{"paper":{"title":"(A Variant of) Clifford Circuit Synthesis is NP-Complete","arxiv_id":"2610.02029","primary_cat":"quant-ph","submitted_at":"2026-10-01T16:47:31+00:00","version_compiled":"v1"},"notice":"Every quoted sentence is the authors' text from the arXiv listing abstract. Labels and probabilities are the outputs of classifier-1.13.0 on the question bank abstract-p1-20260923, exactly as returned. 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design.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.03,"main_claim_p":0.01,"quantitative":false,"quantitative_p":0.08,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.0,"background":1.0,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.0}},{"i":2,"p":0.79,"id":"s2","own":false,"role":"gap","text":"While the classical version is very well understood -- both in terms of heuristics and rigorous hardness assertions -- much less is known about optimal quantum circuit synthesis.","hedge":{"p":0.99,"label":"asserted"},"own_p":0.04,"main_claim_p":0.01,"quantitative":true,"quantitative_p":0.69,"role_probabilities":{"gap":0.79,"method":0.0,"result":0.0,"background":0.21,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.0}},{"i":3,"p":0.95,"id":"s3","own":false,"role":"background","text":"We focus on optimal Clifford circuit synthesis, for which various heuristics are known, e.g. via reduction to 3-SAT.","hedge":{"p":0.98,"label":"asserted"},"own_p":0.06,"main_claim_p":0.03,"quantitative":false,"quantitative_p":0.17,"role_probabilities":{"gap":0.03,"method":0.0,"result":0.0,"background":0.95,"limitation":0.0,"future_work":0.0,"contribution":0.02,"interpretation":0.0}},{"i":4,"p":0.92,"id":"s4","own":true,"role":"result","text":"Our main result supplies a matching hardness result: a variant of optimal Clifford synthesis is NP-hard.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.96,"main_claim_p":0.97,"quantitative":false,"quantitative_p":0.3,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.92,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.08,"interpretation":0.0}},{"i":5,"p":0.99,"id":"s5","own":true,"role":"method","text":"The proof proceeds in two parts: (i) reduce 3-edge colorability on 3-regular graphs to a circuit synthesis problem that only involves CZ gates, (ii) prove that the availability of additional elementary Clifford gates -- most notably: Hadamard, phase and CNOT -- cannot lead to further improvements of the optimal circuit depth.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.93,"main_claim_p":0.09,"quantitative":true,"quantitative_p":0.53,"role_probabilities":{"gap":0.0,"method":0.99,"result":0.0,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.01,"interpretation":0.0}},{"i":6,"p":0.57,"id":"s6","own":true,"role":"interpretation","text":"Our work sharpens the complexity-theoretic understanding of Clifford circuits: simulation and equivalence checking are in P, whereas deciding whether a Clifford unitary admits an implementation within a prescribed depth is NP-complete.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.85,"main_claim_p":0.5,"quantitative":true,"quantitative_p":0.77,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.41,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.02,"interpretation":0.57}}]},"compiled":{"main_claim":{"p":0.91,"text":"Our main 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