In this paper, we focus on the quantum communication complexity of functions of the form $f \circ G = f(G(X_1, Y_1), \ldots, G(X_n, Y_n))$ where $f: \Bset^n \to \Bset$ is a symmetric function, $G: \Bset^j \times \Bset^k \to \Bset$ is any function and Alice (resp. Bob) is given $(X_i)_{i \in [n]}$ (resp. $(Y_i)_{i \in [n]}$). Recently, Chakraborty et al. [STACS 2022] showed that the quantum communication complexity of $f \circ G$ is $O(Q(f)\QCCEX(G))$ when the parties are allowed to use shared entanglement, where $Q(f)$ is the query complexity of $f$ and $\QCCEX(G)$ is the exact communication complexity of $G$. }{In this paper, we first show that the same statement holds \emph{without both shared entanglement and shared randomness}, which generalizes their result. Based on the improved result, we next show tight upper bounds on $f \circ \mathrm{AND}_2$ for any symmetric function $f$ (where $\textrm{AND}_2 : \Bset \times \Bset \to \Bset$ denotes the 2-bit AND function) in both models: with shared entanglement and without shared entanglement. This matches the well-known lower bound by Razborov~[Izv. Math. 67(1) 145, 2003] when shared entanglement is allowed and improves Razborov's bound when shared entanglement is not allowed.
In this paper, we focus on the quantum communication complexity of functions of the form f∘G=f(G(X1,Y1),…,G(Xn,Yn)) where f:\Bsetn→\Bset is a symmetric function, G:\Bsetj×\Bsetk→\Bset is any function and Alice (resp. Bob) is given (Xi)i∈[n] (resp. (Yi)i∈[n]). Recently, Chakraborty et al. [STACS 2022] showed that the quantum communication complexity of f∘G is O(Q(f)\QCCEX(G)) when the parties are allowed to use shared entanglement, where Q(f) is the query complexity of f and \QCCEX(G) is the exact communication complexity of G. }{In this paper, we first show that the same statement holds \emph{without both shared entanglement and shared randomness}, which generalizes their result. Based on the improved result, we next show tight upper bounds on f∘AND2 for any symmetric function f (where AND2:\Bset×\Bset→\Bset denotes the 2-bit AND function) in both models: with shared entanglement and without shared entanglement. This matches the well-known lower bound by Razborov~[Izv. Math. 67(1) 145, 2003] when shared entanglement is allowed and improves Razborov's bound when shared entanglement is not allowed.
Buhrman, Cleve and Wigderson (STOC'98) showed that for every Boolean function f : {-1,1}ⁿ → {-1,1} and G ∈ {AND₂, XOR₂}, the bounded-error quantum communication complexity of the composed function f∘G equals O(𝖰(f) log n), where 𝖰(f) denotes the bounded-error quantum query complexity of f. This is achieved by Alice running the optimal quantum query algorithm for f, using a round of O(log n) qubits of communication to implement each query. This is in contrast with the classical setting, where it is easy to show that 𝖱^{cc}(f∘G) ≤ 2𝖱(f), where 𝖱^{cc} and 𝖱 denote bounded-error communication and query complexity, respectively. Chakraborty et al. (CCC'20) exhibited a total function for which the log n overhead in the BCW simulation is required. This established the somewhat surprising fact that quantum reductions are in some cases inherently more expensive than classical reductions. We improve upon their result in several ways. - We show that the log n overhead is not required when f is symmetric (i.e., depends only on the Hamming weight of its input), generalizing a result of Aaronson and Ambainis for the Set-Disjointness function (Theory of Computing'05). Our upper bound assumes a shared entangled state, though for most symmetric functions the assumed number of entangled qubits is less than the communication and hence could be part of the communication. - In order to prove the above, we design an efficient distributed version of noisy amplitude amplification that allows us to prove the result when f is the OR function. This also provides a different, and arguably simpler, proof of Aaronson and Ambainis’s O(√n) communication upper bound for Set-Disjointness. - In view of our first result above, one may ask whether the log n overhead in the BCW simulation can be avoided even when f is transitive, which is a weaker notion of symmetry. We give a strong negative answer by showing that the log n overhead is still necessary for some transitive functions even when we allow the quantum communication protocol an error probability that can be arbitrarily close to 1/2 (this corresponds to the unbounded-error model of communication). - We also give, among other things, a general recipe to construct functions for which the log n overhead is required in the BCW simulation in the bounded-error communication model, even if the parties are allowed to share an arbitrary prior entangled state for free.
