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Computing the partition function for cliques in a graph
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We present a deterministic algorithm which, given a graph G with n vertices and an integer 1<m < n, computes in n^{O(ln m)} time the sum of weights w(S) over all m-subsets S of the set of vertices of G, where w(S)=exp{gamma t m +O(1/m)} provided exactly t{m choose 2} pairs of vertices of S span an edge of G for some 0 < t < 1. Here gamma >0 is an absolute constant: we can choose gamma=0.06, and if n > 4m and m > 10, we can choose gamma=0.18. This allows us to tell apart the graphs that do not have m-subsets of high density from the graphs that have sufficiently many m-subsets of high density, even when the probability to hit such a subset at random is exponentially small in m.
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When quantum thermal states look classical
Long-range Pauli Gibbs states lose entanglement, magic, and infinite-temperature analyticity at distinct constant inverse temperatures Θ(1/sk), Θ(log(1/ε)/sk), and Θ(1/s√k), with matching classical algorithms.
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