{"paper":{"title":"Logarithmic-Depth Quantum Circuits for Hamming Weight Projections","arxiv_id":"2404.07151","primary_cat":"quant-ph","submitted_at":"2024-04-10T16:35:36+00:00","version_compiled":"v3"},"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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weight.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.08,"main_claim_p":0.03,"quantitative":false,"quantitative_p":0.12,"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":1.0,"id":"s2","own":true,"role":"contribution","text":"In this work, we propose several quantum algorithms that realize a coherent Hamming weight projective measurement on an input pure state, meaning that the post-measurement state of the algorithm is the projection of the input state onto the corresponding subspace of fixed Hamming weight.","hedge":{"p":0.98,"label":"asserted"},"own_p":0.9,"main_claim_p":0.33,"quantitative":false,"quantitative_p":0.04,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.0,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":1.0,"interpretation":0.0}},{"i":3,"p":0.8,"id":"s3","own":true,"role":"contribution","text":"We analyze a depth-width trade-off for the corresponding quantum circuits, allowing for a depth reduction of the circuits at the cost of more control qubits.","hedge":{"p":0.87,"label":"asserted"},"own_p":0.88,"main_claim_p":0.12,"quantitative":false,"quantitative_p":0.23,"role_probabilities":{"gap":0.0,"method":0.16,"result":0.04,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.8,"interpretation":0.0}},{"i":4,"p":0.99,"id":"s4","own":true,"role":"result","text":"For an $n$-qubit input, the depth-optimal algorithm uses $O(n)$ control qubits and the corresponding circuit has depth $O(\\log (n))$, assuming that we have the ability to perform qubit resets.","hedge":{"p":0.99,"label":"asserted"},"own_p":0.9,"main_claim_p":0.56,"quantitative":true,"quantitative_p":0.98,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.99,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.01,"interpretation":0.0}},{"i":5,"p":0.65,"id":"s5","own":true,"role":"method","text":"Furthermore, the proposed 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