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Allerton Conference on Communication, Control, and Computing

Expected query complexity of symmetric boolean functions

Jayadev Acharya, Ashkan Jafarpour, Alon Orlitsky

Published Sep 1, 2011• 5 citations
Publisher

Abstract

The problem of finding optimal querying policy, for expected query complexity of symmetric boolean threshold functions was solved in [1] in the context of collocated networks. In this paper, instead of considering the optimal policy to compute the functions, we define the problem of verification of the function value. We use this idea to provide a simpler proof of the optimal querying policy for threshold functions. The method is more generic and is extended to delta and some other symmetric functions. We also provide some partial results for interval functions and finally address a question posed in [1]. Recently we have extended these results to any symmetric function of boolean inputs, which we mention at the end.

DOI
10.1109/allerton.2011.6120145

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The problem of finding optimal querying policy, for expected query complexity of symmetric boolean threshold functions was solved in [1] in the context of collocated networks. In this paper, instead of considering the optimal policy to compute the functions, we define the problem of verification of the function value. We use this idea to provide a simpler proof of the optimal querying policy for threshold functions. The method is more generic and is extended to delta and some other symmetric functions. We also provide some partial results for interval functions and finally address a question posed in [1]. Recently we have extended these results to any symmetric function of boolean inputs, which we mention at the end.
In the query model of multi-variate function computation, the values of the variables are queried sequentially, in an order that may depend on previously revealed values, until the function's value can be determined. The function's computation query complexity is the lowest expected number of queries required by any query order. Instead of computation, it is often easier to consider verification, where the value of the function is given and the queries aim to verify it. The lowest expected number of queries necessary is the function's verification query complexity. We show that for all symmetric functions of independent binary random variables, the computation and verification complexities coincide. This provides a simple method for finding the query complexity and the optimal query order for computing many functions. We also show that if the symmetry condition is removed, there are functions whose verification complexity is strictly lower than their computation complexity, and mention that the same holds when the independence or binary conditions are removed.
We present several families of total boolean functions which have exact quantum query com-plexity which is a constant multiple (between 1/2 and 2/3) of their classical query complexity, and show that optimal quantum algorithms for these functions cannot be obtained by simply computing parities of pairs of bits. We also characterise the model of nonadaptive exact quantum query complexity in terms of coding theory and completely characterise the query complexity of symmetric boolean functions in this context. These results were originally inspired by nu-merically solving the semidefinite programs characterising quantum query complexity for small problem sizes. We include numerical results giving the optimal success probabilities achievable by quantum algorithms computing all boolean functions on up to 4 bits, and all symmetric boolean functions on up to 6 bits. 1
We give and prove an optimal exact quantum query algorithm with complexity k+1k+1k+1 for computing the promise problem (i.e., symmetric and partial Boolean function) DJnkDJ_n^kDJnk​ defined as: DJnk(x)=1DJ_n^k(x)=1DJnk​(x)=1 for ∣x∣=n/2|x|=n/2∣x∣=n/2, DJnk(x)=0DJ_n^k(x)=0DJnk​(x)=0 for ∣x∣|x|∣x∣ in the set {0,1,…,k,n−k,n−k+1,…,n}\{0, 1,\ldots, k, n-k, n-k+1,\ldots,n\}{0,1,…,k,n−k,n−k+1,…,n}, and it is undefined for the rest cases, where nnn is even, ∣x∣|x|∣x∣ is the Hamming weight of xxx. The case of k=0k=0k=0 is the well-known Deutsch-Jozsa problem. We outline all symmetric (and partial) Boolean functions with degrees 1 and 2, and prove their exact quantum query complexity. Then we prove that any symmetrical (and partial) Boolean function fff has exact quantum 1-query complexity if and only if fff can be computed by the Deutsch-Jozsa algorithm. We also discover the optimal exact quantum 2-query complexity for distinguishing between inputs of Hamming weight {⌊n/2⌋,⌈n/2⌉}\{ \lfloor n/2\rfloor, \lceil n/2\rceil \}{⌊n/2⌋,⌈n/2⌉} and Hamming weight in the set {0,n}\{ 0, n\}{0,n} for all odd nnn. In addition, a method is provided to determine the degree of any symmetrical (and partial) Boolean function.
