A longstanding question in quantum communication complexity is whether some task can be accomplished with a small amount of communication in the presence of entanglement, yet require much more quantum communication in the absence of entanglement. Separations of this nature were previously known for relational problems and, in the simultaneous message passing model, for partial functions. But it has remained unresolved whether any such separation exists for a total Boolean function. We resolve this question with an exponential separation in the one-way setting: we exhibit a family of total Boolean functions $f_n\colon \{0,1\}^n \times \{0,1\}^n \to \{0,1\}$ that can be computed with $O(\log n)$ bits of one-way classical communication given prior entanglement, but that require $Ω(n^{1/3})$ qubits of one-way quantum communication without entanglement. Our function is a special case of the subgroup membership problem, first studied in the communication setting by Aaronson, Le Gall, Russell, and Tani.
A longstanding question in quantum communication complexity is whether some task can be accomplished with a small amount of communication in the presence of entanglement, yet require much more quantum communication in the absence of entanglement. Separations of this nature were previously known for relational problems and, in the simultaneous message passing model, for partial functions. But it has remained unresolved whether any such separation exists for a total Boolean function. We resolve this question with an exponential separation in the one-way setting: we exhibit a family of total Boolean functions fn:{0,1}n×{0,1}n→{0,1} that can be computed with O(logn) bits of one-way classical communication given prior entanglement, but that require Ω(n1/3) qubits of one-way quantum communication without entanglement. Our function is a special case of the subgroup membership problem, first studied in the communication setting by Aaronson, Le Gall, Russell, and Tani.
We study the advantages of quantum communication models over classical communication models that are equipped with a limited number of qubits of entanglement. In this direction, we give explicit partial functions on n bits for which reducing the entanglement increases the classical communication complexity exponentially. Our separations are as follows. For every k≥1: Q∥∗ versus R2∗: We show that quantum simultaneous protocols with Θ~(k5log3n) qubits of entanglement can exponentially outperform two-way randomized protocols with O(k) qubits of entanglement. This resolves an open problem from [Gav08] and improves the state-of-the-art separations between quantum simultaneous protocols with entanglement and two-way randomized protocols without entanglement [Gav19, GRT22]. R∥∗ versus Q∥∗: We show that classical simultaneous protocols with Θ~(klogn) qubits of entanglement can exponentially outperform quantum simultaneous protocols with O(k) qubits of entanglement, resolving an open question from [GKRW06, Gav19]. The best result prior to our work was a relational separation against protocols without entanglement [GKRW06]. R∥∗ versus R1∗: We show that classical simultaneous protocols with Θ~(klogn) qubits of entanglement can exponentially outperform randomized one-way protocols with O(k) qubits of entanglement. Prior to our work, only a relational separation was known [Gav08]. Our techniques can also be used to show advantages of quantum communication models over hybrid classical-quantum models, i.e., models that have a large amount of both classical communication and quantum simultaneous communication.
We study a new type of separations between quantum and classical communication complexity, separations that are obtained using quantum protocols where all parties are efficient , in the sense that they can be implemented by small quantum circuits, with oracle access to their inputs. Our main result qualitatively matches the strongest known separation between quantum and classical communication complexity Gavinsky (2016) and is obtained using a quantum protocol where all parties are efficient. More precisely, we give an explicit partial Boolean function f over inputs of length N , such that: f can be computed by a simultaneous-message quantum protocol with communication complexity polylog( N ) (where at the beginning of the protocol Alice and Bob also have polylog( N ) entangled EPR pairs). Any classical randomized protocol for f , with any number of rounds, has communication complexity at least Ω~(N1/4) . All parties in the quantum protocol of Item (1) (Alice, Bob and the referee) can be implemented by quantum circuits of size polylog( N ) (where Alice and Bob have oracle access to their inputs). Items (1), (2) qualitatively match the strongest known separation between quantum and classical communication complexity, proved by Gavinsky (2016). Item (3) is new. (Our result is incomparable to the one of Gavinsky. While he obtained a quantitatively better lower bound of Ω(N1/2) in the classical case, the referee in his quantum protocol is inefficient). Exponential separations of quantum and classical communication complexity have been studied in numerous previous works, but to the best of our knowledge the efficiency of the parties in the quantum protocol has not been addressed, and in most previous separations the quantum parties seem to be inefficient. The only separations that we know of that have efficient quantum parties are the recent separations that are based on lifting Göös et al. (2017), Chattopadhyay et al. (2019a). However, these separations seem to require quantum protocols with at least two rounds of communication, so they imply a separation of two-way quantum and classical communication complexity, but they do not give the stronger separations of simultaneous-message quantum communication complexity vs. two-way classical communication complexity (or even one-way quantum communication complexity vs. two-way classical communication complexity). Our proof technique is completely new, in the context of communication complexity, and is based on techniques from Raz & Tal (2019). Our function f is based on a lift of the forrelation problem, using xor as a gadget.
