{"paper":{"title":"Efficient unitary paths and quantum computational supremacy: A proof of average-case hardness of Random Circuit Sampling","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cond-mat.str-el","hep-th","math-ph","math.MP"],"primary_cat":"quant-ph","authors_text":"Ramis Movassagh","submitted_at":"2018-10-10T18:00:07Z","abstract_excerpt":"One-parameter interpolations between any two unitary matrices (e.g., quantum gates) $U_1$ and $U_2$ along efficient paths contained in the unitary group are constructed. Motivated by applications, we propose the continuous unitary path $U(\\theta)$ obtained from the QR-factorization \\[ U(\\theta)R(\\theta)=(1-\\theta)A+\\theta B, \\] where $U_1 R_1=A$ and $U_2 R_2=B$ are the QR-factorizations of $A$ and $B$, and $U(\\theta)$ is a unitary for all $\\theta$ with $U(0)=U_1$ and $U(1)=U_2$. The QR-algorithm is modified to, instead of $U(\\theta)$, output a matrix whose columns are proportional to the corre"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1810.04681","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"}