We study the power of quantum fingerprints in the simultaneous message passing (SMP) setting of communication complexity. Yao recently showed how to simulate, with exponential overhead, classical shared-randomness SMP protocols by means of quantum SMP protocols without shared randomness ( $Q^\parallel$ -protocols). Our first result is to extend Yao's simulation to the strongest possible model: every many-round quantum protocol with unlimited shared entanglement can be simulated, with exponential overhead, by $Q^\parallel$ -protocols. We apply our technique to obtain an efficient $Q^\parallel$ -protocol for a function which cannot be efficiently solved through more restricted simulations. Second, we tightly characterize the power of the quantum fingerprinting technique by making a connection to arrangements of homogeneous halfspaces with maximal margin. These arrangements have been well studied in computational learning theory, and we use some strong results obtained in this area to exhibit weaknesses of quantum fingerprinting. In particular, this implies that for almost all functions, quantum fingerprinting protocols are exponentially worse than classical deterministic SMP protocols.We study the power of quantum fingerprints in the simultaneous message passing (SMP) setting of communication complexity. Yao recently showed how to simulate, with exponential overhead, classical shared-randomness SMP protocols by means of quantum SMP protocols without shared randomness ($Q^\parallel$-protocols). Our first result is to extend Yao's simulation to the strongest possible model: every many-round quantum protocol with unlimited shared entanglement can be simulated, with exponential overhead, by $Q^\parallel$-protocols. We apply our technique to obtain an efficient $Q^\parallel$-protocol for a function which cannot be efficiently solved through more restricted simulations. Second, we tightly characterize the power of the quantum fingerprinting technique by making a connection to arrangements of homogeneous halfspaces with maximal margin. These arrangements have been well studied in computational learning theory, and we use some strong results obtained in this area to exhibit weaknesses of quantum fingerprinting. In particular, this implies that for almost all functions, quantum fingerprinting protocols are exponentially worse than classical deterministic SMP protocols.

arXiv:quant-ph/0603173Mar 20, 200613 citations

Quantum Communication Cannot Simulate a Public Coin

Dmitry Gavinsky, Julia Kempe, Ronald de Wolf

We study the simultaneous message passing model of communication complexity. Building on the quantum fingerprinting protocol of Buhrman et al., Yao recently showed that a large class of efficient classical public-coin protocols can be turned into efficient quantum protocols without public coin. This raises the question whether this can be done always, i.e. whether quantum communication can always replace a public coin in the SMP model. We answer this question in the negative, exhibiting a communication problem where classical communication with public coin is exponentially more efficient than quantum communication. Together with a separation in the other direction due to Bar-Yossef et al., this shows that the quantum SMP model is incomparable with the classical public-coin SMP model. In addition we give a characterization of the power of quantum fingerprinting by means of a connection to geometrical tools from machine learning, a quadratic improvement of Yao's simulation, and a nearly tight analysis of the Hamming distance problem from Yao's paper.We study the simultaneous message passing model of communication complexity. Building on the quantum fingerprinting protocol of Buhrman et al., Yao recently showed that a large class of efficient classical public-coin protocols can be turned into efficient quantum protocols without public coin. This raises the question whether this can be done always, i.e. whether quantum communication can always replace a public coin in the SMP model. We answer this question in the negative, exhibiting a communication problem where classical communication with public coin is exponentially more efficient than quantum communication. Together with a separation in the other direction due to Bar-Yossef et al., this shows that the quantum SMP model is incomparable with the classical public-coin SMP model. In addition we give a characterization of the power of quantum fingerprinting by means of a connection to geometrical tools from machine learning, a quadratic improvement of Yao's simulation, and a nearly tight analysis of the Hamming distance problem from Yao's paper.

