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arXiv:2107.11806 · Leibniz international proceedings in informatics

Tight Bounds for the Randomized and Quantum Communication Complexities of Equality with Small Error

Lalonde, Olivier, Nikhil S. Mande, Ronald de Wolf

Published Jul 25, 2021• 0 citations
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Abstract

We investigate the randomized and quantum communication complexities of the well-studied Equality function with small error probability $ε$, getting optimal constant factors in the leading terms in a number of different models. In the randomized model, 1) we give a general technique to convert public-coin protocols to private-coin protocols by incurring a small multiplicative error, at a small additive cost. This is an improvement over Newman's theorem [Inf. Proc. Let.'91] in the dependence on the error parameter. 2) Using this we obtain a $(\log(n/ε^2)+4)$-cost private-coin communication protocol that computes the $n$-bit Equality function, to error $ε$. This improves upon the $\log(n/ε^3)+O(1)$ upper bound implied by Newman's theorem, and matches the best known lower bound, which follows from Alon [Comb. Prob. Comput.'09], up to an additive $\log\log(1/ε)+O(1)$. In the quantum model, 1) we exhibit a one-way protocol of cost $\log(n/ε)+4$, that uses only pure states and computes the $n$-bit Equality function to error $ε$. This bound was implicitly already shown by Nayak [PhD thesis'99]. 2) We show that any $ε$-error one-way protocol for $n$-bit Equality that uses only pure states communicates at least $\log(n/ε)-\log\log(1/ε)-O(1)$ qubits. 3) We exhibit a one-way protocol of cost $\log(\sqrt{n}/ε)+3$, that uses mixed states and computes the $n$-bit Equality function to error $ε$. This is also tight up to an additive $\log\log(1/ε)+O(1)$, which follows from Alon's result. 4) We study the number of EPR pairs required to be shared in an entanglement-assisted one-way protocol. Our upper bounds also yield upper bounds on the approximate rank and related measures of the Identity matrix.

DOI
10.4230/lipics.fsttcs.2023.32

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Tight Bounds for the Randomized and Quantum Communication Complexities of Equality with Small Error

