Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2001.09642

How symmetric is too symmetric for large quantum speedups?

Shalev Ben-David, Supartha Podder

Published Jan 27, 2020• 2 citations
arXivHTMLPDF

Abstract

Suppose a Boolean function $f$ is symmetric under a group action $G$ acting on the $n$ bits of the input. For which $G$ does this mean $f$ does not have an exponential quantum speedup? Is there a characterization of how rich $G$ must be before the function $f$ cannot have enough structure for quantum algorithms to exploit? In this work, we make several steps towards understanding the group actions $G$ which are "quantum intolerant" in this way. We show that sufficiently transitive group actions do not allow a quantum speedup, and that a "well-shuffling" property of group actions -- which happens to be preserved by several natural transformations -- implies a lack of super-polynomial speedups for functions symmetric under the group action. Our techniques are motivated by a recent paper by Chailloux (2018), which deals with the case where $G=S_n$. Our main application is for graph symmetries: we show that any Boolean function $f$ defined on the adjacency matrix of a graph (and symmetric under relabeling the vertices of the graph) has a power $6$ relationship between its randomized and quantum query complexities, even if $f$ is a partial function. In particular, this means no graph property testing problems can have super-polynomial quantum speedups, settling an open problem of Ambainis, Childs, and Liu (2011).

DOI
10.48550/arxiv.2001.09642

Related papers

Symmetries, Graph Properties, and Quantum Speedups

Shalev Ben-David, Andrew M. Childs, Andras Gilyen +3 more

Aaronson and Ambainis (2009) and Chailloux (2018) showed that fully symmetric (partial) functions do not admit exponential quantum query speedups. This raises a natural question: how symmetric must a function be before it cannot exhibit a large quantum speedup? In this work, we prove that hypergraph symmetries in the adjacency matrix model allow at most a polynomial separation between randomized and quantum query complexities. We also show that, remarkably, permutation groups constructed out of these symmetries are essentially the only permutation groups that prevent super-polynomial quantum speedups. We prove this by fully characterizing the primitive permutation groups that allow super-polynomial quantum speedups. In contrast, in the adjacency list model for bounded-degree graphs-where graph symmetry is manifested differently-we exhibit a property testing problem that shows an exponential quantum speedup. These results resolve open questions posed by Ambainis, Childs, and Liu (2010) and Montanaro and de Wolf (2013).

arXiv:2006.12760•Annual Symposium on Foundations of Computer Science•Nov 1, 2020•21 citations

Quantum Query Complexity Beyond the Worst Case

Srinivasan Arunachalam, Yanlin Chen, Amin Shiraz Gilani

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. $(1)$ We show that there is a total function whose smoothed quantum query complexity is exponentially smaller than its classical query complexity. $(2)$ 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]. $(3)$ 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 $\varepsilon$-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.

quant-pharXiv:2609.35580•Sep 29, 2026•0 citations

Symmetries, Graph Properties, and Quantum Speedups

Shalev Ben-David, Andrew M. Childs, András Gilyén +3 more

Abstract. Aaronson and Ambainis [ Theory Comput., 10 (2014), pp. 133–166] and Chailloux [ Proceedings of the 10 th Innovations in Theoretical Computer Science Conference, 2018, pp. 19:1–19:7] showed that fully symmetric (partial) functions do not admit exponential quantum query speedups. This raises a natural question: how symmetric must a function be before it cannot exhibit a large quantum speedup? In this work, we prove that hypergraph symmetries in the adjacency matrix model allow at most a polynomial separation between randomized and quantum query complexities. We also show that, remarkably, permutation groups constructed out of these symmetries are essentially the only permutation groups that prevent superpolynomial quantum speedups. We prove this by fully characterizing the primitive permutation groups that allow superpolynomial quantum speedups. In contrast, in the adjacency list model for bounded-degree graphs—where graph symmetry is manifested differently—we exhibit a property testing problem that shows an exponential quantum speedup. These results resolve open questions posed by Ambainis, Childs, and Liu [ Lecture Notes in Comput. Sci. 6845, Springer, 2011, pp. 365–376] and Montanaro and de Wolf [ Theory Comput., 7 (2016)].

