{"paper":{"title":"The exact LCU sampling overhead of collective diagonal unitaries: resonances and a continued-fraction dichotomy","arxiv_id":"2610.03595","primary_cat":"quant-ph","submitted_at":"2026-10-02T17:00:56+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. No sentence in this record was written by a language model.","schema":"pith.compiled-record/v0","status":"compiled","abstract":{"source":"arXiv listing abstract v1","sentences":[{"i":0,"p":0.77,"id":"s0","own":false,"role":"background","text":"We consider the cost of implementing the quadratic collective phase $e^{-i\\gamma K^2}$, with $K$ a collective observable of evenly spaced spectrum, such as the permutation-symmetric Hamming weight.","hedge":{"p":0.72,"label":"asserted"},"own_p":0.23,"main_claim_p":0.03,"quantitative":false,"quantitative_p":0.07,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.0,"background":0.77,"limitation":0.0,"future_work":0.0,"contribution":0.23,"interpretation":0.0}},{"i":1,"p":0.98,"id":"s1","own":false,"role":"background","text":"This gate is the cardinality-penalty layer of constrained quantum optimization, the one-axis-twisting gate of spin squeezing and the Kerr phase of a bosonic mode.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.07,"main_claim_p":0.02,"quantitative":false,"quantitative_p":0.07,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.0,"background":0.98,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.02}},{"i":2,"p":0.57,"id":"s2","own":false,"role":"method","text":"Instead of two-qubit gates, it can be sampled as a linear combination of single-qubit rotation layers (an LCU) at an overhead $\\Gamma$.","hedge":{"p":0.83,"label":"asserted"},"own_p":0.19,"main_claim_p":0.05,"quantitative":false,"quantitative_p":0.13,"role_probabilities":{"gap":0.0,"method":0.57,"result":0.0,"background":0.36,"limitation":0.0,"future_work":0.0,"contribution":0.07,"interpretation":0.0}},{"i":3,"p":0.91,"id":"s3","own":true,"role":"contribution","text":"We determine the minimal overhead.","hedge":{"p":0.89,"label":"asserted"},"own_p":0.9,"main_claim_p":0.15,"quantitative":false,"quantitative_p":0.05,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.09,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.91,"interpretation":0.0}},{"i":4,"p":1.0,"id":"s4","own":true,"role":"result","text":"Ancilla-free sampling of the layers reproduces the target's outcome probabilities up to this factor, and no smaller factor works for every input; when $K$ is permutation symmetric, arbitrary single-qubit gates do no better.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.86,"main_claim_p":0.2,"quantitative":true,"quantitative_p":0.66,"role_probabilities":{"gap":0.0,"method":0.0,"result":1.0,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.0}},{"i":5,"p":1.0,"id":"s5","own":true,"role":"result","text":"The minimal overhead's growth with the register size $n$ is set by the continued fraction of $\\gamma/\\pi$: rational angles $\\pi p/q$ in lowest terms cost exactly $\\Gamma=q$ for every $n\\ge2q-2$; the cost is $\\Theta(n)$ if and only if $\\gamma/\\pi$ is badly approximable; and a master formula, proved two-sided near low-denominator rationals, reads it off the nearest one.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.93,"main_claim_p":0.74,"quantitative":true,"quantitative_p":0.97,"role_probabilities":{"gap":0.0,"method":0.0,"result":1.0,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.0}},{"i":6,"p":0.99,"id":"s6","own":true,"role":"result","text":"Without the symmetry the cost over single-qubit gates can be exponential: for the binary-weighted $K=\\sum_j2^jx_j$ the squaring phase of Fourier arithmetic on $m$ qubits costs at least $2^{0.557m-O(1)}$, beyond the $2^{m/2}$ ceiling of every cut bound, by a new certificate-lifting argument.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.86,"main_claim_p":0.11,"quantitative":true,"quantitative_p":0.99,"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":7,"p":0.84,"id":"s7","own":true,"role":"result","text":"The Fourier-based LCU of Carrera Vazquez, Egger and Woerner (2026), which guarantees $\\Gamma\\le n+1$, is optimal up to a constant factor at typical angles at fixed $n$; at rational angles the unique optimizer is the fractional-revival decomposition of Kerr optics, a recipe independent of $n$.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.6,"main_claim_p":0.13,"quantitative":true,"quantitative_p":0.98,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.84,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.01,"interpretation":0.15}},{"i":8,"p":0.93,"id":"s8","own":false,"role":"result","text":"In bosonic simulation, the Kerr-gate phase-shifter decomposition of Upreti, Quesada and Chabaud (2026) is the unique optimum over phase shifters, and a finite-cost one exists only at rational $\\gamma/\\pi$.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.45,"main_claim_p":0.07,"quantitative":false,"quantitative_p":0.27,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.93,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.02,"interpretation":0.04}}]},"compiled":{"main_claim":{"p":0.97,"text":"The minimal overhead's growth with the register size $n$ is set by the continued fraction of $\\gamma/\\pi$: rational angles $\\pi p/q$ in lowest terms cost exactly $\\Gamma=q$ for every $n\\ge2q-2$; 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