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.
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, 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=3transmitters (K≥4settings remain open). Coding schemes based on theN-sum box protocol (along with time-sharing and batch-processing) are shown to be information theoretically optimal in all cases.
The problem of entanglement-assisted summation over a quantum multiple access channel (Σ-QMAC) is intro-duced, involvingSservers,Kclassical(Fd)data streams that are replicated arbitrarily across various subsets of servers, and a receiver who wishes to compute the sum of theKdata streams. Independent of the data, entangled quantum systemsQ1,Q2,⋯,QSare prepared in advance and distributed to the corresponding servers. Each servers,s∈[S]locally manipulates its quantum systemQsaccording to its classical data and sendsQsto the receiver. The total communication cost islogd∣Q1∣+logd∣Q2∣+⋯+logd∣QS∣qudits, where∣Qsdenotes the dimension ofQs. Based on a measurement of the composite systemQ1Q2⋯QS, the receiver must recover the desired sum. The rate thus achieved is defined as the number of dits(Fdsymbols) of the desired sum computed by the receiver per qudit (d-dimsional quantum system) of download. The capacityCis the supremum of the set of all achievable rates. As the main result of this work, the precise capacity ofΣ-QMAC is obtained, from which it follows that quantum entanglements allow a factor of 2 gain in capacity (superdense coding gain) relative to capacity with no entanglements, in all cases (anyS,K,Fdand any data replication pattern) provided that the entanglement-assisted capacity does not exceed 1 dit/qudit (Holevo bound). Coding schemes based on a recentN-sum box abstraction are sufficient to achieve capacity.
In this work, we consider the problem of computing a linear combination over a noiseless quantum many-to-one network. There are k senders, Alice1, …, Alicek, and a single receiver, Bob. Each Alicei has a data vector Wi∈Fmi, where F is a finite field. Bob wants to compute the linear combination Y=V1W1+V2W2+⋯+VkWk∈Fm, where Vi is an m×mi matrix over 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 Wi, i=1,…,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.
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. A user aims to compute K linear combinations of these data streams, represented as Y=V1W1+V2W2+⋅s+VSWS∈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.
The Σ-QMAC problem is introduced, involvingSservers,Kclassical (Fd) data streams, andTindependent quantum systems. Data stream Wk,k∈ [K] is replicated at a subset of servers W(k) ⊂ [S], and quantum system Qt,t∈ [T] is distributed among a subset of servers ε(t) ⊂ [S] such that Servers∈ ε(t) receives subsystem Qt,sof Qt= (Qt,s)s∈ε(t). Servers manipulate their quantum subsystems according to their data and send the subsystems to a receiver. The total download cost is Σt∈[T]Σs∈ε(t)logd|Qt,s| qudits, where |Q| is the dimension of Q. The states and measurements of (Qt)t∈[T]are required to be separable acrosst∈ [T] throughout, but for eacht∈ [T], thesubsystemsof Qtcan be prepared initially in an arbitrary (independent of data) entangled state, manipulated arbitrarily by the respective servers, and measured jointly by the receiver. From the measurements, the receiver must recover the sum of all data streams. Rate is defined as the number of dits (Fdsymbols) of the desired sum computed per qudit of download. The capacity of Σ-QMAC, i.e., the supremum of achievable rates is characterized for arbitrary data and entanglement distributions W, ε. For example, in the symmetric setting withK= (Sα) data-streams, each replicated among a distinct α-subset of [S], andT= (Sβ) quantum systems, each distributed among a distinct β-subset of [S], the capacity of the Σ-QMAC is 1/βTΣmin(α,β)γ=(α+β−S)+ min(β, 2γ) · (αγ) · (S−αβ−γ). Coding based on the N-sum box abstraction is optimal in every case. Notably, for everyS≠ 3 there exists an instance of the Σ-QMAC whereS-party entanglement is necessary to achieve the fully entangled capacity.
The optimal quantum communication cost of computing a classical sum of distributed sources is studied over a quantum erasure multiple access channel (QEMAC).Kclassical messages comprised of finite-field symbols are distributed acrossSservers, who also share quantum entanglement in advance. Each servers∈ [S] manipulates its quantum subsystemQsaccording to its own available classical messages and sendsQsto the receiver who then computes the sum of the messages based on a joint quantum measurement. The download cost from Servers∈ [S] is the logarithm of the dimension ofQs. The rateRis defined as the number of instances of the sum computed at the receiver, divided by the total download cost from all the servers. The main focus is on the symmetric setting withK= (Sα) messages where each message is replicated among a unique subset of α servers, and the answers from any β servers may be erased. If no entanglement is initially available to the receiver, then we show that the capacity (maximal rate) is preciselyC= max { min { 2(α−β)/S,S−2β/S}, α−β/S}. The capacity with arbitrary levels of prior entanglement (Δ0) between theSdata-servers and the receiver is also characterized, by including an auxiliary server (Server 0) that has no classical data, so that the communication cost from Server 0 is a proxy for the amount of receiver-side entanglement that is available in advance. The challenge on the converse side resides in the optimal application of the weak monotonicity property, while the achievability combines ideas from classical network coding and treating qudits as classical dits, as well as new constructions based on the N-sum box abstraction that rely on absolutely maximally entangled quantum states.
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 ofNq-ary digits linearly dependent (via anN×2Ntransfer matrix) on2Nclassical 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.