SIAM Journal on Computing•Dec 13, 2024•2 citations

Can graph properties have exponential quantum speedup?

Andrew M. Childs, Daochen Wang

Quantum computers can sometimes exponentially outperform classical ones, but only for problems with sufficient structure. While it is well known that query problems with full permutation symmetry can have at most polynomial quantum speedup -- even for partial functions -- it is unclear how far this condition must be relaxed to enable exponential speedup. In particular, it is natural to ask whether exponential speedup is possible for (partial) graph properties, in which the input describes a graph and the output can only depend on its isomorphism class. We show that the answer to this question depends strongly on the input model. In the adjacency matrix model, we prove that the bounded-error randomized query complexity $R$ of any graph property $\mathcal{P}$ has $R(\mathcal{P}) = O(Q(\mathcal{P})^{6})$, where $Q$ is the bounded-error quantum query complexity. This negatively resolves an open question of Montanaro and de Wolf in the adjacency matrix model. More generally, we prove $R(\mathcal{P}) = O(Q(\mathcal{P})^{3l})$ for any $l$-uniform hypergraph property $\mathcal{P}$ in the adjacency matrix model. In direct contrast, in the adjacency list model for bounded-degree graphs, we exhibit a promise problem that shows an exponential separation between the randomized and quantum query complexities.

arXiv:2001.10520•Jan 28, 2020•2 citations

Quantum-Classical Equivalence for AND-Functions

Sreejata Kishor Bhattacharya, Farzan Byramji, Arkadev Chattopadhyay +2 more

A major open problem in quantum communication complexity is whether quantum protocols can be exponentially more efficient than classical protocols for computing total Boolean functions; the prevailing conjecture is that they cannot be so. In a seminal work, Razborov (2002) resolved this question for AND-functions of the form $$ F(x,y) = f(x_1 \land y_1, \ldots, x_n \land y_n), $$ when the outer function $f$ is symmetric, by proving that their bounded-error quantum and classical communication complexities are polynomially related. Since then, extending this result to all AND-functions has remained open and has been posed by several authors. In this work, we settle this problem in a strong way. We show that for every Boolean function $f$, the bounded-error quantum and classical deterministic communication complexities of the function $f \circ \mathrm{AND}_2$ are polynomially related, up to polylogarithmic factors in $n$. We prove this by showing that both are characterized--up to polynomial loss--by the logarithm of the De Morgan sparsity of $f$. Our results build on the recent work of Chattopadhyay, Dahiya, and Lovett (2025) on structural characterizations of non-sparse Boolean functions, which we extend to resolve the conjecture for general AND-functions.

cs.CCCross-listedarXiv:2606.03249•Jun 3, 2026•0 citations

On the Fine-Grained Query Complexity of Symmetric Functions

Supartha Podder, Penghui Yao, Zekun Ye

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.

