{"paper":{"title":"Low Rank Structure of the Reduced Transition Matrix","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"The reduced transition matrix for local observables in chaotic dual-unitary circuits admits a low-rank approximation because its entropy grows at most logarithmically in time.","cross_cats":[],"primary_cat":"quant-ph","authors_text":"Bruno Bertini, Cathy Li, Katja Klobas, Tianci Zhou","submitted_at":"2026-05-12T19:13:34Z","abstract_excerpt":"The influence-matrix formalism provides an alternative route to the classical simulation of quantum dynamics. Because influence matrices retain information only about the effective bath seen by local observables, they are expected to be easier to simulate than the full wavefunction. Recent work, however, has shown that they carry strong temporal correlations even in maximally chaotic systems, making them difficult to represent efficiently. Here we show that the reduced transition matrix, a suitable combination of influence matrices that directly determines local expectation values, can neverth"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We then prove that, for chaotic dual-unitary circuits, the associated entropy grows at most logarithmically in time. Our conclusions follow from exact results for random dual-unitary circuits and are further supported by numerical results for fixed instances of both dual-unitary and random circuits.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The truncation error is controlled by the singular-value spectrum of the reduced transition matrix, and the systems considered are chaotic dual-unitary circuits where the entropy bound holds.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"The reduced transition matrix in chaotic dual-unitary quantum circuits has low-rank structure with entropy growing at most logarithmically in time, enabling efficient approximation for local expectation values.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"The reduced transition matrix for local observables in chaotic dual-unitary circuits admits a low-rank approximation because its entropy grows at most logarithmically in time.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"4f0cf89f8a8236bec2b7e37569bc8d53c138017296bc14e81dfdbe73fbdd0195"},"source":{"id":"2605.12665","kind":"arxiv","version":1},"verdict":{"id":"c9117170-605f-4579-886f-871f762ab8c9","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-14T20:13:34.555727Z","strongest_claim":"We then prove that, for chaotic dual-unitary circuits, the associated entropy grows at most logarithmically in time. Our conclusions follow from exact results for random dual-unitary circuits and are further supported by numerical results for fixed instances of both dual-unitary and random circuits.","one_line_summary":"The reduced transition matrix in chaotic dual-unitary quantum circuits has low-rank structure with entropy growing at most logarithmically in time, enabling efficient approximation for local expectation values.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The truncation error is controlled by the singular-value spectrum of the reduced transition matrix, and the systems considered are chaotic dual-unitary circuits where the entropy bound holds.","pith_extraction_headline":"The reduced transition matrix for local observables in chaotic dual-unitary circuits admits a low-rank approximation because its entropy grows at most logarithmically in time."},"references":{"count":56,"sample":[{"doi":"","year":2003,"title":"Vidal, Efficient classical simulation of slightly entan- gled quantum computations, Phys","work_id":"5faf5e54-88dc-4fed-b94a-16d202f54115","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2011,"title":"U.Schollwöck,Thedensity-matrixrenormalizationgroup in the age of matrix product states, Ann. Phys.326, 96 (2011)","work_id":"f9883ba2-d110-4c16-8346-2ba948735e77","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2021,"title":"J. I. Cirac, D. Pérez-García, N. Schuch, and F. Ver- straete, Matrix product states and projected entangled pair states: Concepts, symmetries, theorems, Rev. Mod. Phys.93, 045003 (2021)","work_id":"4db45d1f-4e3c-43c4-b4dc-f895d0eed284","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2004,"title":"Vidal, Efficient simulation of one-dimensional quan- tum many-body systems, Phys","work_id":"e3a005e4-ab2b-4efb-a5fa-01bf942696e1","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2004,"title":"A. J. Daley, C. Kollath, U. Schollwöck, and G. Vidal, Time-dependent density-matrix renormalization-group using adaptive effective hilbert spaces, Journal of Statis- tical Mechanics: Theory and Experi","work_id":"c22a68dc-737b-45cf-9d8e-dbd3fa1d2742","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":56,"snapshot_sha256":"3a02384d2005dd1a9cdcffd86ab51c5d9455a1e5cb3c50dc2d9a76c570cfb1e9","internal_anchors":2},"formal_canon":{"evidence_count":1,"snapshot_sha256":"6e277500e7141488cf71b259393ca45c9b652b618f1b584836a36ca3cbdd5ea1"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}