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A standard argument (e.g., the mixed product property of Kronecker products, or a generalization of the Fast Walsh-Hadamard transform) shows that any such $N \\times N$ matrix has a depth-2 circuit of size $O(N^{1.5})$. We improve on this for all such matrices, and especially for some such matrices of particular interest:\n  - For any integer $q > 1$ and any matrix which is the Kronecker power of a fixed $q \\times q$ matrix, we construct a depth-2 circuit "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2211.05217","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2022-11-09T21:58:36Z","cross_cats_sorted":[],"title_canon_sha256":"96e6f71c2d7eaa38e217590686d4daf13abd16fd47c93069f1744d62414c3e32","abstract_canon_sha256":"33f66c1e34be13f99afc9655c4f824dd8426275b5fbd7f05180a47417b4bb1e8"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T05:15:01.448234Z","signature_b64":"gRSaFsdL0ZwraY44u24jz9H/1kLl3J3dUu8jFbolyKoy9RRuPj3ydIWnv/RyJO/Vg5+dz1IKx9GV5vRz5LBUBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2db0f776b2309826cf817083d8f706531705150491db528f3c1d29a5a4964cfc","last_reissued_at":"2026-07-05T05:15:01.447786Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T05:15:01.447786Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Smaller Low-Depth Circuits for Kronecker Powers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.DS","authors_text":"Ashwin Padaki, Josh Alman, Yunfeng Guan","submitted_at":"2022-11-09T21:58:36Z","abstract_excerpt":"We give new, smaller constructions of constant-depth linear circuits for computing any matrix which is the Kronecker power of a fixed matrix. 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