Matching upper bounds on symmetric predicates in quantum communication complexity
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In this paper, we focus on the quantum communication complexity of functions of the form f ∘ G = f(G(X_1, Y_1), …, G(X_n, Y_n)) where f: {0, 1}^n →{0, 1} is a symmetric function, G: {0, 1}^j ×{0, 1}^k →{0, 1} is any function and Alice (resp. Bob) is given (X_i)_i ≤ n (resp. (Y_i)_i ≤ n). Recently, Chakraborty et al. [STACS 2022] showed that the quantum communication complexity of f ∘ G is O(Q(f)QCC_E(G)) when the parties are allowed to use shared entanglement, where Q(f) is the query complexity of f and QCC_E(G) is the exact communication complexity of G. In this paper, we first show that the same statement holds without shared entanglement, which generalizes their result. Based on the improved result, we next show tight upper bounds on f ∘AND_2 for any symmetric function f (where AND_2 : {0, 1}×{0, 1}→{0, 1} denotes the 2-bit AND function) in both models: with shared entanglement and without shared entanglement. This matches the well-known lower bound by Razborov [Izv. Math. 67(1) 145, 2003] when shared entanglement is allowed and improves Razborov's bound when shared entanglement is not allowed.
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