We prove new lower bounds for bounded error quantum communication complexity. Our methods are based on the Fourier transform of the considered functions. First we generalize a method for proving classical communication complexity lower bounds developed by R. Raz (1995) to the quantum case. Applying this method we give an exponential separation between bounded error quantum communication complexity and nondeterministic quantum communication complexity. We develop several other Fourier based lower bound methods, notably showing that /spl radic/(s~(f)/log n) n, for the average sensitivity s~(f) of a function f, yields a lower bound on the bounded error quantum communication complexity of f (x/spl and/y/spl oplus/yz), where x is a Boolean word held by Alice and y, z are Boolean words held by Bob. We then prove the first large lower bounds on the bounded error quantum communication complexity of functions, for which a polynomial quantum speedup is possible. For all the functions we investigate, only the previously applied general lower bound method based on discrepancy yields bounds that are O(log n).We prove new lower bounds for bounded error quantum communication complexity. Our methods are based on the Fourier transform of the considered functions. First we generalize a method for proving classical communication complexity lower bounds developed by R. Raz (1995) to the quantum case. Applying this method we give an exponential separation between bounded error quantum communication complexity and nondeterministic quantum communication complexity. We develop several other Fourier based lower bound methods, notably showing that /spl radic/(s~(f)/log n) n, for the average sensitivity s~(f) of a function f, yields a lower bound on the bounded error quantum communication complexity of f (x/spl and/y/spl oplus/yz), where x is a Boolean word held by Alice and y, z are Boolean words held by Bob. We then prove the first large lower bounds on the bounded error quantum communication complexity of functions, for which a polynomial quantum speedup is possible. For all the functions we investigate, only the previously applied general lower bound method based on discrepancy yields bounds that are O(log n).

Annual Symposium on Foundations of Computer ScienceJan 1, 200130 citations

Lower Bounds for Quantum Communication Complexity

Hartmut Klauck

We prove lower bounds on the bounded error quantum communication complexity. Our methods are based on the Fourier transform of the considered functions. First we generalize a method for proving classical communication complexity lower bounds developed by Raz [Comput. Complexity, 5 (1995), pp. 205–221] to the quantum case. Applying this method, we give an exponential separation between bounded error quantum communication complexity and nondeterministic quantum communication complexity. We develop several other lower bound methods based on the Fourier transform, notably showing that $\sqrt{\bar{s}(f)/\log n}$ , for the average sensitivity $\bar{s}(f)$ of a function f, yields a lower bound on the bounded error quantum communication complexity of $f((x \wedge y)\oplus z)$ , where x is a Boolean word held by Alice and $y,z$ are Boolean words held by Bob. We then prove the first large lower bounds on the bounded error quantum communication complexity of functions, for which a polynomial quantum speedup is possible. For all the functions we investigate, the only previously applied general lower bound method based on discrepancy yields bounds that are $O(\log n)$ .We prove lower bounds on the bounded error quantum communication complexity. Our methods are based on the Fourier transform of the considered functions. First we generalize a method for proving classical communication complexity lower bounds developed by Raz [Comput. Complexity, 5 (1995), pp. 205–221] to the quantum case. Applying this method, we give an exponential separation between bounded error quantum communication complexity and nondeterministic quantum communication complexity. We develop several other lower bound methods based on the Fourier transform, notably showing that $\sqrt{\bar{s}(f)/\log n}$, for the average sensitivity $\bar{s}(f)$ of a function f, yields a lower bound on the bounded error quantum communication complexity of $f((x \wedge y)\oplus z)$, where x is a Boolean word held by Alice and $y,z$ are Boolean words held by Bob. We then prove the first large lower bounds on the bounded error quantum communication complexity of functions, for which a polynomial quantum speedup is possible. For all the functions we investigate, the only previously applied general lower bound method based on discrepancy yields bounds that are $O(\log n)$.

SIAM Journal on ComputingJan 1, 200768 citations

The pattern matrix method for lower bounds on quantum communication

Alexander A. Sherstov

In a breakthrough result, Razborov (2003) gave optimal lower bounds on the communication complexity of every function f of the form f(x,y)=D(|x AND y|) for some D:{0,1,...,n}->{0,1}, in the bounded-error quantum model with and without prior entanglement. This was proved by the multidimensional discrepancy method. We give an entirely different proof of Razborov's result, using the original, one-dimensional discrepancy method. This refutes the commonly held intuition (Razborov 2003) that the original discrepancy method fails for functions such as DISJOINTNESS. More importantly, our communication lower bounds hold for a much broader class of functions for which no methods were available. Namely, fix an arbitrary function f:{0,1}n/4->{0,1} and let A be the Boolean matrix whose columns are each an application of f to some subset of the variables x1,x2,...,xn. We prove that the communication complexity of A in the bounded-error quantum model with and without prior entanglement is Omega(d), where d is the approximate degree of f. From this result, Razborov's lower bounds follow easily. Our result also establishes a large new class of total Boolean functions whose quantum communication complexity (regardless of prior entanglement) is at best polynomially smaller than their classical complexity. Our proof method is a novel combination of two ingredients. The first is a certain equivalence of approximation and orthogonality in Euclidean n-space, which follows by linear-programming duality. The second is a new construction of suitably structured matrices with low spectral norm, the pattern matrices, which we realize using matrix analysis and the Fourier transform over (Z2)n. The method of this paper has recently inspired important progress in multiparty communication complexity.In a breakthrough result, Razborov (2003) gave optimal lower bounds on the communication complexity of every function f of the form f(x,y)=D(|x AND y|) for some D:{0,1,...,n}->{0,1}, in the bounded-error quantum model with and without prior entanglement. This was proved by the multidimensional discrepancy method. We give an entirely different proof of Razborov's result, using the original, one-dimensional discrepancy method. This refutes the commonly held intuition (Razborov 2003) that the original discrepancy method fails for functions such as DISJOINTNESS. More importantly, our communication lower bounds hold for a much broader class of functions for which no methods were available. Namely, fix an arbitrary function f:{0,1}n/4->{0,1} and let A be the Boolean matrix whose columns are each an application of f to some subset of the variables x1,x2,...,xn. We prove that the communication complexity of A in the bounded-error quantum model with and without prior entanglement is Omega(d), where d is the approximate degree of f. From this result, Razborov's lower bounds follow easily. Our result also establishes a large new class of total Boolean functions whose quantum communication complexity (regardless of prior entanglement) is at best polynomially smaller than their classical complexity. Our proof method is a novel combination of two ingredients. The first is a certain equivalence of approximation and orthogonality in Euclidean n-space, which follows by linear-programming duality. The second is a new construction of suitably structured matrices with low spectral norm, the pattern matrices, which we realize using matrix analysis and the Fourier transform over (Z2)n. The method of this paper has recently inspired important progress in multiparty communication complexity.

