Beam Retrieval Framework

• Beam retrieval framework is a unified approach integrating diffractive phase reconstruction and multi-hop evidence search by maintaining multiple hypothesis beams.
• It employs parallel expansion and scoring of candidate chains to improve accuracy and mitigate error propagation in both imaging and QA applications.
• The framework demonstrates significant performance gains, achieving up to 50% improvements in multi-hop QA and robust phase recovery across various beam modalities.

Beam Retrieval Frameworks

Beam retrieval encompasses a set of computational and experimental methodologies for extracting specific information—ranging from wavefront phase to multi-step reasoning paths—by maintaining and expanding sets of hypotheses (“beams”) throughout an iterative process. The term is rooted in both physical sciences (notably diffractive phase retrieval in optics/electron microscopy) and information retrieval (primarily multi-hop passage retrievers for complex question answering), unified by the principle of non-greedy, parallel exploration or reconstruction of plausible solutions. Representative frameworks span electron-optical phase determination (Venturi et al., 2017), multi-hop passage selection for QA (Zhang et al., 2023, Zhao et al., 2021, Wang, 25 Apr 2025, Ghassel et al., 9 Jun 2025), and post-processing algorithms for structured beam synthesis (Kingsley-Smith et al., 2023).

1. Theoretical Foundations and General Principles

Central to all beam retrieval frameworks is the maintenance of multiple partial hypotheses at each expansion stage, thus mitigating error propagation and improving the likelihood of global optimum selection. In diffractive phase retrieval (Venturi et al., 2017), interference between an unknown probe beam and a well-characterized reference beam encodes phase information in an observable intensity pattern; computational reconstruction leverages knowledge of the reference to extract the sample’s phase. In multi-hop retrieval for information systems (Zhang et al., 2023, Zhao et al., 2021, Wang, 25 Apr 2025, Ghassel et al., 9 Jun 2025), beam search—parallel expansion and scoring of candidate chains—systematically explores evidence or reasoning paths, resisting the combinatorial explosion by pruning unlikely trajectories at each stage.

The unifying algorithmic motif is that at each retrieval or inference step $t$, a pool of $B$ candidate chains (the beam) is generated by expanding all beams from the previous step with allowed candidates, scoring each by a task-specific metric (e.g., embedding similarity, phase coherence, or path likelihood), and pruning to retain the top-ranked $B$ for the next step.

2. Beam Retrieval in Diffractive Imaging and Phase Reconstruction

The classic beam retrieval use case in diffractive imaging is phase retrieval, where only beam intensities are directly measurable. In diffraction holography (Venturi et al., 2017), two side-by-side nanofabricated holograms generate a structured sample beam and a "defocused beam" reference with a known quadratic phase. The recorded intensity in the far field, given by

$$I(q) = |Ψ_{s}(q) + Ψ_\mathrm{ref}(q)|^2 = |Ψ_{s}(q)|^2 + |Ψ_\mathrm{ref}(q)|^2 + 2 \Re\{ Ψ_{s}(q) Ψ^*_\mathrm{ref}(q) \},$$

encodes the sample beam’s phase $\phi(q)$ as a phase shift of the reference’s interference fringes. Reconstruction proceeds by Fourier transforming the intensity pattern, isolating off-axis sidebands, inverse transforming, and extracting the argument, followed by compensation for the known reference phase and phase unwrapping if required. The method is robust to beam type, adaptable to electron, optical, X-ray, or acoustic domains, generalizing to any scenario where a known reference can overlap the probe in the far field.

3. Beam Search-Based Retrieval in Information Retrieval and QA Systems

In multi-step information retrieval, especially multi-hop QA, beam retrieval denotes a general framework for maintaining and iteratively expanding multiple partial evidence chains—often modeled as search paths or sequences of documents/statements.

Key architectural elements include:

• Dual or shared encoders generating dense representations for queries and candidates (Zhao et al., 2021, Zhang et al., 2023).
• Beam expansion at each hop: each partial chain in the beam is extended with every allowable next hop, scored (e.g., via similarity in embedding space), and the beam is pruned to the top $B$ expansions.
• End-to-end optimization: modern frameworks such as Beam Retrieval (Zhang et al., 2023) perform joint training over all hops, with losses accounting for correct evidence selection at each stage and optional negative sampling via the beam.

