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arXiv:0906.1399 · Quantum Information and Computation

On quantum-classical equivalence for composed communication problems

Alexander A. Sherstov

Published May 1, 2010• 18 citations
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Abstract

An open problem in communication complexity proposed by several authors is to prove that for every Boolean function $f,$ the task of computing $f(x\wedge y)$ has polynomially related classical and quantum bounded-error complexities. We solve a variant of this question. For every $f,$ we prove that the task of computing, on input \mbox{$x$ and $y,$} \emph{both} of the quantities $f(x\wedge y)$ and $f(x\vee y)$ has polynomially related classical and quantum bounded-error complexities. We further show that the quantum bounded-error complexity is polynomially related to the classical deterministic complexity and the block sensitivity of $f.$ This result holds regardless of prior entanglement.

DOI
10.26421/qic10.5-6-5

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An open problem in communication complexity proposed by several authors is to prove that for every Boolean function f,f,f, the task of computing f(x∧y)f(x\wedge y)f(x∧y) has polynomially related classical and quantum bounded-error complexities. We solve a variant of this question. For every f,f,f, we prove that the task of computing, on input \mbox{xxx and y,y,y,} \emph{both} of the quantities f(x∧y)f(x\wedge y)f(x∧y) and f(x∨y)f(x\vee y)f(x∨y) has polynomially related classical and quantum bounded-error complexities. We further show that the quantum bounded-error complexity is polynomially related to the classical deterministic complexity and the block sensitivity of f.f.f. This result holds regardless of prior entanglement.
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.
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.
A major open problem in communication complexity is whether or not quantum protocols can be exponentially more efficient than classical ones for computing a {\em total} Boolean function in the two-party interactive model. Razborov's result ({\em Izvestiya: Mathematics}, 67(1):145--159, 2002) implies the conjectured negative answer for functions FFF of the following form: F(x,y)=fn(x1⋅y1,x2⋅y2,...,xn⋅yn)F(x, y)=f_n(x_1\cdot y_1, x_2\cdot y_2, ..., x_n\cdot y_n)F(x,y)=fn​(x1​⋅y1​,x2​⋅y2​,...,xn​⋅yn​), where fnf_nfn​ is a {\em symmetric} Boolean function on nnn Boolean inputs, and xix_ixi​, yiy_iyi​ are the iii'th bit of xxx and yyy, respectively. His proof critically depends on the symmetry of fnf_nfn​. We develop a lower-bound method that does not require symmetry and prove the conjecture for a broader class of functions. Each of those functions FFF is the ``block-composition'' of a ``building block'' gk:{0,1}k×{0,1}k→{0,1}g_k : \{0, 1\}^k \times \{0, 1\}^k \rightarrow \{0, 1\}gk​:{0,1}k×{0,1}k→{0,1}, and an fn:{0,1}n→{0,1}f_n : \{0, 1\}^n \rightarrow \{0, 1\}fn​:{0,1}n→{0,1}, such that F(x,y)=fn(gk(x1,y1),gk(x2,y2),...,gk(xn,yn))F(x, y) = f_n( g_k(x_1, y_1), g_k(x_2, y_2), ..., g_k(x_n, y_n) )F(x,y)=fn​(gk​(x1​,y1​),gk​(x2​,y2​),...,gk​(xn​,yn​)), where xix_ixi​ and yiy_iyi​ are the iii'th kkk-bit block of x,y∈{0,1}nkx, y\in\{0, 1\}^{nk}x,y∈{0,1}nk, respectively. We show that as long as g_k itself is "hard'' enough, its block-composition with an arbitrary f_n has polynomially related quantum and classical communication complexities. For example, when g_k is the Inner Product function with k=\Omega(\log n), the deterministic communication complexity of its block-composition with any f_n is asymptotically at most the quantum complexity to the power of 7.
The quantum version of communication complexity allows the two communicating parties to exchange qubits and/or to make use of prior entanglement (shared EPR-pairs). Some lower bound techniques are available for qubit communication complexity, but except for the inner product function, no bounds are known for the model with unlimited prior entanglement. We show that the "log rank" lower bound extends to the strongest variant of quantum communication complexity (qubit communication+unlimited prior entanglement). By relating the rank of the communication matrix to properties of polynomials, we are able to derive some strong bounds for exact protocols. In particular, we prove both the "log rank conjecture" and the polynomial equivalence of quantum and classical communication complexity for various classes of functions. We also derive some weaker bounds for bounded-error quantum protocols.
This dissertation is motivated by the following fundamental questions: (a) are there any exponential gaps between quantum and classical communication complexities? (b) what is the role of entanglement in assisting quantum communications? (c) how to characterize the nonlocality of quantum operations? We study four specific problems below. 1. The communication complexity of the Hamming Distance problem. The Hamming Distance problem is for two parties to determine whether or not the Hamming distance between two n-bit strings is more than a given threshold. We prove tighter quantum lower bounds in the general two-party, interactive communication model. We also construct an efficient classical protocol in the more restricted Simultaneous Message Passing model, improving previous results. 2. The Log-Equivalence Conjecture. A major open problem in communication complexity is whether or not quantum protocols can be exponentially more efficient than classical ones for computing a total Boolean function in the two-party, interactive model. The answer is believed to be No. Razborov proved this conjecture for the most general class of functions so far. We prove this conjecture for a broader class of functions that we called block-composed functions. Our proof appears to be the first demonstration of the dual approach of the polynomial method in proving new results. 3. Classical simulations of bipartite quantum measurement. We define a new ix concept that measures the nonlocality of bipartite quantum operations. From this measure, we derive an upper bound that shows the limitation of entanglement in reducing communication costs. 4. The maximum tensor norm of bipartite superoperators. We define a maximum tensor norm to quantify the nonlocality of bipartite superoperators. We show that a bipartite physically realizable superoperator is bi-local if and only if its maximum tensor norm is 1. Furthermore, the estimation of the maximum tensor norm can also be used to prove quantum lower bounds on communication complexities.
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 sˉ(f)/log⁡n\sqrt{\bar{s}(f)/\log n}sˉ(f)/logn​, for the average sensitivity sˉ(f)\bar{s}(f)sˉ(f) of a function f, yields a lower bound on the bounded error quantum communication complexity of f((x∧y)⊕z)f((x \wedge y)\oplus z)f((x∧y)⊕z), where x is a Boolean word held by Alice and y,zy,zy,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)O(\log n)O(logn).