Geometry- and Topology-Informed Quantum Computing: From States to Real‑Time Control with FPGA Prototypes

ArXiv pre‑print: https://arxiv.org/abs/2601.09556

Quantum computing seeks to harness the principles of quantum mechanics to solve problems intractable for classical computers, yet realizing this potential demands a cohesive approach linking theoretical states to practical, real‑time control systems. Gunhee Cho, alongside colleagues, addresses this challenge in their work exploring the intersection of geometry, topology, and quantum computation. They present a hardware‑aware pathway through quantum information workflows, connecting states, circuits, and measurement to deterministic classical pipelines. This research is significant because it moves beyond purely theoretical frameworks, detailing the implementation of quantum algorithms and error correction using field‑programmable gate array (FPGA) prototypes, and offering a crucial step toward building functional, hybrid quantum systems. The authors demonstrate how understanding quantum phenomena through geometric and topological lenses can facilitate low‑latency control and robust error mitigation, ultimately paving the way for more reliable and scalable quantum technologies.

The research investigates a geometric interpretation of quantum evolution and measurement, framing them as motion within curved spaces and statistical processes respectively. This approach leverages concepts from differential geometry, quantum Fisher information geometry, and applies them to the design of quantum circuits. The methodology translates this geometric viewpoint into an FPGA framework, representing circuits as data‑flow graphs and hybrid loops as streaming pipelines. This allows for the parsing, aggregation, and reduction of measurement outcomes into concise linear‑algebra updates which then dictate subsequent pulse scheduling. Further development explores multi‑qubit structures and entanglement, treating these as geometric and computational elements, demonstrated through applications including quantum teleportation, superdense coding, entanglement detection, and implementation of Shor’s algorithm via quantum phase estimation. The final section concentrates on topological error correction and the development of real‑time decoding techniques.

FPGA Implementation of Stabilizer Surface Code Decoding

The work details a comprehensive approach to quantum computing, spanning theoretical foundations to practical hardware implementation, with a particular focus on field‑programmable gate arrays (FPGAs). Track A of the study centers on stabilizer codes and surface‑code decoding, progressing from topological concepts through graph‑based algorithms to detailed real‑time level (RTL) design considerations. This involves rigorous verification using testbenches, fault‑injection techniques, and a regression methodology, all geared toward achieving product‑level metrics such as bounded latency, predictable performance, fail‑closed policies, and comprehensive observability. Experimental procedures within Track A translate the surface‑code decoding process into a finite‑state‑machine (FSM) architecture suitable for FPGA implementation.

The Union‑Find decoder is examined in detail, with a progression from its underlying data structure to a complete RTL pipeline. Emphasis is placed on the host interface and control‑stack integration, ensuring efficient handling of syndrome input and correction output. Hands‑on examples utilize the Lattice iCEstick FPGA for switch‑to‑bit conditioning and fixed‑point phase arithmetic. Track C explores quantum cryptography (BB84 and E91 protocols), analyzing parameters such as quantum bit error rate (QBER) and defining abort rules for secure communication. Privacy amplification and post‑quantum cryptographic themes are also investigated, highlighting how protocol logic can be naturally expressed using FSMs, counters, and hash pipelines.

The research demonstrates how to move from geometric principles to implementable systems, offering visualization‑driven FPGA examples to facilitate understanding. Further investigation covers multi‑qubit systems and entanglement, employing algebraic geometry (Segre varieties) to explore entanglement properties. Algorithms such as teleportation, superdense coding, entanglement detection, and Shor’s algorithm via quantum phase estimation are examined. The work also details streaming linear‑algebra techniques for hybrid quantum algorithms, bridging the gap between theoretical concepts and practical hardware acceleration using FPGAs, and providing a pathway for deterministic pipelines in hybrid workflows.

Real‑Time Quantum Error Correction via Geometry‑First Pipelines

Scientists have established a geometry‑first, hardware‑aware approach to information workflows in quantum computing, focusing on connecting states, circuits, and measurement to deterministic classical pipelines. The research details a system where evolution is understood as motion on curved spaces and measurement is interpreted through statistical analysis, forming the foundation for hybrid quantum‑classical systems. Experiments demonstrate the critical importance of low‑latency, low‑jitter streaming in managing measurement outcomes, parsing, aggregating, and reducing data to linear‑algebra updates that schedule subsequent pulses. The team developed a real‑time quantum error‑correction (QEC) decoder as a deployable infrastructure component, processing syndrome streams into correction decisions within bounded timeframes.

Core to this decoder is the handling of measurement bit streams indexed by space and time, processed using local update rules and stored in structured memory formats such as arrays and FIFOs. Measurements confirm the decoder satisfies a strict contract encompassing correctness against a golden model, adherence to a defined deadline, backlog stability under load, observability through emitted traces, and explicit interface definitions. Further work built geometry‑aware optimization tools for variational circuits, linking parameters to circuit outputs via measurement statistics and utilizing the quantum Fisher information matrix (QFIM) and quantum natural gradient (QNG) updates. Results show QFIM/QNG provides a principled preconditioner, a diagnostic for barren plateaus, and a bridge between information geometry and hardware calibration.

Reproducible scripts were generated to compute QFIM for small ansätze, demonstrating QNG’s performance advantage over Euclidean updates in certain scenarios. Additionally, scientists treated quantum cryptography and post‑processing as streaming verification tasks, encompassing raw key outcomes, sifting, QBER estimation, error correction, and privacy amplification. This pipeline, designed with strict correctness guarantees, utilizes streaming data, fixed message formats, and versioned schemas. By the end of the work, a complete BB84 simulation with logged metrics and a streaming post‑processing design suitable for hardware implementation were achieved, alongside a test suite with known‑answer tests for each stage.

Quantum Workflows as Geometric Dataflow Networks

This work establishes a geometric framework for understanding quantum information workflows, connecting states, circuits, and measurement processes to deterministic classical pipelines. By leveraging differential geometry—specifically manifolds, Riemannian metrics, and geodesics—the authors demonstrate how quantum evolution can be interpreted as motion on curved spaces and measurement as statistical analysis within that geometric landscape. This approach reframes quantum circuits as data‑flow graphs, emphasizing the importance of low‑latency processing for efficient hybrid systems. The research extends this geometric treatment to multi‑qubit systems, representing entanglement as a geometric and computational property, and successfully applies it to algorithms such as Shor’s phase estimation.

A significant contribution lies in the detailed analysis of topological error correction, particularly surface‑code decoding, presented as a progression from topology to graph algorithms and ultimately to microarchitectural implementation. The authors acknowledge limitations stemming from the complexity of modeling real‑world hardware constraints and noise, and suggest future work could focus on refining these models and exploring more advanced error‑correction schemes. They also detail how measurement sensitivity is maximized by equatorial measurements or equivalent basis changes, offering practical guidance for optimizing quantum systems. While the framework relies on idealized mathematical models, the inclusion of iCEstick labs demonstrates a commitment to bridging the gap between theoretical principles and implementable systems. Future research directions involve further exploration of streaming post‑processing techniques and the development of more robust and efficient decoding algorithms for practical quantum computation.