ACM Symposium on Theory of Computing (STOC)May 17, 200874 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.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:2606.03249Jun 3, 20260 citations

Near-Optimal Bounds on Bounded-Round Quantum Communication Complexity of Disjointness

Mark Braverman, Ankit Garg, Young Kun Ko +2 more

We prove a near optimal round-communication tradeoff for the two-party quantum communication complexity of disjointness. For protocols with r rounds, we prove a lower bound of Omega(n/r) on the communication required for computing disjointness of input size n, which is optimal up to logarithmic factors. The previous best lower bound was Omega(n/r̂2) due to Jain, Radhakrishnan and Sen. Along the way, we develop several tools for quantum information complexity, one of which is a lower bound for quantum information complexity in terms of the generalized discrepancy method. As a corollary, we get that the quantum communication complexity of any boolean function f is at most 2 ̂O(QIC(f)), where QIC(f) is the prior-free quantum information complexity of f (with error 1/3).We prove a near optimal round-communication tradeoff for the two-party quantum communication complexity of disjointness. For protocols with r rounds, we prove a lower bound of Omega(n/r) on the communication required for computing disjointness of input size n, which is optimal up to logarithmic factors. The previous best lower bound was Omega(n/r̂2) due to Jain, Radhakrishnan and Sen. Along the way, we develop several tools for quantum information complexity, one of which is a lower bound for quantum information complexity in terms of the generalized discrepancy method. As a corollary, we get that the quantum communication complexity of any boolean function f is at most 2 ̂O(QIC(f)), where QIC(f) is the prior-free quantum information complexity of f (with error 1/3).

arXiv:1505.03110Annual Symposium on Foundations of Computer ScienceOct 1, 201525 citations

Efficient quantum protocols for XOR functions

Shengyu Zhang

We show that for any Boolean function f : {0,1}n → {0,1}, the bounded-error quantum communication complexity Q∊(f ○ ⊕) of XOR functions f(x ⊕ y) satisfies that where d = deg2(f) is the ℱ2-degree of f, and ‖ ‖1,∊ = ming:‖f–g‖∞≤∊ ‖ĝ‖1. This implies that the previous lower bound Q∊(f ○ ⊕) = Ω(log ‖ ‖1,∊) by Lee and Shraibman [LS09] is tight for f with low 2-degree. The result also confirms the quantum version of the Log-rank Conjecture for low-degree XOR functions. In addition, we show that the exact quantum communication complexity satisfies where ‖ ‖0 is the number of nonzero Fourier coefficients of f. This matches the previous lower bound QE(f(x, y)) = Ω(logrank(Mf)) by Buhrman and de Wolf [BdW01] for low-degree XOR functions.We show that for any Boolean function f : {0,1}n → {0,1}, the bounded-error quantum communication complexity Q∊(f ○ ⊕) of XOR functions f(x ⊕ y) satisfies that where d = deg2(f) is the ℱ2-degree of f, and ‖ ‖1,∊ = ming:‖f–g‖∞≤∊ ‖ĝ‖1. This implies that the previous lower bound Q∊(f ○ ⊕) = Ω(log ‖ ‖1,∊) by Lee and Shraibman [LS09] is tight for f with low 2-degree. The result also confirms the quantum version of the Log-rank Conjecture for low-degree XOR functions. In addition, we show that the exact quantum communication complexity satisfies where ‖ ‖0 is the number of nonzero Fourier coefficients of f. This matches the previous lower bound QE(f(x, y)) = Ω(logrank(Mf)) by Buhrman and de Wolf [BdW01] for low-degree XOR functions.

arXiv:1307.6738ACM-SIAM Symposium on Discrete Algorithms (SODA)Dec 18, 201317 citations

\sqrt{\bar{s}(f)/\log n}

, for the average sensitivity

\bar{s}(f)

of a function f, yields a lower bound on the bounded error quantum communication complexity of

f((x \wedge y)\oplus z)

, where x is a Boolean word held by Alice and

y,z

are Boolean words held by Bob. We then prove the first large lower bounds on the bounded error quantum communication complexity of functions, for which a polynomial quantum speedup is possible. For all the functions we investigate, the only previously applied general lower bound method based on discrepancy yields bounds that are

O(\log n)

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.