Project Description

Abstract

The main objective of the project is to investigate quantum computing approaches for phase unwrapping in Multi-Temporal Interferometric Synthetic Aperture Radar (MT-InSAR) Persistent Scatterer (PS) analysis. Temporal processing is particularly important in MT-InSAR because PSs do not form a regular spatial grid. Some points have reliable neighbors, whereas others are weakly connected or isolated and must be reconstructed mainly from their temporal information. We start from the classical temporal phase-unwrapping problem to understand its mathematical structure, identify the information available for phase reconstruction, and determine which formulations and decomposition strategies can support quantum computational approaches.

On the classical side, we developed new temporal phase-unwrapping methods, including an adaptive Minimum-Path-Length (MPL)-based approach and a model-based strategy specifically designed for MT-InSAR time-series reconstruction. A further methodological contribution is a reference-free self-diagnostic strategy that assesses the reliability of the reconstructed solution without requiring an externally unwrapped reference. This provides a practical way to identify potentially incorrect solutions and to apply additional processing only when needed. We also developed a synthetic MT-InSAR time-series simulator and a custom cascaded LSTM model to investigate data-driven temporal reconstruction under controlled conditions. Beyond the temporal dimension, we studied when spatial information can effectively complement the temporal solution. The PS distribution was therefore represented as a graph, and edge, node, and connected-component statistics were used to characterize the available spatial support. This analysis also led to Domino Unwrapping, a graph-based propagation method that exploits reliable connections while accounting for path quality and critical bottlenecks.

Building on this classical analysis, we then investigated different routes toward quantum phase unwrapping, including hybrid classical-quantum optimization and gate-based variational approaches. We selected quantum annealing as the proof-of-concept approach because the phase-ambiguity problem can be expressed as a discrete quadratic optimization problem and naturally mapped to a QUBO/Ising formulation. We implemented the complete software workflow and tested it on real quantum-annealing hardware. The experiments produced high-quality reconstructions for tractable instances and exposed the main present scaling limitation: the rapid growth of binary variables with time-series length and ambiguity bit depth. To address this limitation, we introduced a blockwise formulation that decomposes long time series into smaller QUBO subproblems and naturally supports parallel execution. The project therefore provides both new classical phase-unwrapping tools and an experimentally validated path toward quantum phase unwrapping

Detailed description

Research objective: from classical phase unwrapping to quantum optimization

Persistent Scatterer MT-InSAR reconstructs deformation histories from repeated radar acquisitions. A central processing step is temporal phase unwrapping: each observed phase sample is known only within a principal interval, while the physically useful time series requires the correct integer number of phase cycles to be assigned at each epoch. The long-term objective of this project is not only to improve this classical reconstruction but to understand how the problem can be expressed in a form that quantum optimization methods can solve. For this reason, the activity starts with the mathematical structure of classical phase unwrapping and progressively isolates the elements relevant to a quantum formulation: integer ambiguity variables, quadratic continuity or model-consistency costs, admissible ranges, coupling terms, and opportunities for problem decomposition.

Why temporal processing remains essential in MT-InSAR

As illustrated in Fig. 1, Persistent Scatterers do not lie on a regular spatial grid. Their density depends on the availability of stable radar targets and can vary strongly across the scene. Some PSs have several reliable neighbors, whereas others belong to sparse or isolated components. Spatial support, therefore, cannot be assumed everywhere, and temporal phase unwrapping remains the fundamental reconstruction mechanism for individual PSs. The same irregular geometry makes connectivity itself a relevant quantity to study before adding spatial couplings to classical or quantum formulations, since the graph structure indicates where spatial information can reliably complement temporal unwrapping and where temporal information must remain dominant.

Figure 1. Example of the spatial connectivity of Persistent Scatterers in an MT-InSAR dataset. PS color represents estimated deformation velocity, while blue edges indicate spatial connections. The network includes dense regions, sparse areas, and isolated components, illustrating the non-uniform availability of spatial support.