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.
Boolean function F(x,y) for x,y ∈ {0,1}ⁿ is an XOR function if F(x,y) = f(x⊕ y) for some function f on n input bits, where ⊕ is a bit-wise XOR. XOR functions are relevant in communication complexity, partially for allowing the Fourier analytic technique. For total XOR functions, it is known that deterministic communication complexity of F is closely related to parity decision tree complexity of f. Montanaro and Osbourne (2009) observed that one-way communication complexity D_{cc}^{→}(F) of F is exactly equal to non-adaptive parity decision tree complexity NADT^{⊕}(f) of f. Hatami et al. (2018) showed that unrestricted communication complexity of F is polynomially related to parity decision tree complexity of f. We initiate the study of a similar connection for partial functions. We show that in the case of one-way communication complexity whether these measures are equal, depends on the number of undefined inputs of f. More precisely, if D_{cc}^{→}(F) = t and f is undefined on at most O((2^{n-t})/(√{n-t})) inputs, then NADT^{⊕}(f) = t. We also provide stronger bounds in extreme cases of small and large complexity. We show that the restriction on the number of undefined inputs in these results is unavoidable. That is, for a wide range of values of D_{cc}^{→}(F) and NADT^{⊕}(f) (from constant to n-2) we provide partial functions (with more than Ω((2^{n-t})/(√{n-t})) undefined inputs, where t = D_{cc}^{→}) for which D_{cc}^{→}(F) < NADT^{⊕}(f). In particular, we provide a function with an exponential gap between the two measures. Our separation results translate to the case of two-way communication complexity as well, in particular showing that the result of Hatami et al. (2018) cannot be generalized to partial functions. Previous results for total functions heavily rely on the Boolean Fourier analysis and thus, the technique does not translate to partial functions. For the proofs of our results we build a linear algebraic framework instead. Separation results are proved through the reduction to covering codes.
Numbers-on-Forehead (NOF) communication model is a central model in communication complexity. As a restricted variant, one-way NOF model is of particular interest. Establishing strong one-way NOF lower bounds would imply circuit lower bounds, resolve well-known problems in additive combinatorics, and yield wide-ranging applications in areas such as cryptography and distributed computing. However, proving strong lower bounds in one-way NOF communication remains highly challenging; many fundamental questions in one-way NOF communication remain wide open. One of the fundamental questions, proposed by Gavinsky and Pudlák (CCC 2008), is to establish an explicit exponential separation between quantum and classical one-way NOF communication.
In this paper, we resolve this open problem by establishing the first exponential separation between quantum and randomized communication complexity in one-way NOF model. Specifically, we define a lifted variant of the Hidden Matching problem of Bar-Yossef, Jayram, and Kerenidis (STOC 2004) and show that it admits an (O(logn))-cost quantum protocol in the one-way NOF setting. By contrast, we prove that any k-party one-way randomized protocol for this problem requires communication Ω(2k/3n1/3). Notably, our separation applies even to a generalization of k-player one-way communication, where the first player speaks once, and all other k−1 players can communicate freely.
Boolean function F(x,y) for x,y∈{0,1}n is an XOR function if F(x,y)=f(x⊕y) for some function f on n input bits, where ⊕ is a bit-wise XOR. XOR functions are relevant in communication complexity, partially for allowing Fourier analytic technique. For total XOR functions it is known that deterministic communication complexity of F is closely related to parity decision tree complexity of f. Montanaro and Osbourne (2009) observed that one-sided communication complexity Dcc→(F) of F is exactly equal to nonadaptive parity decision tree complexity NADT⊕(f) of f. Hatami et al. (2018) showed that unrestricted communication complexity of F is polynomially related to parity decision tree complexity of f. We initiate the studies of a similar connection for partial functions. We show that in case of one-sided communication complexity whether these measures are equal, depends on the number of undefined inputs of f. On the one hand, if Dcc→(F)=t and f is undefined on at most O(n−t2n−t), then NADT⊕(f)=t. On the other hand, for a wide range of values of Dcc→(F) and NADT⊕(f) (from constant to n−2) we provide partial functions for which Dcc→(F)<NADT⊕(f). In particular, we provide a function with an exponential gap between the two measures. Our separation results translate to the case of two-sided communication complexity as well, in particular showing that the result of Hatami et al. (2018) cannot be generalized to partial functions. Previous results for total functions heavily rely on Boolean Fourier analysis and the technique does not translate to partial functions. For the proofs of our results we build a linear algebraic framework instead. Separation results are proved through the reduction to covering codes.