Watrous conjectured that the randomized and quantum query complexities of symmetric functions are polynomially equivalent, which was resolved by Aaronson & Ambainis (2014) and was later improved by Chailloux (2019) and Ben-David et al. (2020). This paper explores a fine-grained version of the Watrous conjecture, including the randomized and quantum algorithms with success probabilities arbitrarily close to 1/21/21/2 1 / 2 . Our contributions include the following: 1. We analyze the optimal success probabilities of quantum and randomized query algorithms of two fundamental partial symmetric Boolean functions given a fixed number of queries. 2. We establish that for any total symmetric Boolean function fff f , if a quantum algorithm uses TTT T queries to compute fff f with success probability 1/2+β1/2+\beta1/2+β 1 / 2 + β , then there exists a randomized algorithm using O(T2)O(T^2)O(T2) O ( T 2 ) queries to compute fff f with success probability 1/2+Ωδβ21/2+\Omega{\delta\beta^2}1/2+Ωδβ2 1 / 2 + Ω δ β 2 on a 1−δ1-\delta1−δ 1 - δ fraction of inputs, where β,δ\beta,\deltaβ,δ β , δ can be arbitrarily small positive values. Moreover, we prove a randomized version of Aaronson-Ambainis Conjecture (Aaronson & Ambainis 2014) for symmetric Boolean functions in the regime where the success probability of algorithms can be arbitrarily close to 1/2. 3. We present tight polynomial equivalence for several fundamental complexity measures of partial symmetric Boolean functions.
We study the query complexity of Boolean functions f:{0,1}n→{0,1}f: \{0, 1\}^n \rightarrow \{0, 1\}f:{0,1}n→{0,1} in the noisy query model introduced by Feige, Raghavan, Peleg and Upfal [SICOMP 1994]. In this model, an algorithm can adaptively query the bits of an input vector, but each query result is independently flipped with constant probability p∈(0,1/2)p \in (0, 1/2)p∈(0,1/2); repeated queries are allowed. The noisy query complexity Np(f)\mathsf{N}_p(f)Np​(f) of a function fff is defined as the minimum expected number of queries needed to compute f(x)f(x)f(x) with error probability at most 1/31/31/3, for the worst case input xxx. We prove a general lower bound on Np(f)\mathsf{N}_p(f)Np​(f) based on degree statistics of certain subgraphs of the Boolean hypercube. This is the first general lower bound beyond those implied by the simple observation that Np(f)\mathsf{N}_p(f)Np​(f) is lower bounded by the randomized query complexity. We show that this recovers (up to a constant factor) most previously known lower bounds on the noisy query complexity of Boolean functions, providing a unified framework for understanding these results and simplifying the proofs in several cases. Furthermore, this resolves in the affirmative an open problem of Gu, Li and Xu [COLT 2025] that Np(f)=Ω(I(f)log⁡I(f))\mathsf{N}_p(f) = Ω(\mathsf{I}(f) \log \mathsf{I}(f))Np​(f)=Ω(I(f)logI(f)), where I(f)\mathsf{I}(f)I(f) denotes the total influence of fff. We also apply our general lower bound to obtain tight bounds on the noisy query complexity for several new functions.
The query model (or black-box model) has attracted much attention from the communities of both classical and quantum computing. Usually, quantum advantages are revealed by presenting a quantum algorithm that has a better query complexity than its classical counterpart. For example, the well-known quantum algorithms including Deutsch-Jozsa algorithm, Simon algorithm and Grover algorithm all show a considerable advantage of quantum computing from the viewpoint of query complexity. Recently we have considered in (Phys. Rev. A. {\bf 101}, 02232 (2020)) the problem: what functions can be computed by an exact one-query quantum algorithm? This problem has been addressed for total Boolean functions but still open for partial Boolean functions. Thus, in this paper we continue to characterize the computational power of exact one-query quantum algorithms for partial Boolean functions by giving several necessary and sufficient conditions. By these conditions, we construct some new functions that can be computed exactly by one-query quantum algorithms but have essential difference from the already known ones. Note that before our work, the known functions that can be computed by exact one-query quantum algorithms are all symmetric functions, whereas the ones constructed in this papers are generally asymmetric.