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:2607.13756

Precoding-based protocols for entanglement assisted linear computation over a quantum many-to-one network

Ruoyu Meng, Aditya Ramamoorthy

cs.ITquant-phSubmitted Jul 15, 2026• Announced Jul 16, 2026• 0 citations
arXivHTMLPDF

Abstract

In this work, we consider the problem of computing a linear combination over a noiseless quantum many-to-one network. There are $k$ senders, Alice$_1$, $\ldots$, Alice$_k$, and a single receiver, Bob. Each Alice$_i$ has a data vector $W_i \in \mathbb{F}^{m_i}$, where $\mathbb{F}$ is a finite field. Bob wants to compute the linear combination $Y = V_1 W_1 + V_2 W_2 + \cdots + V_k W_k \in \mathbb{F}^m$, where $V_i$ is an $m \times m_i$ matrix over $\mathbb{F}$. The senders transmit quantum states to Bob through a noiseless many-to-one quantum network, but they are not allowed to communicate with each other. The senders share entanglement among themselves, while Bob does not share this entanglement. They encode their classical information $W_i$, $i=1,\ldots,k$, into their local subsystems and transmit them to Bob so that he can recover $Y$ through a quantum measurement and subsequent post-processing. The N-Sum Box protocol proposed by Allaix et al. (2025) considers this problem under certain constraints on the linear combination and the distribution of the data vectors among the senders. We present protocols that support the computation of a more general class of linear transformations by giving the senders access to more qudits and allowing them to judiciously precode their input symbols. The communication cost of our schemes is at most that of the best-known prior results in this area and is strictly lower in certain cases. Finally, we demonstrate that the communication cost is subadditive across instances. Specifically, we identify two linear functions for which the total cost of computing them individually is strictly larger than the cost of computing them jointly.

Related papers

On the Capacity of Vector Linear Computation Over a Noiseless Quantum Multiple-Access Channel With Entangled Transmitters

Yuhang Yao, Syed A. Jafar

Network function computation is an active topic in network coding, with much recent progress for linear (over a finite field) computations over broadcast (LCBC) and multiple access (LCMAC) channels. Over a quantum multiple access channel (QMAC) with quantum-entanglement shared among transmitters, the linear computation problem (LC-QMAC) is non-trivial even when the channel is noiseless, because of the challenge of optimally exploiting transmit-side entanglement through distributed coding. Given an arbitrary linear function of data streams defined in a finite field$\mathbb {F}_{d}$, the LC-QMAC problem seeks the optimal communication cost (minimum number of qudits that need to be sent by the transmitters to the receiver, per computation instance) over a noise-free QMAC, when the independent input data streams originate at the corresponding transmitters, who share quantum entanglement in advance. As our main result, we fully solve this problem for$K=3$transmitters ($K\geq 4$settings remain open). Coding schemes based on the$N$-sum box protocol (along with time-sharing and batch-processing) are shown to be information theoretically optimal in all cases.

arXiv:2407.01498•IEEE Transactions on Quantum Engineering•Jan 1, 2025•2 citations

Entanglement-Assisted Coding for Arbitrary Linear Computations Over a Quantum MAC

Lei Hu, Mohamed W. Nomeir, Alptug Aytekin +3 more

We study a linear computation problem over a quantum multiple access channel (LC-QMAC), where S servers share an entangled state and separately store classical data streams W1,⋯,WSover a finite field ${\mathbb{F}_d}$. A user aims to compute K linear combinations of these data streams, represented as $Y = {{\mathbf{V}}_1}{W_1} + {{\mathbf{V}}_2}{W_2} + \cdot s + {{\mathbf{V}}_S}{W_S} \in \mathbb{F}_d^{K \times 1}$. To this end, each server encodes its classical information into its local quantum subsystem and transmits it to the user, who retrieves the desired computations via quantum measurements. In this work, we propose an achievable scheme for LC-QMAC based on the stabilizer formalism and the ideas from entanglement-assisted quantum error–correcting codes (EAQECC). Specifically, given any linear computation matrix, we construct a self-orthogonal matrix that can be implemented using the stabilizer formalism. Also, we apply precoding matrices to minimize the number of auxiliary qudits required. Our scheme achieves more computations per qudit, i.e., a higher computation rate, compared to the best-known methods in the literature, and attains the capacity in certain cases.

arXiv:2501.16296•IEEE Information Theory Workshop (ITW)•Sep 29, 2025•2 citations

N-Sum Box: An Abstraction for Linear Computation over Many-to-one Quantum Networks