Table 1: Characteristic Steps in Multi-Hop Beam Retrieval for QA

Step	Description	Referenced Papers
Encoder computation	Map query and passage(s) to dense vectors	(Zhao et al., 2021, Zhang et al., 2023)
Beam expansion	Generate all valid next-hops/extensions for each hypothesis in beam	(Zhang et al., 2023, Wang, 25 Apr 2025)
Scoring	Assign score to each new hypothesis via neural model/composed embedding	(Zhao et al., 2021, Wang, 25 Apr 2025)
Pruning	Retain top B candidate chains for next iteration	(Zhang et al., 2023, Ghassel et al., 9 Jun 2025)
Termination	Stop after max hops or when no new improvement is observed	(Zhang et al., 2023)

The effectiveness of beam size is empirically confirmed; small increases in $B$ (e.g., $B=1\rightarrow2$) yield substantial gains in evidence chain accuracy, with diminishing returns at higher beam widths due to increased cost and marginal benefit (Zhang et al., 2023, Zhao et al., 2021, Wang, 25 Apr 2025). Notably, consistent use of the same $B$ during training and inference is crucial for optimal performance (Zhang et al., 2023).

4. Extensions: Graph and Propositional Path Beam Search

Sophisticated variants operate not just over lists of passages, but over proposition graphs or entity-linked statements. PropRAG (Wang, 25 Apr 2025) and StatementGraphRAG (Ghassel et al., 9 Jun 2025) exemplify retrieval over a structured space of propositions with explicit entity and topic connectivity.

In PropRAG, beam search is performed over a hypergraph whose nodes are propositions linked by shared entities; the retrieval objective is to maximize the aggregated embedding similarity between the query and the constructed multi-step proposition path. Each expansion considers only entity-connected neighbors and coherence is enforced via pairwise similarity thresholds. StatementGraphRAG further integrates keyword search, entity normalization, and path-attentive scoring with attention weights determined by query-statement similarity, anchoring beam expansion in both symbolic and vector space proximity.

5. Performance Outcomes and Key Empirical Trends

Across both imaging and information retrieval domains, beam retrieval frameworks substantially improve accuracy over greedy or local search alternatives. In multi-hop QA, beam-based retrievers consistently surpass prior systems not only on headline metrics such as EM (exact match) and F1, but also on supporting evidence precision—for example, Beam Retrieval achieves nearly 50% improvement over strong baselines on MuSiQue-Ans and establishes state-of-the-art on HotpotQA and 2WikiMultiHopQA at $B=2$ (Zhang et al., 2023). In proposition-path retrieval, PropRAG and StatementGraphRAG attain multi-point recall and F1 gains over triple/KG-based RAG frameworks (Wang, 25 Apr 2025, Ghassel et al., 9 Jun 2025). In diffractive phase retrieval, the beam-based interference protocol enables recovery of the full diffracted wavefront phase, unlocking physics analyses inaccessible to intensity-only data (Venturi et al., 2017).

6. Domain Generalization and Applicability

The beam retrieval paradigm is directly extensible to diverse scientific contexts:

• Phase and amplitude recovery in electron/vortex/X-ray beams via holographic interference with tailored references (Venturi et al., 2017).
• Multi-step reasoning in open- and closed-domain QA, legal analytics (hyper-relational clause chains), biological mechanisms (biomedical QA), troubleshooting, and planning (workflow synthesis from precondition-action graphs) (Wang, 25 Apr 2025, Ghassel et al., 9 Jun 2025, Zhang et al., 2023).
• Dense simulation post-processing for computational optics and nanophotonics, where multiple hypothesized incident fields are synthesized from a compact plane-wave basis (Kingsley-Smith et al., 2023).

7. Limitations and Future Directions

Known constraints include increasing computational and memory costs with beam width and chain depth; diminishing returns for large $B$; and, for current multi-hop QA frameworks, reliance on fixed candidate sets or the need for an external first-stage retriever to scale to massive corpora (Zhang et al., 2023, Zhao et al., 2021). In phase retrieval, the method’s success depends on reference beam fabrication and precise optical alignment (Venturi et al., 2017). Future work is directed at dynamic beam resizing, integrated dense retrieval plus beam selection for truly open-ended inference, and adaptive reference shaping to optimize overlap and SNR in wavefield imaging (Venturi et al., 2017, Wang, 25 Apr 2025). Extensions to new domains, including continuous learning and knowledge-intensive reasoning, are actively being explored with the explicit surrogate “chain-of-thought” formed by beam-constructed paths (Wang, 25 Apr 2025).

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References:

• “Phase retrieval of an electron vortex beam using diffraction holography” (Venturi et al., 2017)
• “End-to-End Beam Retrieval for Multi-Hop Question Answering” (Zhang et al., 2023)
• “Multi-Step Reasoning Over Unstructured Text with Beam Dense Retrieval” (Zhao et al., 2021)
• “PropRAG: Guiding Retrieval with Beam Search over Proposition Paths” (Wang, 25 Apr 2025)
• “Hierarchical Lexical Graph for Enhanced Multi-Hop Retrieval” (Ghassel et al., 9 Jun 2025)
• “Efficient post-processing of electromagnetic plane wave simulations…” (Kingsley-Smith et al., 2023)