arXiv:quant-ph/0411051Nov 8, 200421 citations

Quantum Versus Classical Simultaneity in Communication Complexity

Dmitry Gavinsky

This paper addresses two problems in the context of two-party communication complexity of functions. First, it concludes the line of research which can be viewed as demonstrating qualitative advantage of quantum communication in the three most common communication “layouts”: two-way interactive communication, one-way communication and simultaneous message passing (SMP). I demonstrate a functional (problem c̅E̅q̅T̅, whose communication complexity is O (log n)2) in the quantum version of the SMP and Ω̃ (√n) in the classical (randomized) version of SMP. Second, this paper contributes to understanding the power of the weakest commonly studied regime of quantum communication-SMP with quantum messages and without shared randomness (the latter restriction can be viewed as a somewhat artificial way of making the quantum model “as weak as possible”). Our function c̅E̅q̅T̅ has an efficient solution in this regime as well, which means that even lacking shared randomness, quantum SMP can be exponentially stronger than its classical counterpart with shared randomness.This paper addresses two problems in the context of two-party communication complexity of functions. First, it concludes the line of research which can be viewed as demonstrating qualitative advantage of quantum communication in the three most common communication “layouts”: two-way interactive communication, one-way communication and simultaneous message passing (SMP). I demonstrate a functional (problem c̅E̅q̅T̅, whose communication complexity is O (log n)2) in the quantum version of the SMP and Ω̃ (√n) in the classical (randomized) version of SMP. Second, this paper contributes to understanding the power of the weakest commonly studied regime of quantum communication-SMP with quantum messages and without shared randomness (the latter restriction can be viewed as a somewhat artificial way of making the quantum model “as weak as possible”). Our function c̅E̅q̅T̅ has an efficient solution in this regime as well, which means that even lacking shared randomness, quantum SMP can be exponentially stronger than its classical counterpart with shared randomness.

arXiv:1705.07211IEEE Transactions on Information TheoryMay 22, 20196 citations

A Note on Shared Randomness and Shared Entanglement in Communication

Dmitry Gavinsky

We consider several models of 1-round classical and quantum communication, some of these models have not been defined before. We "almost separate" the models of simultaneous quantum message passing with shared entanglement and the model of simultaneous quantum message passing with shared randomness. We define a relation which can be efficiently exactly solved in the first model but cannot be solved efficiently, either exactly or in 0-error setup in the second model. In fact, our relation is exactly solvable even in a more restricted model of simultaneous classical message passing with shared entanglement. As our second contribution we strengthen a result by Yao that a "very short" protocol from the model of simultaneous classical message passing with shared randomness can be simulated in the model of simultaneous quantum message passing: for a boolean function f, QII(f) \in exp(O(RIIp(f))) log n. We show a similar result for protocols from a (stronger) model of 1-way classical message passing with shared randomness: QII(f) \in exp(O(RIp(f))) log n. We demonstrate a problem whose efficient solution in the model of simultaneous quantum message passing follows from our result but not from Yao's.We consider several models of 1-round classical and quantum communication, some of these models have not been defined before. We "almost separate" the models of simultaneous quantum message passing with shared entanglement and the model of simultaneous quantum message passing with shared randomness. We define a relation which can be efficiently exactly solved in the first model but cannot be solved efficiently, either exactly or in 0-error setup in the second model. In fact, our relation is exactly solvable even in a more restricted model of simultaneous classical message passing with shared entanglement. As our second contribution we strengthen a result by Yao that a "very short" protocol from the model of simultaneous classical message passing with shared randomness can be simulated in the model of simultaneous quantum message passing: for a boolean function f, QII(f) \in exp(O(RIIp(f))) log n. We show a similar result for protocols from a (stronger) model of 1-way classical message passing with shared randomness: QII(f) \in exp(O(RIp(f))) log n. We demonstrate a problem whose efficient solution in the model of simultaneous quantum message passing follows from our result but not from Yao's.

arXiv:quant-ph/0505088May 12, 20053 citations

Exact Asymptotic Rates and an Exponential Strong Converse for quantum SMP and One-Way Communication

