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The pattern matrix method for lower bounds on quantum communication

By Alexander A. Sherstov


In a breakthrough result, Razborov (2003) gave optimal lower bounds on the communication complexity of every function f of the form f (x, y) = D(|x ∧ y|) for some D: {0, 1,..., n} → {0, 1}, in the bounded-error quantum model with and without prior entanglement. This was proved by the multidimensional discrepancy method. We give an entirely different proof of Razborov’s result, using the original, one-dimensional discrepancy method. This refutes the commonly held intuition (Razborov 2003) that the original discrepancy method fails for functions such as disjointness. More importantly, our communication lower bounds hold for a much broader class of functions for which no methods were available. Namely, fix an arbitrary function f: {0, 1} n/4 → {0, 1} and let A be the Boolean matrix whose columns are each an application of f to some subset of the variables x1, x2,..., xn. We prove that the communication complexity of A in the bounded-error quantum model with and without prior entanglement is Ω(d), where d is the approximate degree of f. From this result, Razborov’s lower bounds follow easily. Our result also establishes a large new class of total Boolean functions whose quantum communication complexity (regardless of prior entanglement) is at best polynomially smaller than their classical complexity. Our proof method is a novel combination of two ingredients. The first is a certain equivalence of approximation and orthogonality in Euclidean n-space, which follows by linear-programming duality. The second is a new construction of suitably structured matrices with low spectral norm, the pattern matrices, which we realize using matrix analysis and the Fourier transform over Z n 2. The method of this paper has recently inspired important progress in multiparty communication complexity.

Year: 2007
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