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We investigate the randomized and quantum communication complexities of the well-studied Equality function with small error probability εεε, getting optimal constant factors in the leading terms in a number of different models. In the randomized model, 1) we give a general technique to convert public-coin protocols to private-coin protocols by incurring a small multiplicative error, at a small additive cost. This is an improvement over Newman's theorem [Inf. Proc. Let.'91] in the dependence on the error parameter. 2) Using this we obtain a (log⁡(n/ε2)+4)(\log(n/ε^2)+4)(log(n/ε2)+4)-cost private-coin communication protocol that computes the nnn-bit Equality function, to error εεε. This improves upon the log⁡(n/ε3)+O(1)\log(n/ε^3)+O(1)log(n/ε3)+O(1) upper bound implied by Newman's theorem, and matches the best known lower bound, which follows from Alon [Comb. Prob. Comput.'09], up to an additive log⁡log⁡(1/ε)+O(1)\log\log(1/ε)+O(1)loglog(1/ε)+O(1). In the quantum model, 1) we exhibit a one-way protocol of cost log⁡(n/ε)+4\log(n/ε)+4log(n/ε)+4, that uses only pure states and computes the nnn-bit Equality function to error εεε. This bound was implicitly already shown by Nayak [PhD thesis'99]. 2) We show that any εεε-error one-way protocol for nnn-bit Equality that uses only pure states communicates at least log⁡(n/ε)−log⁡log⁡(1/ε)−O(1)\log(n/ε)-\log\log(1/ε)-O(1)log(n/ε)−loglog(1/ε)−O(1) qubits. 3) We exhibit a one-way protocol of cost log⁡(n/ε)+3\log(\sqrt{n}/ε)+3log(n​/ε)+3, that uses mixed states and computes the nnn-bit Equality function to error εεε. This is also tight up to an additive log⁡log⁡(1/ε)+O(1)\log\log(1/ε)+O(1)loglog(1/ε)+O(1), which follows from Alon's result. 4) We study the number of EPR pairs required to be shared in an entanglement-assisted one-way protocol. Our upper bounds also yield upper bounds on the approximate rank and related measures of the Identity matrix.
We investigate the randomized and quantum communication complexities of the well-studied Equality function with small error probability ϵ\epsilonϵ, getting the optimal constant factors in the leading terms in a number of different models. In the randomized model, 1) we give a general technique to convert public-coin protocols to private-coin protocols by incurring a small multiplicative error, at a small additive cost. This is an improvement over Newman's theorem [Inf. Proc. Let.'91] in the dependence on the error parameter. 2) Using this we obtain a (log⁡(n/ϵ2)+4)(\log(n/\epsilon^2)+4)(log(n/ϵ2)+4)-cost private-coin communication protocol that computes the nnn-bit Equality function, to error ϵ\epsilonϵ. This improves upon the log⁡(n/ϵ3)+O(1)\log(n/\epsilon^3)+O(1)log(n/ϵ3)+O(1) upper bound implied by Newman's theorem, and matches the best known lower bound, which follows from Alon [Comb. Prob. Comput.'09], up to an additive log⁡log⁡(1/ϵ)+O(1)\log\log(1/\epsilon)+O(1)loglog(1/ϵ)+O(1). In the quantum model, 1) we exhibit a one-way protocol of cost log⁡(n/ϵ)+4\log(n/\epsilon)+4log(n/ϵ)+4, that uses only pure states and computes the nnn-bit Equality function to error ϵ\epsilonϵ. This bound was implicitly already shown by Nayak [PhD thesis'99]. 2) We show that any ϵ\epsilonϵ-error one-way protocol for nnn-bit Equality that uses only pure states communicates at least log⁡(n/ϵ)−log⁡log⁡(1/ϵ)−O(1)\log(n/\epsilon)-\log\log(1/\epsilon)-O(1)log(n/ϵ)−loglog(1/ϵ)−O(1) qubits. 3) We exhibit a one-way protocol of cost log⁡(n/ϵ)+3\log(\sqrt{n}/\epsilon)+3log(n​/ϵ)+3, that uses mixed states and computes the nnn-bit Equality function to error ϵ\epsilonϵ. This is also tight up to an additive log⁡log⁡(1/ϵ)+O(1)\log\log(1/\epsilon)+O(1)loglog(1/ϵ)+O(1), which follows from Alon's result. Our upper bounds also yield upper bounds on the approximate rank and related measures of the Identity matrix. This also implies improved upper bounds on these measures for the distributed SINK function, which was recently used to refute the randomized and quantum versions of the log-rank conjecture.
We study the relative power of public randomness and quantum communication in the multiparty Simultaneous Message Passing (SMP) model. By analyzing a total Boolean function and a relational problem, we establish polynomial and exponential separations between classical public-coin protocols and quantum protocols without shared randomness or prior entanglement. We first introduce XOR Equality, a total Boolean function in which each of kkk players receives an unrestricted string xj∈{0,1}nx_j\in\{0,1\}^nxj​∈{0,1}n, and the referee must decide whether x1⊕⋯⊕xk=0nx_1\oplus\cdots\oplus x_k=0^nx1​⊕⋯⊕xk​=0n. For k≥8k\ge8k≥8 and worst-case error at most 1/31/31/3, we prove quantum lower bounds of Ω(min⁡{n,k2})Ω(\min\{n,k^2\})Ω(min{n,k2}) on total communication and Ω(min⁡{n/k,k})Ω(\min\{n/k,k\})Ω(min{n/k,k}) on maximum message length. A classical public-coin protocol uses two bits per player with error at most 1/41/41/4. For n=k2n=k^2n=k2 and worst-case error at most 1/31/31/3, we prove that quantum protocols require maximum message length Ω(n)Ω(\sqrt n)Ω(n​) qubits, whereas two public-coin bits per player suffice. We then turn to relations, for which we obtain exponential separations. Specifically, we study Multiparty Index Coordination, a kkk-player generalization of the two-player relation studied by Gavinsky, Kempe, Regev, and de Wolf (STOC 2006). We prove an unambiguous quantum lower bound of Ω(n1−1/k)Ω(n^{1-1/k})Ω(n1−1/k) on maximum message length, matched by a classical private-coin protocol. For any fixed error ε∈(0,1/8)\varepsilon\in(0,1/8)ε∈(0,1/8), we also prove a bounded-error quantum lower bound of Ω(n(k−1)/(k+1))Ω(n^{(k-1)/(k+1)})Ω(n(k−1)/(k+1)) while we show that public-coin protocols solve the relation unambiguously with maximum message length O(log⁡n)O(\log n)O(logn), giving exponential separations in both regimes. The implicit constants are independent of kkk, and both quantum lower bounds become linear when k≥clog⁡nk\ge c\log nk≥clogn for any fixed c>0c>0c>0.
