{"paper":{"title":"Complexity scaling and optimal policy degeneracy in quantum reinforcement learning via analytically solvable unitary-control-then-measure models","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Quantum RL with unitary control and measurement reduces expected return complexity from exponential to power-law scaling while revealing distinct optimal policy degeneracy patterns.","cross_cats":[],"primary_cat":"math.GM","authors_text":"Alessandro Michelangeli, Andrea Cintio, Dmitrii Tsutskov","submitted_at":"2026-04-09T17:42:37Z","abstract_excerpt":"We propose and analyse a class of analytically solvable models of quantum reinforcement learning (QRL), formulated as finite-horizon Markov decision processes in finite-dimensional Hilbert spaces. The models are built around a `unitary-control-then-measure' protocol, in which a learning agent applies unitary transformations to a quantum state and interleaves each control step with a projective measurement onto a prescribed reference basis. Exact closed-form expressions for trajectory probabilities, rewards, and the expected return are derived for four concrete realisations: a closed-chain and "},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"we identify and quantify a two-level reduction in the computational complexity of the expected return, from the nominally exponential O(e^N) scaling in the trajectory length N to an explicit power-law O(N^I): a trajectory-based level, arising from equivalence classes of paths sharing the same unordered state counts and transition frequencies, and a policy-based level, arising from the sparsity of the transition graph enforced by constrained unitary actions.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The specific choice of unitary controls and projective measurements onto a fixed reference basis in finite-dimensional spaces is assumed to permit closed-form derivations and to capture the essential complexity and degeneracy features relevant to broader quantum RL.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Analytically solvable QRL models reduce expected-return computation from O(e^N) to O(N^I) via path equivalence and transition sparsity, while exhibiting unique optima governed by Zeno effect or discrete/plateau degeneracy at critical energies.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Quantum RL with unitary control and measurement reduces expected return complexity from exponential to power-law scaling while revealing distinct optimal policy degeneracy patterns.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"7118ef9dfdb8a08702cf8d946336aa667c86e0ba72b74cbf4f22f60da15fbfc6"},"source":{"id":"2604.13096","kind":"arxiv","version":2},"verdict":{"id":"4d6a153c-2976-4c6e-916a-139fd195dc8f","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-10T17:04:50.361582Z","strongest_claim":"we identify and quantify a two-level reduction in the computational complexity of the expected return, from the nominally exponential O(e^N) scaling in the trajectory length N to an explicit power-law O(N^I): a trajectory-based level, arising from equivalence classes of paths sharing the same unordered state counts and transition frequencies, and a policy-based level, arising from the sparsity of the transition graph enforced by constrained unitary actions.","one_line_summary":"Analytically solvable QRL models reduce expected-return computation from O(e^N) to O(N^I) via path equivalence and transition sparsity, while exhibiting unique optima governed by Zeno effect or discrete/plateau degeneracy at critical energies.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The specific choice of unitary controls and projective measurements onto a fixed reference basis in finite-dimensional spaces is assumed to permit closed-form derivations and to capture the essential complexity and degeneracy features relevant to broader quantum RL.","pith_extraction_headline":"Quantum RL with unitary control and measurement reduces expected return complexity from exponential to power-law scaling while revealing distinct optimal policy degeneracy patterns."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.13096/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}