Matteo Allaix, Yuxiang Lu, Yuhang Yao +3 more

Linear computations over quantum many-to-one communication networks offer opportunities for communication cost improvements through schemes that exploit quantum entanglement among transmitters to achieve superdense coding gains, combined with classical techniques such as interference alignment. The problem becomes much more broadly accessible if suitable abstractions can be found for the underlying quantum functionality via classical black box models. This work formalizes such an abstraction in the form of an “N-sum box”, a black box generalization of a two-sum protocol of Song et al. with recent applications to$N$-server private information retrieval. The N- sum box has a communication cost of$N$qudits and classical output of a vector of$N$q-ary digits linearly dependent (via an N x 2N transfer matrix) on 2N classical inputs distributed among$N$transmitters. We characterize which transfer matrices are feasible by our construction, both with and without the possibility of additional locally invertible classical operations at the transmitters and receivers.

Globecom•Dec 4, 2023•16 citations

N-Sum Box: An Abstraction for Linear Computation Over Many-to-One Quantum Networks

Matteo Allaix, Yuxiang Lu, Yuhang Yao +3 more

Linear computations over quantum many-to-one communication networks offer opportunities for communication cost improvements through schemes that exploit quantum entanglement among transmitters to achieve superdense coding gains, combined with classical techniques such as interference alignment. The problem becomes much more broadly accessible if suitable abstractions can be found for the underlying quantum functionality via classical black box models. This work formalizes such an abstraction in the form of an “N-sum box”, a black box generalization of a two-sum protocol of Song et al. with recent applications to N-server private information retrieval. The N-sum box has a communication cost of N qudits and classical output of a vector of$N~q$-ary digits linearly dependent (via an$N \times 2N$transfer matrix) on$2N$classical inputs distributed among N transmitters. We characterize which transfer matrices are feasible by our construction, both with and without the possibility of additional locally invertible classical operations at the transmitters and receivers. Furthermore, we provide a sample application to Cross-Subspace Alignment (CSA) schemes to obtain efficient instances of Quantum Private Information Retrieval (QPIR) and Quantum Secure Distributed Batch Matrix Multiplication (QSDBMM). We first describe N-sum boxes based on maximal stabilizers and we then consider non-maximal-stabilizer-based constructions to obtain an instance of Quantum Symmetric Private Information Retrieval.

IEEE Transactions on Information Theory•Dec 11, 2024•8 citations

$N$-Sum Box: An Abstraction for Linear Computation over Many-to-one Quantum Networks

Matteo Allaix, Yuxiang Lu, Yuhang Yao +3 more

Linear computations over quantum many-to-one communication networks offer opportunities for communication cost improvements through schemes that exploit quantum entanglement among transmitters to achieve superdense coding gains, combined with classical techniques such as interference alignment. The problem becomes much more broadly accessible if suitable abstractions can be found for the underlying quantum functionality via classical black box models. This work formalizes such an abstraction in the form of an "$N$-sum box", a black box generalization of a two-sum protocol of Song \emph{et al.} with recent applications to $N$-server private information retrieval. The $N$-sum box has a communication cost of $N$ qudits and classical output of a vector of $N$ $q$-ary digits linearly dependent (via an $N \times 2N$ transfer matrix) on $2N$ classical inputs distributed among $N$ transmitters. We characterize which transfer matrices are feasible by our construction, both with and without the possibility of additional locally invertible classical operations at the transmitters and receivers. Furthermore, we provide a sample application to Cross-Subspace Alignment (CSA) schemes to obtain efficient instances of Quantum Private Information Retrieval (QPIR) and Quantum Secure Distributed Batch Matrix Multiplication (QSDBMM). We first describe $N$-sum boxes based on maximal stabilizers and we then consider non-maximal-stabilizer-based constructions to obtain an instance of Quantum Symmetric Private Information Retrieval.

arXiv:2304.07561•Apr 15, 2023•0 citations

Correlation-assisted quantum communication

Mário Ziman, Vladimír Bužek

Quantum dense coding is considered to be one of the most remarkable demonstrations of the existence of quantum entanglement. In this paper, we analyze the role of correlations (both classical and quantum) in quantum communication protocols. We study how the capacity of a single-qubit quantum channel depends on correlations between Alice and Bob. In particular, we consider a scenario when Alice and Bob share a pair of qubits in a state such that Alice's qubit is in the state described by the density operator $\ensuremath{\varrho}=\frac{1}{2}1+\stackrel{\ensuremath{\rightarrow}}{n}\ensuremath{\cdot}\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\sigma}}.$ We evaluate a capacity of the noiseless channel for two cases: (1) when Bob performs measurement just on the particle received from Alice, and (2) in the case when he utilizes correlations that have been established prior to the communication. We present a simple classical scenario which might serve as a prototype of quantum dense coding. We generalize our results also for qudits. In addition, we address some aspects of the issue of security in correlation-assisted communication protocols.

