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Computational Complexity

On the Fine-Grained Query Complexity of Symmetric Functions

Supartha Podder, Penghui Yao, Zekun Ye

Published Feb 6, 2025• 0 citations
Publisher

Abstract

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 $$1/2$$ 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 $$f$$ f , if a quantum algorithm uses $$T$$ T queries to compute $$f$$ f with success probability $$1/2+\beta$$ 1 / 2 + β , then there exists a randomized algorithm using $$O(T^2)$$ O ( T 2 ) queries to compute $$f$$ f with success probability $$1/2+\Omega{\delta\beta^2}$$ 1 / 2 + Ω δ β 2 on a $$1-\delta$$ 1 - δ fraction of inputs, where $$\beta,\delta$$ 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.

DOI
10.1007/s00037-025-00263-w

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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.
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. Our contributions include the following: i) An analysis of the optimal success probability of quantum and randomized query algorithms of two fundamental partial symmetric Boolean functions given a fixed number of queries. We prove that for any quantum algorithm computing these two functions using TTT queries, there exist randomized algorithms using poly(T)\mathsf{poly}(T)poly(T) queries that achieve the same success probability as the quantum algorithm, even if the success probability is arbitrarily close to 1/2. ii) We establish that for any total symmetric Boolean function fff, if a quantum algorithm uses TTT queries to compute fff with success probability 1/2+β1/2+β1/2+β, then there exists a randomized algorithm using O(T2)O(T^2)O(T2) queries to compute fff with success probability 1/2+Ω(δβ2)1/2+Ω(δβ^2)1/2+Ω(δβ2) on a 1−δ1-δ1−δ fraction of inputs, where β,δβ,δβ,δ can be arbitrarily small positive values. As a corollary, we prove a randomized version of Aaronson-Ambainis Conjecture for total symmetric Boolean functions in the regime where the success probability of algorithms can be arbitrarily close to 1/2. iii) We present polynomial equivalences for several fundamental complexity measures of partial symmetric Boolean functions. Specifically, we first prove that for certain partial symmetric Boolean functions, quantum query complexity is at most quadratic in approximate degree for any error arbitrarily close to 1/2. Next, we show exact quantum query complexity is at most quadratic in degree. Additionally, we give the tight bounds of several complexity measures, indicating their polynomial equivalence.
A longstanding conjecture in quantum complexity theory asserts that, under the uniform input distribution, quantum query algorithms can be polynomially simulated by classical query algorithms. More precisely, the acceptance probability of any quantum query algorithm can be approximated, on average over uniformly random inputs, by a classical query algorithm, with only polynomial query overhead. The conjecture is central to understanding whether exponential quantum advantages for decision problems necessarily rely on additional structure. We study analogues of this conjecture under other natural input distributions and prove that they are all equivalent to the original uniform-distribution conjecture. We first consider the product distribution μpμ_pμp​, where the input bits are independent Bernoulli variables with fixed bias ppp. We show that for every fixed p∈(0,1)p \in (0, 1)p∈(0,1), quantum query algorithms under the μpμ_pμp​ distribution admit polynomial-overhead classical simulations if and only if the same holds under the uniform distribution. Second, we consider the distribution νpν_pνp​ that is uniform over the slice of strings with Hamming weight ⌊pn⌋\lfloor pn \rfloor⌊pn⌋ and prove a similar equivalence for the νpν_pνp​ distribution and the uniform distribution. The Aaronson-Ambainis conjecture is a stronger statement that implies the above-mentioned conjecture and is formulated in terms of bounded low-degree polynomials on the Boolean hypercube. It asserts that under the uniform distribution, any such polynomial with nonnegligible variance must have an influential variable. We formulate analogues of this conjecture, where the underlying distribution is a biased product distribution or a uniform distribution over a slice, and prove that all these variants are equivalent to the original Aaronson-Ambainis conjecture.
A folklore conjecture in quantum computing is that the acceptance probability of a quantum query algorithm can be approximated by a classical decision tree, with only a polynomial increase in the number of queries. Motivated by this conjecture, Aaronson and Ambainis (Theory of Computing, 2014) conjectured that this should hold more generally for any bounded function computed by a low degree polynomial. In this work we prove two new results towards establishing this conjecture: first, that any such polynomial has a small fractional certificate complexity; and second, that many inputs have a small sensitive block. We show that these would imply the Aaronson and Ambainis conjecture, assuming a conjectured extension of Talagrand’s concentration inequality. On the technical side, many classical techniques used in the analysis of Boolean functions seem to fail when applied to bounded functions. Here, we develop a new technique, based on a mix of combinatorics, analysis and geometry, and which in part extends a recent technique of Knop et al. (STOC 2021) to bounded functions.