Daiki Suruga

Computing many instances jointly can reduce communication per instance. We ask whether such savings occur in the simultaneous-message-passing (SMP) model and how they depend on quantum messages and shared resources. For every finite total function $f$ and fixed error $0\le\varepsilon<1$ , we prove that the optimal worst-case communication per instance for $f^n$ converges to $D^\parallel(f)=\log|\mathrm{Row}(f)|+\log|\mathrm{Col}(f)|,$ where $\mathrm{Row}(f)$ and $\mathrm{Col}(f)$ are the sets of distinct rows and columns of the table $(f(x,y))_{x,y}$ . This holds for classical and quantum SMP without shared entanglement, with or without shared randomness. Thus, transmitting row and column indices is asymptotically optimal, even with bounded error, joint computation, and quantum messages. In particular, the exponential advantage of quantum fingerprinting for a single instance of equality without shared randomness disappears in the asymptotic rate. We also prove that the same threshold governs an exponential strong converse: every fixed worst-case communication rate below it forces the probability of computing all $n$ outputs correctly to decay as $2^{-Ω(n)}$ , even on average under a fixed product input distribution. With shared randomness, the optimal expected communication rate is $(1-\varepsilon)D^\parallel(f)$ . Entanglement shared between each sender and the referee halves both rates; arbitrary tripartite entanglement gives no further reduction. The results extend to a class of total relations and yield one-way characterizations with $D^\parallel(f)$ replaced by $\log|\mathrm{Row}(f)|$ . Our proof reduces SMP to one-way communication and relates successful computation to sampling based on quantum min-entropy; we also give a simpler proof of the key lemma using the pretty-good measurement.Computing many instances jointly can reduce communication per instance. We ask whether such savings occur in the simultaneous-message-passing (SMP) model and how they depend on quantum messages and shared resources. For every finite total function $f$ and fixed error $0\le\varepsilon<1$, we prove that the optimal worst-case communication per instance for $f^n$ converges to \[ D^\parallel(f)=\log|\mathrm{Row}(f)|+\log|\mathrm{Col}(f)|, \] where $\mathrm{Row}(f)$ and $\mathrm{Col}(f)$ are the sets of distinct rows and columns of the table $(f(x,y))_{x,y}$. This holds for classical and quantum SMP without shared entanglement, with or without shared randomness. Thus, transmitting row and column indices is asymptotically optimal, even with bounded error, joint computation, and quantum messages. In particular, the exponential advantage of quantum fingerprinting for a single instance of equality without shared randomness disappears in the asymptotic rate. We also prove that the same threshold governs an exponential strong converse: every fixed worst-case communication rate below it forces the probability of computing all $n$ outputs correctly to decay as $2^{-Ω(n)}$, even on average under a fixed product input distribution. With shared randomness, the optimal expected communication rate is $(1-\varepsilon)D^\parallel(f)$. Entanglement shared between each sender and the referee halves both rates; arbitrary tripartite entanglement gives no further reduction. The results extend to a class of total relations and yield one-way characterizations with $D^\parallel(f)$ replaced by $\log|\mathrm{Row}(f)|$. Our proof reduces SMP to one-way communication and relates successful computation to sampling based on quantum min-entropy; we also give a simpler proof of the key lemma using the pretty-good measurement.