Physical Review A•Apr 25, 2003•38 citations
In this work, we consider the problem of computing a linear combination over a noiseless quantum many-to-one network. There are kkk senders, Alice1_11​, …\ldots…, Alicek_kk​, and a single receiver, Bob. Each Alicei_ii​ has a data vector Wi∈FmiW_i \in \mathbb{F}^{m_i}Wi​∈Fmi​, where F\mathbb{F}F is a finite field. Bob wants to compute the linear combination Y=V1W1+V2W2+⋯+VkWk∈FmY = V_1 W_1 + V_2 W_2 + \cdots + V_k W_k \in \mathbb{F}^mY=V1​W1​+V2​W2​+⋯+Vk​Wk​∈Fm, where ViV_iVi​ is an m×mim \times m_im×mi​ matrix over F\mathbb{F}F. The senders transmit quantum states to Bob through a noiseless many-to-one quantum network, but they are not allowed to communicate with each other. The senders share entanglement among themselves, while Bob does not share this entanglement. They encode their classical information WiW_iWi​, i=1,…,ki=1,\ldots,ki=1,…,k, into their local subsystems and transmit them to Bob so that he can recover YYY through a quantum measurement and subsequent post-processing. The N-Sum Box protocol proposed by Allaix et al. (2025) considers this problem under certain constraints on the linear combination and the distribution of the data vectors among the senders.
We present protocols that support the computation of a more general class of linear transformations by giving the senders access to more qudits and allowing them to judiciously precode their input symbols. The communication cost of our schemes is at most that of the best-known prior results in this area and is strictly lower in certain cases. Finally, we demonstrate that the communication cost is subadditive across instances. Specifically, we identify two linear functions for which the total cost of computing them individually is strictly larger than the cost of computing them jointly.
Network function computation is an active topic in network coding, with much recent progress for linear (over a finite field) computations over broadcast (LCBC) and multiple access (LCMAC) channels. Over a quantum multiple access channel (QMAC) with quantum-entanglement shared among transmitters, the linear computation problem (LC-QMAC) is non-trivial even when the channel is noiseless, because of the challenge of optimally exploiting transmit-side entanglement through distributed coding. Given an arbitrary linear function of data streams defined in a finite fieldFd\mathbb {F}_{d}Fd​, the LC-QMAC problem seeks the optimal communication cost (minimum number of qudits that need to be sent by the transmitters to the receiver, per computation instance) over a noise-free QMAC, when the independent input data streams originate at the corresponding transmitters, who share quantum entanglement in advance. As our main result, we fully solve this problem forK=3K=3K=3transmitters (K≥4K\geq 4K≥4settings remain open). Coding schemes based on theNNN-sum box protocol (along with time-sharing and batch-processing) are shown to be information theoretically optimal in all cases.
We study a linear computation problem over a quantum multiple access channel (LC-QMAC), where S servers share an entangled state and separately store classical data streams W1,⋯,WSover a finite field Fd{\mathbb{F}_d}Fd​. A user aims to compute K linear combinations of these data streams, represented as Y=V1W1+V2W2+⋅s+VSWS∈FdK×1Y = {{\mathbf{V}}_1}{W_1} + {{\mathbf{V}}_2}{W_2} + \cdot s + {{\mathbf{V}}_S}{W_S} \in \mathbb{F}_d^{K \times 1}Y=V1​W1​+V2​W2​+⋅s+VS​WS​∈FdK×1​. To this end, each server encodes its classical information into its local quantum subsystem and transmits it to the user, who retrieves the desired computations via quantum measurements. In this work, we propose an achievable scheme for LC-QMAC based on the stabilizer formalism and the ideas from entanglement-assisted quantum error–correcting codes (EAQECC). Specifically, given any linear computation matrix, we construct a self-orthogonal matrix that can be implemented using the stabilizer formalism. Also, we apply precoding matrices to minimize the number of auxiliary qudits required. Our scheme achieves more computations per qudit, i.e., a higher computation rate, compared to the best-known methods in the literature, and attains the capacity in certain cases.
Linear computations over quantum many-to-one communication networks offer opportunities for communication cost improvements through schemes that exploit quantum entanglement among transmitters to achieve superdense coding gains, combined with classical techniques such as interference alignment. The problem becomes much more broadly accessible if suitable abstractions can be found for the underlying quantum functionality via classical black box models. This work formalizes such an abstraction in the form of an “N-sum box”, a black box generalization of a two-sum protocol of Song et al. with recent applications toNNN-server private information retrieval. The N- sum box has a communication cost ofNNNqudits and classical output of a vector ofNNNq-ary digits linearly dependent (via an N x 2N transfer matrix) on 2N classical inputs distributed amongNNNtransmitters. We characterize which transfer matrices are feasible by our construction, both with and without the possibility of additional locally invertible classical operations at the transmitters and receivers.