Smoothed analysis is a central framework in classical algorithms for explaining the performance of algorithms beyond the worst case, often explaining why algorithms perform well in practice. We initiate a systematic study of its quantum counterpart and show the following results. (1)(1)(1) We show that there is a total function whose smoothed quantum query complexity is exponentially smaller than its classical query complexity. (2)(2)(2) We give near-tight characterizations of smoothed randomized and quantum query complexities for symmetric Boolean functions, unifying the worst-case complexity results of [Beals et al, FOCS'98] and average-case complexity results of [Ambainis and de Wolf, STACS'00]. (3)(3)(3) We study string problems such as pattern matching and edit distance and, in various regimes, give polynomial to superpolynomial quantum speedups. Our main technical ingredients include a near-tight quantum algorithm for ε\varepsilonε-approximating the number of collisions between two non-repetitive strings, improving the result of Le Gall and Ng [QIC'22]. Together, our results show that smoothing can reveal larger quantum speedups than worst-case analysis suggests, opening a path towards quantum advantage on more realistic inputs.
The query model offers a concrete setting where quantum algorithms are provably superior to randomized algorithms. Beautiful results by Bernstein-Vazirani, Simon, Aaronson, and others presented partial Boolean functions that can be computed by quantum algorithms making much fewer queries compared to their randomized analogs. To date, separations of O(1) vs. √N between quantum and randomized query complexities remain the state-of-the-art (where N is the input length), leaving open the question of whether O(1) vs. N1/2+Ω(1)separations are possible? We answer this question in the affirmative. Our separating problem is a variant of the Aaronson-Ambainis k-fold Forrelation problem. We show that our variant: 1)Can be solved by a quantum algorithm making 2O(k)queries to the inputs. 2)Requires at least ~Ω(N2(k-1)/(3k-1)) queries for any randomized algorithm. For any constant , this gives a O(1) vs. N1/2-εseparation between the quantum and randomized query complexities of partial Boolean functions. Our proof is Fourier analytical and uses new bounds on the Fourier spectrum of classical decision trees, which could be of independent interest. Looking forward, we conjecture that the Fourier bounds could be further improved in a precise manner, and show that such conjectured bounds imply optimal O(1) vs. N1-εseparations between the quantum and randomized query complexities of partial Boolean functions.
We consider the problem of establishing limitations on the power of quantum query algorithms via the completely bounded polynomial method. In particular, we prove several optimal functional inequalities involving different notions of completely bounded polynomials. These inequalities lead to limiting theorems for the power of quantum query algorithms that improve on prior works. 1. An optimal root-influence bound for block-multilinear polynomials. Prior work showed that block-multilinear polynomials ppp of degree ttt satisfy a root-influence bound, ∥p∥cb≥∑iInfi[p]/t2\|p\|_{\text{cb}}\geq \sum_i \sqrt{\mathrm{Inf}_i[p]}/t^2∥p∥cb​≥∑i​Infi​[p]​/t2, which is stronger than the bound appearing in the Aaronson-Ambainis conjecture. We find the optimal constant in that inequality: ∥p∥cb≥∑iInfi[p]/t\|p\|_{\text{cb}}\geq \sum_i \sqrt{\mathrm{Inf}_i[p]}/t∥p∥cb​≥∑i​Infi​[p]​/t. Since the amplitudes of quantum algorithms that query disjoint blocks of inputs-such as ttt-fold forrelation- are block-multilinear polynomials with ∥p∥cb≤1,\|p\|_{\text{cb}}\leq 1,∥p∥cb​≤1, our inequality shows that they satisfy t≥∑iInfi[p]t\geq \sum_i\sqrt{\mathrm{Inf}_i[p]}t≥∑i​Infi​[p]​. We prove that this inequality yields both a more efficient classical simulation than prior results based on the Aaronson-Ambainis argument, and a qualitative improvement: all classical queries are nonadaptive. 2. Optimal Fourier growth of the highest level of quantum query algorithms. We show that for every polynomial ppp defined on {−1,1}n\{-1,1\}^n{−1,1}n of degree 2t2t2t, the Fourier Growth at the level 2t,2t,2t, namely ∥p^2t∥ℓ1\|\widehat p_{2t}\|_{\ell_1}∥p​2t​∥ℓ1​​, satisfies ∥p^2t∥ℓ1≤(en/(2t−1))2t−12∥p∥cb\|\widehat p_{2t}\|_{\ell_1}\leq (en/(2t-1))^{\frac{2t-1}{2}}\|p\|_{\text{cb}}∥p​2t​∥ℓ1​​≤(en/(2t−1))22t−1​∥p∥cb​. This is optimal up to the factor eee, as witnessed by 2t2t2t-fold forrelation. As quantum query algorithms that make ttt queries (to the whole input) satisfy ∥p∥cb≤1\|p\|_{\text{cb}}\leq 1∥p∥cb​≤1, this yields a Fourier growth bound for these algorithms, partially resolving a question by Girish (STOC, 2026).