Rubik-Inspired GraphRAG Loops: Autonomous AI Planning and Quantum Gate Optimization
Keywords:
Agentic AI, Cayley graphs, Cyber resilience, NISQ computing, Permutation groups, Quantum gate optimisation, Zero-trust securityAbstract
Autonomous artificial intelligence agents deployed in nondeterministic operational environments routinely encounter NP-hard planning challenges in which state permutation spaces grow exponentially with problem scale. Autoregressive large language model (LLM) reasoning chains degrade under such conditions, exhibiting compounding token drift, context-window saturation, and elevated rates of invalid state transitions. This paper introduces the Group-Theoretic GraphRAG Loop (GT-GRL) framework, a mathematically grounded architecture that unifies group-theoretic Cayley graph representations, structural Retrieval-Augmented Generation (GraphRAG), and cyclic multi-agent state machines implemented with LangGraph. The central analogy drawn from Rubik’s Cube combinatorics is that a permutation space of configurations is navigable in at most twenty deterministic moves by exploiting the algebraic structure of the underlying group. GT-GRL models decision domains as algebraic permutation groups and projects high-dimensional problem topologies onto structured Cayley graphs. A cyclic, self-correcting agent loop executes subgroup decomposition and coset navigation, collapsing state spaces exceeding permutations into deterministic trajectories of ten or fewer operational steps when the stabiliser subgroup chain is precomputed from a stable generator set; dynamic recomputation when generators change introduces additional polynomial-time overhead. Framework evaluation is presented via simulated proof-of-concept models across two test domains: an enterprise zero-trust cybersecurity network and a quantum circuit gate compilation benchmark on noisy intermediate-scale quantum (NISQ) hardware. In a controlled synthetic cybersecurity simulation, GT-GRL demonstrated theoretical threat containment in 185 ms (simulation-internal clock time, excluding network and API propagation overhead) with a simulated false-positive rate of 0.02%. In simulated quantum compilation models, the framework achieved up to a 42% reduction in circuit depth relative to standard heuristic compilers on the tested benchmark suite. In conclusion, this paper further outlines theoretical and conceptual frameworks for extending algebraic planning to four additional candidate domains: multi-hop financial fraud interdiction, retrosynthetic pharmaceutical pathway optimisation, post-quantum cryptography migration, and industrial robotics trajectory planning.
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