The Dual Polynomial of Bipartite Perfect Matching
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We obtain a description of the Boolean dual function of the Bipartite Perfect Matching decision problem, as a multilinear polynomial over the Reals. We show that in this polynomial, both the number of monomials and the magnitude of their coefficients are at most exponential in 𝒪(n log n). As an application, we obtain a new upper bound of 𝒪(n^1.5√(log n)) on the approximate degree of the bipartite perfect matching function, improving the previous best known bound of 𝒪(n^1.75). We deduce that, beyond a 𝒪(√(log n)) factor, the polynomial method cannot be used to improve the lower bound on the bounded-error quantum query complexity of bipartite perfect matching.
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