Figure 2 summarizes the research path followed during the project. The activity starts from the classical MT-InSAR phase-unwrapping problem, uses classical and complementary investigations to identify the relevant mathematical structure, and progressively moves toward an optimization formulation, QUBO encoding, and quantum annealing implementation. At the same time, this quantum-oriented investigation generated classical outcomes with independent value, including self-diagnostic tools, graph analysis, Domino Unwrapping, and machine-learning exploration.

Figure 2. Research path toward quantum phase unwrapping. Classical analysis and complementary investigations support the transition from the MT-InSAR ambiguity problem to an optimization/QUBO formulation and quantum annealing proof of concept, while also generating additional classical outcomes.

Classical results emerging from the quantum-oriented analysis

An important outcome of the project is that the investigation into quantum phase unwrapping also generated new results on the classical side. From the continuity-based analysis, we developed an adaptive Minimum-Path-Length (MPL) method. At the same time, the model-based investigation led to an adaptive piecewise strategy able to better follow changes in the temporal evolution of the phase. These methods provide practical classical solutions while retaining a clear connection with the underlying optimization structure.

A further contribution is a reference-free self-diagnostic strategy that evaluates the internal consistency of the reconstructed solution without requiring an externally unwrapped ground truth (Fig. 3). Depending on the processing branch, the diagnostic can exploit residual consistency, coherence-related information, or disagreement between complementary reconstruction principles. Its purpose is operational: it identifies potentially unreliable PS time series or temporal intervals and allows more expensive processing to be concentrated only on uncertain cases.

Figure 3. Qualitative illustration of the self-diagnostic refinement process. Panel (a) shows the wrapped input, while panels (b)-(d) show successive reconstruction stages in which inconsistencies are detected and the phase trajectory is progressively corrected.

Simulation and machine-learning exploration

As part of the classical analysis preceding the quantum formulation, we also explored a data-driven approach to temporal phase unwrapping. We developed a simulator that generates synthetic MT-InSAR time series under controlled conditions, including different deformation trends, nonlinear temporal evolutions, seasonal components, phase wrapping, and variable noise levels. Because the underlying unwrapped trajectory is known by construction, the simulator provides reference data for systematic training, testing, and comparison across progressively more difficult conditions. On these data, we developed a custom cascaded LSTM architecture trained end-to-end as a single model. It contains two sequential components with complementary roles: a regression component followed by a classification component, each with 96 units. Numerical tests produced very good results when the time series exhibited a sufficiently well-defined temporal trend, and the model remained effective even at high noise levels. When the temporal evolution was weak or poorly defined, however, unnecessary ambiguity corrections could occasionally be introduced, producing over-unwrapping. The experiments, therefore, showed that data-driven reconstruction depends not only on noise level but also on the presence of identifiable temporal structure. This activity complements the optimization-based work and provides a common controlled environment for studying classical, machine-learning, and quantum-oriented phase-unwrapping strategies.

Spatial support between neighboring Persistent Scatterers

The project also investigated how spatial information can complement temporal phase unwrapping without making it a mandatory assumption. Candidate neighbors are selected based on geometric proximity and the consistency of their wrapped phase histories, and pairwise contributions can be weighted by circular coherence. Strongly related PSs, therefore, provide a larger spatial constraint than weakly related ones, while isolated or poorly connected PSs continue to rely mainly on temporal processing. From the quantum perspective, this analysis is also useful because spatial coupling increases the number of interactions that must be represented in the optimization problem. Understanding when and where spatial links are actually informative is therefore important for deciding where spatial support can complement the temporal solution, while avoiding unnecessary couplings and supporting scalable classical and quantum formulations.

Graph representation and network statistics

To make the spatial structure explicit, the PS configuration was represented as a graph in which nodes correspond to Persistent Scatterers and edges encode admissible spatial relations. The graph is first used as a characterization tool (Fig. 4). Edge statistics describe the available relations through quantities such as edge length, phase coherence, and edge cost. Node statistics describe local connectivity through degree, number of usable neighbors, and the presence of isolated or weakly connected PSs. Connected-component statistics describe the global fragmentation of the network through the number and size of components and the fraction of PSs contained in the main or smaller components. These statistics have a direct practical role: they can guide the choice of maximum linking distance, coherence threshold, neighborhood size, and spatial weights, and they indicate where spatial phase unwrapping is meaningful and where temporal information must dominate. From a longer-term quantum perspective, the same connectivity analysis helps identify sparse interaction structures and natural subdivisions of the problem that may reduce unnecessary couplings and facilitate decomposition.

