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arXiv:2401.11274 · Random Structures and Algorithms

Unambiguous Parity‐Query Complexity

Dmitry Gavinsky

Published May 1, 2025• 1 citation
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Abstract

ABSTRACT We give a lower bound of on the unambiguous randomized parity‐query complexity of the approximate majority problem—that is, on the lowest randomized parity‐query complexity of any function over whose value is “” if the Hamming weight of the input is at most , is “” if the weight is at least , and may be arbitrary otherwise.

DOI
10.1002/rsa.70010

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ABSTRACT We give a lower bound of on the unambiguous randomized parity‐query complexity of the approximate majority problem—that is, on the lowest randomized parity‐query complexity of any function over whose value is “” if the Hamming weight of the input is at most , is “” if the weight is at least , and may be arbitrary otherwise.
We study the query complexity of Weak Parity: the problem of computing the parity of an n-bit input string, where one only has to succeed on a 1/2+eps fraction of input strings, but must do so with high probability on those inputs where one does succeed. It is well-known that n randomized queries and n/2 quantum queries are needed to compute parity on all inputs. But surprisingly, we give a randomized algorithm for Weak Parity that makes only O(n/log^0.246(1/eps)) queries, as well as a quantum algorithm that makes only O(n/sqrt(log(1/eps))) queries. We also prove a lower bound of Omega(n/log(1/eps)) in both cases; and using extremal combinatorics, prove lower bounds of Omega(log n) in the randomized case and Omega(sqrt(log n)) in the quantum case for any eps>0. We show that improving our lower bounds is intimately related to two longstanding open problems about Boolean functions: the Sensitivity Conjecture, and the relationships between query complexity and polynomial degree.
We give improved separations for the query complexity analogue of the log-approximate-rank conjecture i.e. we show that there are a plethora of total Boolean functions on n input bits, each of which has approximate Fourier sparsity at most O(n³) and randomized parity decision tree complexity Θ(n). This improves upon the recent work of Chattopadhyay, Mande and Sherif [Chattopadhyay et al., 2020] both qualitatively (in terms of designing a large number of examples) and quantitatively (shrinking the gap from quartic to cubic). We leave open the problem of proving a randomized communication complexity lower bound for XOR compositions of our examples. A linear lower bound would lead to new and improved refutations of the log-approximate-rank conjecture. Moreover, if any of these compositions had even a sub-linear cost randomized communication protocol, it would demonstrate that randomized parity decision tree complexity does not lift to randomized communication complexity in general (with the XOR gadget).
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 give improved separations for the query complexity analogue of the log-approximate-rank conjecture i.e. we show that there are a plethora of total Boolean functions on nnn input bits, each of which has approximate Fourier sparsity at most O(n3)O(n^3)O(n3) and randomized parity decision tree complexity Θ(n)Θ(n)Θ(n). This improves upon the recent work of Chattopadhyay, Mande and Sherif (JACM '20) both qualitatively (in terms of designing a large number of examples) and quantitatively (improving the gap from quartic to cubic). We leave open the problem of proving a randomized communication complexity lower bound for XOR compositions of our examples. A linear lower bound would lead to new and improved refutations of the log-approximate-rank conjecture. Moreover, if any of these compositions had even a sub-linear cost randomized communication protocol, it would demonstrate that randomized parity decision tree complexity does not lift to randomized communication complexity in general (with the XOR gadget).
We prove a very general lower bound technique for quantum and randomized query complexity that is easy to prove as well as to apply. To achieve this, we introduce the use of Kolmogorov complexity to query complexity. Our technique generalizes the weighted and unweighted methods of Ambainis and the spectral method of Barnum, Saks, and Szegedy. As an immediate consequence of our main theorem, it can be shown that adversary methods can only prove lower bounds for Boolean functions f in O(min⁡(nC0(f),nC1(f)))O(\min(\sqrt{n C_0(f)},\sqrt{n C_1(f)}))O(min(nC0​(f)​,nC1​(f)​)), where C0,C1C_0, C_1C0​,C1​ is the certificate complexity and n is the size of the input.
We prove a very general lower bound technique for quantum and randomized query complexity, that is easy to prove as well as to apply. To achieve this, we introduce the use of Kolmogorov complexity to query complexity. Our technique generalizes the weighted, unweighted methods of Ambainis, and the spectral method of Barnum, Saks and Szegedy. As an immediate consequence of our main theorem, adversary methods can only prove lower bounds for boolean functions fff in O(min⁡(nC0(f),nC1(f)))O(\min(\sqrt{n C_0(f)},\sqrt{n C_1(f)}))O(min(nC0​(f)​,nC1​(f)​)), where C0,C1C_0, C_1C0​,C1​ is the certificate complexity, and nnn is the size of the input. We also derive a general form of the ad hoc weighted method used by Hoyer, Neerbek and Shi to give a quantum lower bound on ordered search and sorting.