Buhrman, Cleve and Wigderson (STOC'98) observed that for every Boolean function f:{-1,1}ⁿ → {-1,1} and •:{-1,1}² → {-1,1} the two-party bounded-error quantum communication complexity of (f ∘ •) is O(Q(f) log n), where Q(f) is the bounded-error quantum query complexity of f. Note that the bounded-error randomized communication complexity of (f ∘ •) is bounded by O(R(f)), where R(f) denotes the bounded-error randomized query complexity of f. Thus, the BCW simulation has an extra O(log n) factor appearing that is absent in classical simulation. A natural question is if this factor can be avoided. Razborov (IZV MATH'03) showed that the bounded-error quantum communication complexity of Set-Disjointness is Ω(√n). The BCW simulation yields an upper bound of O(√n log n). Høyer and de Wolf (STACS'02) showed that this can be reduced to c^(log^* n) for some constant c, and subsequently Aaronson and Ambainis (FOCS'03) showed that this factor can be made a constant. That is, the quantum communication complexity of the Set-Disjointness function (which is NOR_n ∘ ∧) is O(Q(NOR_n)). Perhaps somewhat surprisingly, we show that when • = ⊕, then the extra log n factor in the BCW simulation is unavoidable. In other words, we exhibit a total function F:{-1,1}ⁿ → {-1,1} such that Q^{cc}(F ∘ ⊕) = Θ(Q(F) log n). To the best of our knowledge, it was not even known prior to this work whether there existed a total function F and 2-bit function •, such that Q^{cc}(F ∘ •) = ω(Q(F)).
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), 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∘AND2 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.
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, …, x_n ∧ 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∘And₂ 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 [Arkadev Chattopadhyay et al., 2026] on structural characterizations of non-sparse Boolean functions, which we extend to resolve the conjecture for general And-functions.
Buhrman, Cleve and Wigderson (STOC'98) showed that for every Boolean function f : {-1,1}^n to {-1,1} and G in {AND_2, XOR_2}, the bounded-error quantum communication complexity of the composed function f o G equals O(Q(f) log n), where Q(f) denotes the bounded-error quantum query complexity of f. This is achieved by Alice running the optimal quantum query algorithm for f, using a round of O(log n) qubits of communication to implement each query. This is in contrast with the classical setting, where it is easy to show that R^{cc}(f o G) is at most 2R(f), where R^{cc} and R denote bounded-error communication and query complexity, respectively. We show that the O(log n) overhead is required for some functions in the quantum setting, and thus the BCW simulation is tight. We note here that prior to our work, the possibility of Q^{cc}(f o G) = O(Q(f)), for all f and all G in {AND_2, XOR_2}, had not been ruled out. More specifically, we show the following. - We show that the log n overhead is *not* required when f is symmetric, generalizing a result of Aaronson and Ambainis for the Set-Disjointness function (Theory of Computing'05). - In order to prove the above, we design an efficient distributed version of noisy amplitude amplification that allows us to prove the result when f is the OR function. - In view of our first result above, one may ask whether the log n overhead in the BCW simulation can be avoided even when f is transitive, which is a weaker notion of symmetry. We give a strong negative answer by showing that the log n overhead is still necessary for some transitive functions even when we allow the quantum communication protocol an error probability that can be arbitrarily close to 1/2. - We also give, among other things, a general recipe to construct functions for which the log n overhead is required in the BCW simulation in the bounded-error communication model.
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.