Quantum Query Complexity of Persistence Statistics in Graph Zigzags

• URL: http://arxiv.org/abs/2609.05813v1
• Date: Sat, 05 Sep 2026 02:16:28 GMT
• Title: Quantum Query Complexity of Persistence Statistics in Graph Zigzags
• Abstract summary: We study the query complexity of estimating scalar summaries of zigzag bar lifetimes from snapshot-adjacency bits.<n>For graphs that rank is the circuit rank of a graph, so a nonlinear barcode functional becomes an average of edge and component counts.<n>For every fixed power weight $xr$, $rge2$, and for the uniform-window mean, the worst-case complexities are $widetilde(nsqrt m/varepsilon)$ quantum and $(n2m)$ classical.
• Score: 2.895030292351388
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We study the query complexity of estimating scalar summaries of zigzag bar lifetimes from snapshot-adjacency bits. For graphs $G_1,\ldots,G_m$ on $n$ labeled vertices, let $\ell_b$ be the snapshot lifetime of a degree-one bar $b$ of the intersection zigzag. For a probability generating function $φ(x)=\mathbb{E}[x^R]$, the statistic $F_φ=\sum_bφ(\ell_b/m)$ includes normalized degree-$r$ total persistence and the mean generalized rank over a uniform time window. An exact identity underlies our algorithm: sample $R$ uniform times; the expected generalized rank between their minimum and maximum equals $F_φ$. For graphs that rank is the circuit rank of an intersection graph, so a nonlinear barcode functional becomes an average of edge and component counts, and no barcode is computed. Without spectral-gap, homology-state, or QRAM assumptions, this gives a quantum estimator with additive error $\varepsilon n$ and $\widetilde O(\sqrt{m(K+n)}/\varepsilon)$ queries when a bound $K\ge F_φ$ is supplied, against $\widetilde O(m\min\{n^2,(K+n)/\varepsilon^2\})$ classically, and an adaptive quantum variant with the same instance dependence. These estimators are optimal in two regimes. For every fixed power weight $x^r$, $r\ge2$, and for the uniform-window mean, the worst-case complexities are $\widetildeΘ(n\sqrt m/\varepsilon)$ quantum and $Θ(n^2m)$ classical. On sparse instances, under an explicit split-leakage promise met by power and binomial weights of logarithmic degree and the promise $F_φ\le K$, they are $\widetildeΘ(\sqrt{mK}/\varepsilon)$ and $\widetildeΘ(m\min\{n^2,K/\varepsilon^2\})$. The classical lower bounds hold against fully adaptive algorithms, and fewer than $m$ such statistics cannot determine the positive-lifetime histogram. All bounds concern snapshot access; with an explicit update stream, near-linear full-barcode algorithms are known.

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