Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1007.3587 · Quantum Information and Computation

A new exponential separation between quantum and classical one-way communication complexity

Ashley Montanaro

Published Jul 1, 2011• 11 citations
arXivHTMLPDF

Abstract

We present a new example of a partial boolean function whose one-way quantum communication complexity is exponentially lower than its one-way classical communication complexity. The problem is a natural generalisation of the previously studied Subgroup Membership problem: Alice receives a bit string $x$, Bob receives a permutation matrix $M$, and their task is to determine whether $Mx=x$ or $Mx$ is far from $x$. The proof uses Fourier analysis and an inequality of Kahn, Kalai and Linial.

DOI
10.26421/qic11.7-8-3

Related papers

An exponential separation between entanglement-assisted and unassisted one-way quantum communication

Ryan Anselm, Srijita Kundu, Olivier Lalonde +1 more

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.

quant-pharXiv:2610.02099•Oct 2, 2026•0 citations

Exponential Separation for One-Way Quantum Communication Complexity, with Applications to Cryptography

Dmitry Gavinsky, Julia Kempe, Iordanis Kerenidis +2 more

We give an exponential separation between one-way quantum and classical communication protocols for a partial Boolean function (a variant of the Boolean hidden matching problem of Bar-Yossef et al.). Previously, such an exponential separation was known only for a relational problem. The communication problem corresponds to a strong extractor that fails against a small amount of quantum information about its random source. Our proof uses the Fourier coefficients inequality of Kahn, Kalai, and Linial. We also give a number of applications of this separation. In particular, we show that there are privacy amplification schemes that are secure against classical adversaries but not against quantum adversaries; and we give the first example of a key-expansion scheme in the model of bounded-storage cryptography that is secure against classical memory-bounded adversaries but not against quantum ones.

SIAM Journal on Computing•Dec 19, 2008•80 citations

The one-way communication complexity of the Boolean Hidden Matching Problem

Iordanis Kerenidis, Ran Raz

We give a tight lower bound of Omega(\sqrt{n}) for the randomized one-way communication complexity of the Boolean Hidden Matching Problem [BJK04]. Since there is a quantum one-way communication complexity protocol of O(\log n) qubits for this problem, we obtain an exponential separation of quantum and classical one-way communication complexity for partial functions. A similar result was independently obtained by Gavinsky, Kempe, de Wolf [GKdW06]. Our lower bound is obtained by Fourier analysis, using the Fourier coefficients inequality of Kahn Kalai and Linial [KKL88].

arXiv:quant-ph/0607173•Jul 25, 2006•9 citations

The One-Way Communication Complexity of Group Membership

Scott Aaronson, François Le Gall, Alexander C. Russell +1 more

This paper studies the one-way communication complexity of the subgroup membership problem, a classical problem closely related to basic questions in quantum computing.Here Alice receives, as input, a subgroup H of a finite group G; Bob receives an element y ∈ G. Alice is permitted to send a single message to Bob, after which he must decide if his input y is an element of H.We establish the following bounds on the classical communication complexity of this problem in the bounded-error setting:1.The problem can be solved with O(log |G|)-bit communication with the promise that H is normal.2. The problem can be solved with O(d max • log |G|)-bit communication, where d max is the maximum degree of an irreducible complex representation of G.3. For any prime p not dividing |G|, the problem can be solved with O(log |G| + d max • log p)-bit communication, where d max is the maximum degree of an irreducible F p -representation of G.

arXiv:0902.3175•Chicago Journal of Theoretical Computer Science•Jan 1, 2011•3 citations

Exponential separations for one-way quantum communication complexity, with applications to cryptography

Dmitry Gavinsky, Julia Kempe, Iordanis Kerenidis +2 more

We give an exponential separation between one-way quantum and classical communication protocols for twopartial Boolean functions, both of which are variants of the Boolean Hidden Matching Problem of Bar-Yossef et al. Earlier such an exponential separation was known only for a relational version of the Hidden Matching Problem. Our proofs use the Fourier coefficients inequality of Kahn, Kalai, and Linial. We give a number of applications of this separation. In particular, in the bounded-storage model of cryptography we exhibita scheme that is secure against adversaries with a certain amount of classical storage, but insecure against adversaries with a similar (or even much smaller) amount of quantum storage; in the setting of privacy amplification, we show that there are strong extractors that yield a classically secure key, but are insecure against a quantum adversary.

arXiv:quant-ph/0611209•ACM Symposium on Theory of Computing (STOC)•Jun 11, 2007•128 citations

Exponential Separation of Quantum and Classical One-Way Communication Complexity

Ziv Bar-Yossef, T. S. Jayram, Iordanis Kerenidis

We give the first exponential separation between quantum and bounded-error randomized one-way communication complexity. Specifically, we define the Hidden Matching Problem HM$_n$: Alice gets as input a string ${\bf x}\in\{0, 1\}^n$, and Bob gets a perfect matching M on the n coordinates. Bob's goal is to output a tuple $\langle i,j,b \rangle$ such that the edge $(i,j)$ belongs to the matching M and $b=x_i\oplus x_j$. We prove that the quantum one-way communication complexity of HM$_n$ is $O(\log n)$, yet any randomized one-way protocol with bounded error must use $\Omega({\sqrt{n}})$ bits of communication. No asymptotic gap for one-way communication was previously known. Our bounds also hold in the model of Simultaneous Messages (SM), and hence we provide the first exponential separation between quantum SM and randomized SM with public coins. For a Boolean decision version of HM$_n$, we show that the quantum one-way communication complexity remains $O(\log n)$ and that the 0-error randomized one-way communication complexity is $\Omega(n)$. We prove that any randomized linear one-way protocol with bounded error for this problem requires $\Omega(\sqrt[3]{n \log n})$ bits of communication.