quant-pharXiv:2609.14658Sep 15, 20260 citations

Simultaneous Communication Protocols with Quantum and Classical Messages

Dmitry Gavinsky, Oded Regev, Ronald de Wolf

We study the simultaneous message passing (SMP) model of communication complexity, for the case where one party is quantum and the other is classical. We show that in an SMP protocol that computes some function with the first party sending q qubits and the second sending c classical bits, the quantum message can be replaced by a randomized message of O(qc) classical bits, as well as by a deterministic message of O(q c log q) classical bits. Our proofs rely heavily on earlier results due to Scott Aaronson. In particular, our results imply that quantum-classical protocols need to send Omega(sqrt{n/log n}) bits/qubits to compute Equality on n-bit strings, and hence are not significantly better than classical-classical protocols (and are much worse than quantum-quantum protocols such as quantum fingerprinting). This essentially answers a recent question of Wim van Dam. Our results also imply, more generally, that there are no superpolynomial separations between quantum-classical and classical-classical SMP protocols for functional problems. This contrasts with the situation for relational problems, where exponential gaps between quantum-classical and classical-classical SMP protocols are known. We show that this surprising situation cannot arise in purely classical models: there, an exponential separation for a relational problem can be converted into an exponential separation for a functional problem.We study the simultaneous message passing (SMP) model of communication complexity, for the case where one party is quantum and the other is classical. We show that in an SMP protocol that computes some function with the first party sending q qubits and the second sending c classical bits, the quantum message can be replaced by a randomized message of O(qc) classical bits, as well as by a deterministic message of O(q c log q) classical bits. Our proofs rely heavily on earlier results due to Scott Aaronson. In particular, our results imply that quantum-classical protocols need to send Omega(sqrt{n/log n}) bits/qubits to compute Equality on n-bit strings, and hence are not significantly better than classical-classical protocols (and are much worse than quantum-quantum protocols such as quantum fingerprinting). This essentially answers a recent question of Wim van Dam. Our results also imply, more generally, that there are no superpolynomial separations between quantum-classical and classical-classical SMP protocols for functional problems. This contrasts with the situation for relational problems, where exponential gaps between quantum-classical and classical-classical SMP protocols are known. We show that this surprising situation cannot arise in purely classical models: there, an exponential separation for a relational problem can be converted into an exponential separation for a functional problem.

arXiv:0807.2758Chicago Journal of Theoretical Computer ScienceJul 17, 200815 citations

Q^\parallel

-protocols). Our first result is to extend Yao's simulation to the strongest possible model: every many-round quantum protocol with unlimited shared entanglement can be simulated, with exponential overhead, by

Q^\parallel

-protocols. We apply our technique to obtain an efficient

Q^\parallel

-protocol for a function which cannot be efficiently solved through more restricted simulations. Second, we tightly characterize the power of the quantum fingerprinting technique by making a connection to arrangements of homogeneous halfspaces with maximal margin. These arrangements have been well studied in computational learning theory, and we use some strong results obtained in this area to exhibit weaknesses of quantum fingerprinting. In particular, this implies that for almost all functions, quantum fingerprinting protocols are exponentially worse than classical deterministic SMP protocols.

f

and fixed error

0\le\varepsilon<1

, we prove that the optimal worst-case communication per instance for

f^n

converges to

D^\parallel(f)=\log|\mathrm{Row}(f)|+\log|\mathrm{Col}(f)|,

where

\mathrm{Row}(f)

and

\mathrm{Col}(f)

are the sets of distinct rows and columns of the table

(f(x,y))_{x,y}

. This holds for classical and quantum SMP without shared entanglement, with or without shared randomness. Thus, transmitting row and column indices is asymptotically optimal, even with bounded error, joint computation, and quantum messages. In particular, the exponential advantage of quantum fingerprinting for a single instance of equality without shared randomness disappears in the asymptotic rate. We also prove that the same threshold governs an exponential strong converse: every fixed worst-case communication rate below it forces the probability of computing all

n

outputs correctly to decay as

2^{-Ω(n)}

, even on average under a fixed product input distribution. With shared randomness, the optimal expected communication rate is

(1-\varepsilon)D^\parallel(f)

. Entanglement shared between each sender and the referee halves both rates; arbitrary tripartite entanglement gives no further reduction. The results extend to a class of total relations and yield one-way characterizations with

D^\parallel(f)

replaced by

\log|\mathrm{Row}(f)|

. Our proof reduces SMP to one-way communication and relates successful computation to sampling based on quantum min-entropy; we also give a simpler proof of the key lemma using the pretty-good measurement.