One of the main reasons for query model's prominence in quantum complexity is the presence of concrete lower bounding techniques: polynomial method and adversary method. There have been considerable efforts to not just give lower bounds using these methods but even to compare and relate them. We explore the value of these bounds on quantum query complexity for the class of symmetric functions, arguably one of the most natural and basic set of Boolean functions. We show (using the recently introduced spectral sensitivity) that both these bounds (positive adversary and approximate degree) give the same value for every total symmetric Boolean function. We also look at the quantum query complexity of Gap Majority, a partial symmetric function. It has gained importance recently in regard to understanding the composition of randomized query complexity. We characterize the quantum query complexity of Gap Majority and show a lower bound on noisy randomized query complexity (Ben-David and Blais, FOCS 2020) in terms of quantum query complexity. In addition, we study how large certificate complexity and block sensitivity can be as compared to sensitivity (even up to constant factors) for symmetric functions. We show tight separations, i.e., give upper bound on possible separations and construct functions achieving the same.
We study a natural complexity measure of Boolean functions known as the rational degree. Denoted rdeg(f), it is the minimal degree of a rational function that is equal to f on the Boolean hypercube. For total functions f, it is conjectured that rdeg(f) is polynomially related to the Fourier degree of f, deg(f). Towards this conjecture, we show that: - Symmetric functions have rational degree at least Ω(deg(f)) and unate functions have rational degree at least deg(f). We observe that both of these lower bounds are asymptotically tight. - Read-once AC and TC formulae have rational degree at least Ω(deg(f)). If these formulae contain parity gates, we show a lower bound of Ω(deg(f)1/2d), where d is the depth. - Almost every Boolean function on n variables has rational degree at least n/2−O(n). In contrast, we exhibit partial functions that witness unbounded separations between rational and approximate degree, in both directions. As a consequence, we show that for quantum computers, post-selection and bounded-error are incomparable resources in the black-box model. In addition, we show AND and OR composition lemmas for the rational degree and exhibit new polynomial separations between the rational degree and other well-studied complexity measures, such as sensitivity and spectral sensitivity.
In card-based cryptography, which uses a physical deck of cards to realize secure multiparty computations, a one-bit value is usually encoded by a pair of cards. Thus, when performing a secure computation of an n-input Boolean function, a sequence of 2n cards representing n bits is needed for input, and some helping cards are typically added to form a protocol. In 2020, Ruangwises and Itoh constructed a card-based protocol for a symmetric Boolean function with an arbitrary range using two helping cards. (Note that a symmetric Boolean function depends only on the number of 1s in its input). At the same time, they showed that the helping cards can be eliminated if the target function is limited to “doubly symmetric” Boolean functions (also known as symmetric self-anti-dual functions). A doubly symmetric Boolean function satisfies the following for all k: when inputting exactly a number k of 1s, the output is the same as the output when inputting exactly a number n − k of 1s. In this paper, we loosen the restriction on doubly symmetric Boolean functions by fixing k = 0, and construct new protocols which require less than two helping cards for that wider class of symmetric Boolean functions. Specifically, we design a one-helping-card protocol for any n > 4, and helping-card-free protocols for n = 3 and n = 4.