SIAM Journal on Computing•Jan 1, 2008•51 citations
We present a new example of a partial boolean function whose one-way quantum communication complexity is exponentially lower than its one-way classical communication complexity. The problem is a natural generalisation of the previously studied Subgroup Membership problem: Alice receives a bit string xxx, Bob receives a permutation matrix MMM, and their task is to determine whether Mx=xMx=xMx=x or MxMxMx is far from xxx. The proof uses Fourier analysis and an inequality of Kahn, Kalai and Linial.
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}f_n\colon \{0,1\}^n \times \{0,1\}^n \to \{0,1\}fn​:{0,1}n×{0,1}n→{0,1} that can be computed with O(log⁡n)O(\log n)O(logn) bits of one-way classical communication given prior entanglement, but that require Ω(n1/3)Ω(n^{1/3})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 give an exponential separation between one-way quantum and classical communication protocols for a partial Boolean function (a variant of the Boolean hidden matching problem of Bar-Yossef et al.). Previously, such an exponential separation was known only for a relational problem. The communication problem corresponds to a strong extractor that fails against a small amount of quantum information about its random source. Our proof uses the Fourier coefficients inequality of Kahn, Kalai, and Linial. We also give a number of applications of this separation. In particular, we show that there are privacy amplification schemes that are secure against classical adversaries but not against quantum adversaries; and we give the first example of a key-expansion scheme in the model of bounded-storage cryptography that is secure against classical memory-bounded adversaries but not against quantum ones.
We give a tight lower bound of Omega(\sqrt{n}) for the randomized one-way communication complexity of the Boolean Hidden Matching Problem [BJK04]. Since there is a quantum one-way communication complexity protocol of O(\log n) qubits for this problem, we obtain an exponential separation of quantum and classical one-way communication complexity for partial functions. A similar result was independently obtained by Gavinsky, Kempe, de Wolf [GKdW06]. Our lower bound is obtained by Fourier analysis, using the Fourier coefficients inequality of Kahn Kalai and Linial [KKL88].
This paper studies the one-way communication complexity of the subgroup membership problem, a classical problem closely related to basic questions in quantum computing.Here Alice receives, as input, a subgroup H of a finite group G; Bob receives an element y ∈ G. Alice is permitted to send a single message to Bob, after which he must decide if his input y is an element of H.We establish the following bounds on the classical communication complexity of this problem in the bounded-error setting:1.The problem can be solved with O(log |G|)-bit communication with the promise that H is normal.2. The problem can be solved with O(d max • log |G|)-bit communication, where d max is the maximum degree of an irreducible complex representation of G.3. For any prime p not dividing |G|, the problem can be solved with O(log |G| + d max • log p)-bit communication, where d max is the maximum degree of an irreducible F p -representation of G.
We give an exponential separation between one-way quantum and classical communication protocols for twopartial Boolean functions, both of which are variants of the Boolean Hidden Matching Problem of Bar-Yossef et al. Earlier such an exponential separation was known only for a relational version of the Hidden Matching Problem. Our proofs use the Fourier coefficients inequality of Kahn, Kalai, and Linial. We give a number of applications of this separation. In particular, in the bounded-storage model of cryptography we exhibita scheme that is secure against adversaries with a certain amount of classical storage, but insecure against adversaries with a similar (or even much smaller) amount of quantum storage; in the setting of privacy amplification, we show that there are strong extractors that yield a classically secure key, but are insecure against a quantum adversary.
We give the first exponential separation between quantum and bounded-error randomized one-way communication complexity. Specifically, we define the Hidden Matching Problem HMn_nn​: Alice gets as input a string x∈{0,1}n{\bf x}\in\{0, 1\}^nx∈{0,1}n, and Bob gets a perfect matching M on the n coordinates. Bob's goal is to output a tuple ⟨i,j,b⟩\langle i,j,b \rangle⟨i,j,b⟩ such that the edge (i,j)(i,j)(i,j) belongs to the matching M and b=xi⊕xjb=x_i\oplus x_jb=xi​⊕xj​. We prove that the quantum one-way communication complexity of HMn_nn​ is O(log⁡n)O(\log n)O(logn), yet any randomized one-way protocol with bounded error must use Ω(n)\Omega({\sqrt{n}})Ω(n​) bits of communication. No asymptotic gap for one-way communication was previously known. Our bounds also hold in the model of Simultaneous Messages (SM), and hence we provide the first exponential separation between quantum SM and randomized SM with public coins. For a Boolean decision version of HMn_nn​, we show that the quantum one-way communication complexity remains O(log⁡n)O(\log n)O(logn) and that the 0-error randomized one-way communication complexity is Ω(n)\Omega(n)Ω(n). We prove that any randomized linear one-way protocol with bounded error for this problem requires Ω(nlog⁡n3)\Omega(\sqrt[3]{n \log n})Ω(3nlogn​) bits of communication.