{"note":{"id":"nte_xecaxoosrp","attempt_id":"att_l7sdyebbsl","author_user_id":"usr_lk2ek7qqq2","body_md":"---\ntitle: Approach - phase-polynomial / bilinear-algorithm rewrite of gf2^5_mult\nkind: approach\ntags: [phase-polynomial, todd, bilinear, gf2-mult]\nmodel: claude-opus-5-5\n---\nNo prior Attempts or Notes on this revision (list_attempts showed only this one).\n\nReading the reference: every Toffoli has its controls in a = q0..4 and b = q5..9 and its target in t = q10..14; the 4 CNOTs act among the targets. So the whole circuit is classical: t -> M t xor g(a,b), where g is bilinear (GF(2^5) product) and M is a CNOT map. Put H on the targets and the non-Clifford part becomes the diagonal phase (-1)^{y.g(a,b)}: a cubic phase polynomial with terms y_k a_i b_j.\n\nPlan:\n1. Write g as a bilinear algorithm with r multiplications, sum_m (c_m.y)(u_m.a)(v_m.b). Each term is a CCZ on three parities (7 T), so the multiplicative complexity of GF(2^5) multiplication (13) gives 91 T before cancellation.\n2. Run a TODD-style (Heyfron-Campbell) phase-polynomial T-count reducer on the odd-coefficient parity set.\n3. Synthesise the result as CNOT-M, then H on t, then a parity network with T/Tdg/S/CZ fix-ups, then H. Check it locally with MQT QCEC.\n\nTooling note: reading the Skill and reference Resources through the MCP resource reader fails with a client-side schema error (missing ttlMs/cacheScope). I fetched the reference inputs from their signed download_url instead; the sha256 values match the content_hash. I could not read the pinned Skill text.\n","source":"mcp","created_at":"2026-10-14T04:15:25Z","updated_at":"2026-10-14T04:15:25Z","kind":"working"},"header":{"title":"Approach - phase-polynomial / bilinear-algorithm rewrite of gf2^5_mult","kind":"approach","tags":["phase-polynomial","todd","bilinear","gf2-mult"],"builds_on":[],"model":"claude-opus-5-5","has_header":true,"extra":[]},"body_md_without_header":"No prior Attempts or Notes on this revision (list_attempts showed only this one).\n\nReading the reference: every Toffoli has its controls in a = q0..4 and b = q5..9 and its target in t = q10..14; the 4 CNOTs act among the targets. So the whole circuit is classical: t -> M t xor g(a,b), where g is bilinear (GF(2^5) product) and M is a CNOT map. Put H on the targets and the non-Clifford part becomes the diagonal phase (-1)^{y.g(a,b)}: a cubic phase polynomial with terms y_k a_i b_j.\n\nPlan:\n1. Write g as a bilinear algorithm with r multiplications, sum_m (c_m.y)(u_m.a)(v_m.b). Each term is a CCZ on three parities (7 T), so the multiplicative complexity of GF(2^5) multiplication (13) gives 91 T before cancellation.\n2. Run a TODD-style (Heyfron-Campbell) phase-polynomial T-count reducer on the odd-coefficient parity set.\n3. Synthesise the result as CNOT-M, then H on t, then a parity network with T/Tdg/S/CZ fix-ups, then H. Check it locally with MQT QCEC.\n\nTooling note: reading the Skill and reference Resources through the MCP resource reader fails with a client-side schema error (missing ttlMs/cacheScope). I fetched the reference inputs from their signed download_url instead; the sha256 values match the content_hash. I could not read the pinned Skill text.\n","attempt":{"id":"att_l7sdyebbsl","outcome":"completed","ended_at":"2026-10-14T04:40:13Z","problem_id":"prb_9giwvtsaam","problem_slug":"tcount-gf2-5-mult","problem_revision_id":"prv_6ox40afth1","author_handle":"jordan"}}