Figure 4. Example of PS-graph representation together with representative statistics for edges, nodes, and connected components. The panels show a connectivity graph of Persistent Scatterers and illustrative quantities related to edge coherence, edge distance, isolated nodes, and component structure.

Domino Unwrapping: a graph-based classical result

A second result that emerged from the graph analysis is Domino Unwrapping. This is a classical propagation strategy rather than a quantum solver, but it originated from the same effort to understand how to decompose a large phase-unwrapping problem into smaller, controlled operations. Starting from one or more reliable seed PSs, the method propagates unwrapped information across selected graph paths. The central issue is not only whether two PSs are connected, but how reliable the complete propagation route is. For this reason, Domino Unwrapping introduces path-dependent quantities that are distinct from the general edge, node, and component statistics: number of hops, accumulated path cost, minimum or bottleneck coherence, weak or critical bridges, and the size of the downstream region that depends on a given connection. These indicators can support seed selection, path comparison, component splitting, and the detection of routes where a local error could propagate to many PSs. Domino Unwrapping is therefore an example of a useful classical innovation generated while studying decomposition and connectivity for the broader quantum objective.

Selection of the quantum-computing approach

Before fixing the hardware implementation, the project examined different routes for transferring phase unwrapping to quantum computing. Candidate directions included hybrid classical-quantum optimizers and gate-based variational methods, including VQE-style minimization of a cost Hamiltonian and QAOA-like combinatorial optimization. These approaches remain scientifically relevant, but they require a parameterized circuit, repeated measurements, and a classical optimization loop. At the same time, the size and connectivity of the binary phase-ambiguity problem can rapidly increase with time-series length and ambiguity range. Quantum annealing was therefore selected as the most direct platform for the present proof of concept. After binary encoding, the unknown phase ambiguities are discrete variables, and the main temporal, model, and spatial consistency terms can be written as quadratic interactions. The resulting objective can be transferred directly to a QUBO and then to the corresponding Ising form used by the annealer. This route avoids circuit design and makes the principal scaling issues explicit through logical variable count, coupling structure, embedding complexity, and sampling quality. Hybrid strategies remain important in the overall architecture, particularly for decomposition, diagnostics, preprocessing, and postprocessing.

Once quantum annealing was selected, the implementation followed the hybrid processing scheme illustrated in Fig. 5. Classical preprocessing adapts the optimization problem and performs the logical-to-physical embedding required by the QPU; the annealing stage samples candidate solutions; and classical post-processing reconstructs and evaluates the phase-unwrapping result. The relative-time bar is schematic and emphasizes that the QPU call is only one component of the complete computational chain.

Figure 5. A hybrid quantum annealing workflow was adopted for the implementation. Problem adaptation and embedding are performed before QPU execution, while classical post-processing reconstructs and evaluates the returned solution. The relative processing times are shown schematically.

QUBO formulation and implementation on quantum annealing hardware

The selected quantum branch reformulates temporal phase unwrapping as a bounded integer quadratic optimization problem. The unknown cycle ambiguities are represented by integer variables, while temporal continuity and, where appropriate, model or spatial-consistency terms are expressed as quadratic contributions to the objective. The bounded integers are converted into binary variables, yielding the QUBO used by the quantum annealer. A complete software workflow was implemented to build the QUBO, submit it to quantum-annealing hardware, retrieve samples, reconstruct the integer ambiguities, and compare the resulting phase histories with classical solutions and samplers. The code was therefore tested beyond simulation: real quantum annealing runs were performed as part of the proof-of-concept activity. For compact instances, the annealer produced high-quality reconstructions consistent with the expected unwrapped trajectories. The tests included time series with strong non-linear deformation, seasonal behavior, and substantial noise, showing that the formulation can represent realistic temporal structures rather than only simplified synthetic cases (Fig. 8). These results provide a successful proof of concept for quantum phase unwrapping. They are not interpreted as evidence of quantum advantage, but as experimental validation that the MT-InSAR ambiguity problem can be mapped, embedded, solved, and reconstructed on current quantum-annealing hardware.