Linear computations over quantum many-to-one communication networks offer opportunities for communication cost improvements through schemes that exploit quantum entanglement among transmitters to achieve superdense coding gains, combined with classical techniques such as interference alignment. The problem becomes much more broadly accessible if suitable abstractions can be found for the underlying quantum functionality via classical black box models. This work formalizes such an abstraction in the form of an “N-sum box”, a black box generalization of a two-sum protocol of Song et al. with recent applications to N-server private information retrieval. The N-sum box has a communication cost of N qudits and classical output of a vector ofN qN~qN q-ary digits linearly dependent (via anN×2NN \times 2NN×2Ntransfer matrix) on2N2N2Nclassical inputs distributed among N transmitters. We characterize which transfer matrices are feasible by our construction, both with and without the possibility of additional locally invertible classical operations at the transmitters and receivers. Furthermore, we provide a sample application to Cross-Subspace Alignment (CSA) schemes to obtain efficient instances of Quantum Private Information Retrieval (QPIR) and Quantum Secure Distributed Batch Matrix Multiplication (QSDBMM). We first describe N-sum boxes based on maximal stabilizers and we then consider non-maximal-stabilizer-based constructions to obtain an instance of Quantum Symmetric Private Information Retrieval.
NNN-Sum Box: An Abstraction for Linear Computation over Many-to-one Quantum Networks
Linear computations over quantum many-to-one communication networks offer opportunities for communication cost improvements through schemes that exploit quantum entanglement among transmitters to achieve superdense coding gains, combined with classical techniques such as interference alignment. The problem becomes much more broadly accessible if suitable abstractions can be found for the underlying quantum functionality via classical black box models. This work formalizes such an abstraction in the form of an "NNN-sum box", a black box generalization of a two-sum protocol of Song \emph{et al.} with recent applications to NNN-server private information retrieval. The NNN-sum box has a communication cost of NNN qudits and classical output of a vector of NNN qqq-ary digits linearly dependent (via an N×2NN \times 2NN×2N transfer matrix) on 2N2N2N classical inputs distributed among NNN transmitters. We characterize which transfer matrices are feasible by our construction, both with and without the possibility of additional locally invertible classical operations at the transmitters and receivers. Furthermore, we provide a sample application to Cross-Subspace Alignment (CSA) schemes to obtain efficient instances of Quantum Private Information Retrieval (QPIR) and Quantum Secure Distributed Batch Matrix Multiplication (QSDBMM). We first describe NNN-sum boxes based on maximal stabilizers and we then consider non-maximal-stabilizer-based constructions to obtain an instance of Quantum Symmetric Private Information Retrieval.
Quantum dense coding is considered to be one of the most remarkable demonstrations of the existence of quantum entanglement. In this paper, we analyze the role of correlations (both classical and quantum) in quantum communication protocols. We study how the capacity of a single-qubit quantum channel depends on correlations between Alice and Bob. In particular, we consider a scenario when Alice and Bob share a pair of qubits in a state such that Alice's qubit is in the state described by the density operator $\ensuremath{\varrho}=\frac{1}{2}1+\stackrel{\ensuremath{\rightarrow}}{n}\ensuremath{\cdot}\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\sigma}}.$ We evaluate a capacity of the noiseless channel for two cases: (1) when Bob performs measurement just on the particle received from Alice, and (2) in the case when he utilizes correlations that have been established prior to the communication. We present a simple classical scenario which might serve as a prototype of quantum dense coding. We generalize our results also for qudits. In addition, we address some aspects of the issue of security in correlation-assisted communication protocols.