{"paper":{"title":"Optimal Bounds, Barriers, and Extensions for Non-Hermitian Bivariate Quantum Signal Processing","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Bivariate quantum signal processing establishes a tight query complexity of Θ(β_I T + log(1/ε)/log log(1/ε)) for anti-Hermitian Hamiltonian simulation.","cross_cats":["cs.CC","cs.DS"],"primary_cat":"quant-ph","authors_text":"Joshua M. Courtney","submitted_at":"2026-05-12T19:03:36Z","abstract_excerpt":"Multivariate quantum signal processing (M-QSP) has recently been shown to be applicable for non-Hermitian Hamiltonian simulation, opening several problems regarding the optimization landscape, angle-finding, and constant-factor analysis. We resolve several of these problems here. We find the anti-Hermitian query complexity $d_I = \\Theta(\\betaI T + \\log(1/\\varepsilon)/\\log\\log(1/\\varepsilon))$ to be tight, established via Chebyshev coefficient bounds, modified Bessel function asymptotics, and Lambert~$W$ inversion. Fast-forwarding to $d_I = \\mathcal{O}(\\sqrt{\\betaI T})$ is impossible in the biv"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We find the anti-Hermitian query complexity d_I = Θ(β_I T + log(1/ε)/log log(1/ε)) to be tight, established via Chebyshev coefficient bounds, modified Bessel function asymptotics, and Lambert W inversion.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The bivariate polynomial model and walk-operator oracle model assumptions continue to apply without modification to the non-Hermitian setting and that the constant-ratio condition extends to non-identical signal operators.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Tight anti-Hermitian query complexity d_I = Θ(β_I T + log(1/ε)/log log(1/ε)) is established for non-Hermitian M-QSP, with impossibility of √(β_I T) fast-forwarding, new angle-finding algorithms, and extensions to time-dependent cases.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Bivariate quantum signal processing establishes a tight query complexity of Θ(β_I T + log(1/ε)/log log(1/ε)) for anti-Hermitian Hamiltonian simulation.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"944996f23a8620ace1863c1b1afd5942097d5fcccc82f17d2dc9ff012455f896"},"source":{"id":"2605.12656","kind":"arxiv","version":1},"verdict":{"id":"d8110df5-4f00-4d96-b8c0-b39b929cae14","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-14T20:17:45.022815Z","strongest_claim":"We find the anti-Hermitian query complexity d_I = Θ(β_I T + log(1/ε)/log log(1/ε)) to be tight, established via Chebyshev coefficient bounds, modified Bessel function asymptotics, and Lambert W inversion.","one_line_summary":"Tight anti-Hermitian query complexity d_I = Θ(β_I T + log(1/ε)/log log(1/ε)) is established for non-Hermitian M-QSP, with impossibility of √(β_I T) fast-forwarding, new angle-finding algorithms, and extensions to time-dependent cases.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The bivariate polynomial model and walk-operator oracle model assumptions continue to apply without modification to the non-Hermitian setting and that the constant-ratio condition extends to non-identical signal operators.","pith_extraction_headline":"Bivariate quantum signal processing establishes a tight query complexity of Θ(β_I T + log(1/ε)/log log(1/ε)) for anti-Hermitian Hamiltonian simulation."},"references":{"count":48,"sample":[{"doi":"","year":null,"title":"Standard recursive:O(d·d R ·d I) =O(d 3)","work_id":"3cf8e6f1-2753-49c5-a17e-167243e62c1f","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"CRC-exploiting block peeling:O(dR ·d I) =O(d 2)","work_id":"23e6884e-3587-4126-9026-45ebeb29378e","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"ap- proximately2","work_id":"02db4a62-8830-442d-b522-2f2115d37e14","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"The warm-start basin guarantee (Theorem 10) requires a full-rank Jacobian, and the analytical basin radius de- cays polynomially with degree asρ analytical ∼d −4.62 (Eq","work_id":"f5152b39-36a5-4076-9bdf-12102436de2f","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"Direct-access polynomial construction.A poly- nomial designed ab initio for the direct-access model that achievesλ=e ωT (1 +o(1))on the full bitorus (Problem 42) would establish exponential advantage ","work_id":"cac23478-fd4e-4a63-a260-31063398e060","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":48,"snapshot_sha256":"46ff21561e160a104da89fb271b3498b1500a68ad93dc82fbd25c60744a66bc0","internal_anchors":1},"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"}