Scaling limitation: ambiguity bit depth, QUBO size, and blockwise decomposition

The hardware experiments identified the principal current limitation: the number of binary variables grows with both time-series length and the number of bits used to represent each integer ambiguity. Higher bit depth enlarges the logical problem and can make physical embedding difficult even when the nominal variable count appears compatible with the processor; larger formulations can also reduce sampling robustness. This observation motivated a blockwise formulation in which long curves are divided into shorter temporal segments, each mapped to a smaller QUBO, solved independently, and then merged into the complete trajectory. The approach reduces the number of variables handled in each quantum call and naturally enables parallel execution across temporal blocks and across different PSs. Blockwise decomposition, therefore, provides a scalable architecture for quantum phase unwrapping while separating algorithmic limitations from current hardware-resource limitations.

The resulting scaling strategy is illustrated in Fig. 6. A long wrapped time series is divided into shorter temporal blocks, each associated with a smaller QUBO subproblem. The independent blocks can be assigned to separate QPU runs and recombined classically after solution. This reduces the logical size of each quantum instance and creates a natural level of parallelism across both temporal blocks and different Persistent Scatterers.

Figure 6. Conceptual blockwise and parallel quantum phase-unwrapping architecture. A long, wrapped time series is decomposed into smaller QUBO blocks, solved in parallel on QPU runs, and then recomposed classically into the complete unwrapped trajectory.

Figure 7 complements this algorithmic view with the hardware-mapping perspective. The logical QUBO connectivity must be represented on the physical connectivity graph of the quantum processor, typically through chains of physical qubits. The example also illustrates how multiple compact embeddings can occupy different regions of the QPU, thereby supporting a parallel execution strategy when the hardware topology and available resources permit.

Figure 7. Example of logical-to-physical embedding on the quantum annealer. The logical QUBO graph is mapped onto connected physical-qubit structures, and multiple compact embeddings can be distributed over the available QPU topology to support parallel solution of independent subproblems.

Figure 8. Representative phase-unwrapping examples from the proof-of-concept tests. For each case, the upper panel shows the wrapped input and the lower panel shows the corresponding unwrapped estimate.

Project Outcomes

Development outputs

The main development outputs are organized as four complementary software repositories. They collect the software developed during the collaboration and separate the project's main methodological components. Repository links can be added when the corresponding releases are made available.

• Classical Temporal Phase-Unwrapping Repository: MPL-based and model-based temporal phase-unwrapping methods, classical benchmarking utilities, and the reference-free self-diagnostic method. Repository: [link to be added].
• Graph-Based Spatial and Domino Unwrapping Repository: PS-graph construction, coherence-weighted spatial support, edge/node/connected-component statistics, spatial-parameter analysis, and Domino Unwrapping tools, including path and bottleneck diagnostics. Repository: [link to be added].
• MT-InSAR Simulation and Machine-Learning Repository: a synthetic MT-InSAR time-series simulator and the custom cascaded LSTM model combining jointly trained regression and classification components. Repository: [link to be added].
• Quantum Phase-Unwrapping Repository: bounded-integer and QUBO formulations, binary encoding, classical-sampler validation, quantum-annealing execution, reconstruction tools, and blockwise/parallel processing strategies. Repository: [link to be added].

Preprints, manuscripts, and publications

Scientific manuscripts have been prepared or are being prepared from the activities undertaken during the collaboration. Until a public preprint or peer-reviewed publication is available, they are listed here as manuscripts in preparation.

• PLL-Inspired Adaptive Temporal Phase Unwrapping for Persistent-Scatterer MT-InSAR Time Series — manuscript in preparation; preprint/publication link to be added when available.
• Temporal Phase Unwrapping as a QUBO Problem: a Quantum-Annealing Formulation for InSAR Time Series — manuscript in preparation; preprint/publication link to be added when available.