Computational Complexity•Feb 6, 2025•0 citations
Suppose a Boolean function fff is symmetric under a group action GGG acting on the nnn bits of the input. For which GGG does this mean fff does not have an exponential quantum speedup? Is there a characterization of how rich GGG must be before the function fff cannot have enough structure for quantum algorithms to exploit? In this work, we make several steps towards understanding the group actions GGG which are "quantum intolerant" in this way. We show that sufficiently transitive group actions do not allow a quantum speedup, and that a "well-shuffling" property of group actions -- which happens to be preserved by several natural transformations -- implies a lack of super-polynomial speedups for functions symmetric under the group action. Our techniques are motivated by a recent paper by Chailloux (2018), which deals with the case where G=SnG=S_nG=Sn​. Our main application is for graph symmetries: we show that any Boolean function fff defined on the adjacency matrix of a graph (and symmetric under relabeling the vertices of the graph) has a power 666 relationship between its randomized and quantum query complexities, even if fff is a partial function. In particular, this means no graph property testing problems can have super-polynomial quantum speedups, settling an open problem of Ambainis, Childs, and Liu (2011).
Aaronson and Ambainis (2009) and Chailloux (2018) showed that fully symmetric (partial) functions do not admit exponential quantum query speedups. This raises a natural question: how symmetric must a function be before it cannot exhibit a large quantum speedup? In this work, we prove that hypergraph symmetries in the adjacency matrix model allow at most a polynomial separation between randomized and quantum query complexities. We also show that, remarkably, permutation groups constructed out of these symmetries are essentially the only permutation groups that prevent super-polynomial quantum speedups. We prove this by fully characterizing the primitive permutation groups that allow super-polynomial quantum speedups. In contrast, in the adjacency list model for bounded-degree graphs-where graph symmetry is manifested differently-we exhibit a property testing problem that shows an exponential quantum speedup. These results resolve open questions posed by Ambainis, Childs, and Liu (2010) and Montanaro and de Wolf (2013).
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.
Abstract. Aaronson and Ambainis [ Theory Comput., 10 (2014), pp. 133–166] and Chailloux [ Proceedings of the 10 th Innovations in Theoretical Computer Science Conference, 2018, pp. 19:1–19:7] showed that fully symmetric (partial) functions do not admit exponential quantum query speedups. This raises a natural question: how symmetric must a function be before it cannot exhibit a large quantum speedup? In this work, we prove that hypergraph symmetries in the adjacency matrix model allow at most a polynomial separation between randomized and quantum query complexities. We also show that, remarkably, permutation groups constructed out of these symmetries are essentially the only permutation groups that prevent superpolynomial quantum speedups. We prove this by fully characterizing the primitive permutation groups that allow superpolynomial quantum speedups. In contrast, in the adjacency list model for bounded-degree graphs—where graph symmetry is manifested differently—we exhibit a property testing problem that shows an exponential quantum speedup. These results resolve open questions posed by Ambainis, Childs, and Liu [ Lecture Notes in Comput. Sci. 6845, Springer, 2011, pp. 365–376] and Montanaro and de Wolf [ Theory Comput., 7 (2016)].
Quantum computers can sometimes exponentially outperform classical ones, but only for problems with sufficient structure. While it is well known that query problems with full permutation symmetry can have at most polynomial quantum speedup -- even for partial functions -- it is unclear how far this condition must be relaxed to enable exponential speedup. In particular, it is natural to ask whether exponential speedup is possible for (partial) graph properties, in which the input describes a graph and the output can only depend on its isomorphism class. We show that the answer to this question depends strongly on the input model. In the adjacency matrix model, we prove that the bounded-error randomized query complexity RRR of any graph property P\mathcal{P}P has R(P)=O(Q(P)6)R(\mathcal{P}) = O(Q(\mathcal{P})^{6})R(P)=O(Q(P)6), where QQQ is the bounded-error quantum query complexity. This negatively resolves an open question of Montanaro and de Wolf in the adjacency matrix model. More generally, we prove R(P)=O(Q(P)3l)R(\mathcal{P}) = O(Q(\mathcal{P})^{3l})R(P)=O(Q(P)3l) for any lll-uniform hypergraph property P\mathcal{P}P in the adjacency matrix model. In direct contrast, in the adjacency list model for bounded-degree graphs, we exhibit a promise problem that shows an exponential separation between the randomized and quantum query complexities.
A major open problem in quantum communication complexity is whether quantum protocols can be exponentially more efficient than classical protocols for computing total Boolean functions; the prevailing conjecture is that they cannot be so. In a seminal work, Razborov (2002) resolved this question for AND-functions of the form F(x,y)=f(x1∧y1,…,xn∧yn), F(x,y) = f(x_1 \land y_1, \ldots, x_n \land y_n), F(x,y)=f(x1​∧y1​,…,xn​∧yn​), when the outer function fff is symmetric, by proving that their bounded-error quantum and classical communication complexities are polynomially related. Since then, extending this result to all AND-functions has remained open and has been posed by several authors. In this work, we settle this problem in a strong way. We show that for every Boolean function fff, the bounded-error quantum and classical deterministic communication complexities of the function f∘AND2f \circ \mathrm{AND}_2f∘AND2​ are polynomially related, up to polylogarithmic factors in nnn. We prove this by showing that both are characterized--up to polynomial loss--by the logarithm of the De Morgan sparsity of fff. Our results build on the recent work of Chattopadhyay, Dahiya, and Lovett (2025) on structural characterizations of non-sparse Boolean functions, which we extend to resolve the conjecture for general AND-functions.
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.