We investigate the randomized and quantum communication complexities of the well-studied Equality function with small error probability ε, getting the optimal constant factors in the leading terms in various different models. The following are our results in the randomized model: - We give a general technique to convert public-coin protocols to private-coin protocols by incurring a small multiplicative error at a small additive cost. This is an improvement over Newman’s theorem [Inf. Proc. Let.'91] in the dependence on the error parameter. - As a consequence we obtain a (log(n/ε²) + 4)-cost private-coin communication protocol that computes the n-bit Equality function, to error ε. This improves upon the log(n/ε³) + O(1) upper bound implied by Newman’s theorem, and matches the best known lower bound, which follows from Alon [Comb. Prob. Comput.'09], up to an additive log log(1/ε) + O(1). The following are our results in various quantum models: - We exhibit a one-way protocol with log(n/ε) + 4 qubits of communication for the n-bit Equality function, to error ε, that uses only pure states. This bound was implicitly already shown by Nayak [PhD thesis'99]. - We give a near-matching lower bound: any ε-error one-way protocol for n-bit Equality that uses only pure states communicates at least log(n/ε) - log log(1/ε) - O(1) qubits. - We exhibit a one-way protocol with log(√n/ε) + 3 qubits of communication that uses mixed states. This is tight up to additive log log(1/ε) + O(1), which follows from Alon’s result. - We exhibit a one-way entanglement-assisted protocol achieving error probability ε with ⌈log(1/ε)⌉ + 1 classical bits of communication and ⌈log(√n/ε)⌉ + 4 shared EPR-pairs between Alice and Bob. This matches the communication cost of the classical public coin protocol achieving the same error probability while improving upon the amount of prior entanglement that is needed for this protocol, which is ⌈log(n/ε)⌉ + O(1) shared EPR-pairs. Our upper bounds also yield upper bounds on the approximate rank, approximate nonnegative-rank, and approximate psd-rank of the Identity matrix. As a consequence we also obtain improved upper bounds on these measures for a function that was recently used to refute the randomized and quantum versions of the log-rank conjecture (Chattopadhyay, Mande and Sherif [J. ACM'20], Sinha and de Wolf [FOCS'19], Anshu, Boddu and Touchette [FOCS'19]).
We show optimal Direct Sum result for the one-way entanglement-assisted quantum com- munication complexity for any relation fX × Y × Z. We show: Q 1,pub (f ⊕m ) = (m · Q 1,pub (f)), where Q 1,pub (f), represents the one-way entanglement-assisted quantum communication complexity of f with error at most 1/3 and f ⊕m represents m-copies of f. Similarly for the one-way public-coin classical communication complexity we show: R 1,pub (f ⊕m ) = (m · R 1,pub (f)), where R 1,pub (f), represents the one-way public-coin classical communication complexity of f with error at most 1/3. We show similar optimal Direct Sum results for the Simultaneous Message Passing (SMP) quantum and classical models. For two-party two-way protocols we present optimal Privacy Trade-off results leading to a Weak Direct Sum result for such protocols. We show our Direct Sum and Privacy Trade-off results via message compression arguments. These arguments also imply a new round elimination lemma in quantum communication, which allows us to extend classical lower bounds on the cell probe complexity of some data structure problems, e.g. Approximate Nearest Neighbor Searching (ANN) on the Hamming cube {0,1} n and Predecessor Search to the quantum setting. In a separate result we show that Newman's (New91) technique of reducing the number of public-coins in a classical protocol cannot be lifted to the quantum setting. We do this by defining a general notion of black-box reduction of prior entanglement that subsumes Newman's technique. We prove that such a black-box reduction is impossible for quantum protocols by exhibiting a particular one-round quantum protocol for the Equality function where the black-box technique fails to reduce the amount of prior entanglement by more than a constant factor.
Private simultaneous quantum messages (PSQM) provide a minimal model of noninteractive secure multiparty quantum computation, yet their communication complexity for concrete functions remains poorly understood. We show that familiar total functions already exhibit both genuine quantum advantages and strong structural limitations. For three-party equality on nnn-bit inputs, which determines whether the players' nnn-bit strings are identical, we construct an entanglement-assisted protocol with asymptotic communication (1.633…)n(1.633\ldots)n(1.633…)n qubits, whereas every perfectly correct protocol without prior entanglement requires at least 3n3n3n qubits. For three-bit majority, one shared GHZ state and one transmitted qubit per player suffice for perfect correctness and perfect privacy, reducing the communication from the optimal classical cost of six bits to three qubits. Three-bit AND behaves differently. No perfectly correct and perfectly private protocol exists when the players share an arbitrary three-qubit state, apply arbitrary local unitary encodings, and transmit one qubit each. However, increasing the local dimension overcomes this obstruction: one shared three-qutrit GHZ state and one transmitted qutrit per player yield a one-shot protocol with total communication 3log⁡233\log_2 33log2​3. Together with the six-bit classical lower bound, this gives a strict quantum communication advantage. More generally, for every k≥3k\geq 3k≥3, we construct a perfectly correct and perfectly private one-shot protocol for kkk-bit AND using one qudit of dimension at most kkk per player, and hence at most klog⁡2kk\log_2 kklog2​k qubits in total.
Shared entanglement is a resource available to parties communicating over a quantum channel, much akin to public coins in classical communication protocols. Whereas shared randomness does not help in the transmission of information, or significantly reduce the classical complexity of computing functions (as compared to private-coin protocols), shared entanglement leads to startling phenomena such as ``quantum teleportation'' and ``superdense coding.'' The problem of characterising the power of prior entanglement has puzzled many researchers. In this paper, we revisit the problem of transmitting classical bits over an entanglement-assisted quantum channel. We derive a new, optimal bound on the number of quantum bits required for this task, for any given probability of error. All known lower bounds in the setting of bounded error entanglement-assisted communication are based on sophisticated information theoretic arguments. In contrast, our result is derived from first principles, using a simple linear algebraic technique.