent effects; longer durations sometimes coincided with less passive J2 use when temporal constraints favored more impulsive transfers.Source: §9 · p. 84

10. Limitations and Future Research Directions

• Static surrogate-based transfer costs cannot fully capture evolving orbital geometry or higher-order perturbative effects.The proposed direction is dynamic orbital-transfer networks with edge weights informed by high-fidelity propagation, including J2, drag, solar radiation pressure, luni-solar perturbations, and covariance growth.Source: §10 · p. 84
• The adjacency matrix assumes deterministic orbital-state knowledge and neglects tracking, conjunction-prediction, maneuver-execution, and navigation uncertainty.Proposed extensions include stochastic adjacency matrices and a multiplex graph representing transfer accessibility, collision risk, fragmentation ancestry, mission priority, and economic value.Source: §10 · p. 84
• Because evaluations use TLE snapshots whose prediction uncertainty can grow with propagation interval, the reported trajectories and schedules are computational mission-design results, not operational flight plans.Operational planning would require higher-fidelity orbit determination and ephemeris products with state-estimation and covariance information.Source: §10 · p. 85
• Closed-loop quantum-evolutionary optimization could use PyGMO feedback to adaptively reweight the sampling graph, while GBS scalability and theoretical performance bounds remain unestablished.The proposed theoretical work would relate graph topology, Hafnian statistics, entropy, and subset quality.Source: §10 · p. 85
• The framework assumes complete prior knowledge of the debris catalogue and omits operational factors such as capture complexity, communication constraints, servicing risk, and economic cost.Future directions include online, receding-horizon planning and extensions to low-thrust, hybrid, finite-burn, and autonomous-rendezvous dynamics.Source: §10 · p. 85
• The largest experiments remain modest relative to campaigns involving hundreds or thousands of objects, leaving catalog-scale optimization and coupling to long-term debris-evolution models as open steps.The text identifies NASA LEGEND and ESA MASTER as examples of such models.Source: §10 · p. 85

11. Conclusion

• The work investigates using orbital-environment topology in target selection and quantum sampling before computationally intensive trajectory optimization.This addresses formulations that separate object-level prioritization from trajectory optimization or jointly explore both decisions without fully exploiting orbital-transfer relationships.Source: §11 · p. 85
• The framework treats each debris object’s position and compatibility within the orbital-transfer network as relevant to mission-level target selection.Transfer accessibility, orbital proximity, community structure, and network centrality complement conventional debris-prioritization criteria.Source: §11 · p. 86
• GBS samples a limited set of candidate target subsets before multi-objective evolutionary optimization evaluates their physical mission trade-offs.This decomposition avoids solving target selection and trajectory design as one monolithic optimization and reduces the downstream search burden.Source: §11 · pp. 85–86
• The workflow assigns distinct roles to its stages: the orbital-transfer graph represents structure, GBS samples target selections, and evolutionary optimization resolves mission design.The quantum stage operates on structured target-selection space, while the classical stage focuses on physical execution.Source: §11 · p. 86
• The framework is presented as a basis for studying quantum sampling with domain-specific graph structure beyond a particular ADR catalogue or optimization configuration.Potential extensions named by the authors include orbital servicing, constellation maintenance, and in-orbit logistics.Source: §11 · p. 86