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arXiv:2505.20284

Entanglement Cost of Erasure Correction in Quantum MDS Codes

Kaushik Senthoor

Published May 26, 2025• 0 citations
arXivHTMLPDF

Abstract

In distributed quantum storage, physical qubits of a code will be stored across the network. When qubits in one of the nodes are lost i.e. when the node is erased, the remaining nodes need to communicate with a new node to replace the lost qubits. Here, we look at the problem of how much entanglement cost is needed to perform such a distributed quantum erasure correction. We focus on distributed quantum storage based on quantum maximum distance separable (MDS) codes. We derive lower bounds on the entanglement cost when the quantum network used for the erasure correction has a star topology. We show that the simple method of downloading the non-erased qudits and performing operations at a single node is optimal when the minimal number of non-erased nodes are accessed. It remains to be seen what the entanglement cost will be when a non-minimal number of non-erased nodes are accessed. The techniques used in this work can be developed further to study the entanglement cost of quantum erasure correction in more general code families and network topologies.

DOI
10.48550/arxiv.2505.20284

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In distributed quantum storage, physical qubits of a code will be stored across the network. When qubits in one of the nodes are lost i.e. when the node is erased, the remaining nodes need to communicate with a new node to replace the lost qubits. Here, we look at the problem of how much entanglement cost is needed to perform such a distributed quantum erasure correction. We focus on distributed quantum storage based on quantum maximum distance separable (MDS) codes. We derive lower bounds on the entanglement cost when the quantum network used for the erasure correction has a star topology. We show that the simple method of downloading the non-erased qudits and performing operations at a single node is optimal when the minimal number of non-erased nodes are accessed. It remains to be seen what the entanglement cost will be when a non-minimal number of non-erased nodes are accessed. The techniques used in this work can be developed further to study the entanglement cost of quantum erasure correction in more general code families and network topologies.
Distributed quantum information stored over nodes within a quantum network are prone to node failures, motivating the need for coded quantum networks. We encode kkk qubits of quantum information over an nnn-node network in a distributed fashion, with one qubit per node, using modified graph-state codes. We devise a procedure to recover the encoded quantum information in any given arbitrary network after a detected node fails. The recovery procedure is locally within four edges of the failed node. For a network with even number of nodes, we provide a class of scalable modified graph-state codes with rate $(n\ensuremath{-}2)/n$, saturating the quantum Singleton bound. This yields higher coding rates than Grassl et al.'s well-known erasure codes and has asymptotic rate 1. For odd node size nnn, we design the network links to obtain an $(n\ensuremath{-}3)/n$ rate-optimal modified graph state code.
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A quantum message is encoded into NNN storage nodes (quantum systems Q1…QNQ_1\dots Q_NQ1​…QN​) with assistance from NBN_BNB​ maximally entangled bi-partite quantum systems A1B1,…,ANBBNBA_1B_1, \dots, A_{N_B}B_{N_B}A1​B1​,…,ANB​​BNB​​, that are prepared in advance such that B1…BNBB_1\dots B_{N_B}B1​…BNB​​ are stored separately as entanglement assistance (EA) nodes, while A1…ANBA_1\dots A_{N_B}A1​…ANB​​ are made available to the encoder. Both the storage nodes and EA nodes are erasure-prone. The quantum message must be recoverable given any KKK of the NNN storage nodes along with any KBK_BKB​ of the NBN_BNB​ EA nodes. The capacity for this setting is the maximum size of the quantum message, given that the size of each EA node is λBλ_BλB​. All node sizes are relative to the size of a storage node, which is normalized to unity. The exact capacity is characterized as a function of N,K,NB,KB,λBN,K,N_B,K_B, λ_BN,K,NB​,KB​,λB​ in all cases, with one exception. The capacity remains open for an intermediate range of λBλ_BλB​ values when a strict majority of the NNN storage nodes, and a strict non-zero minority of the NBN_BNB​ EA nodes, are erased. As a key stepping stone, an analogous classical storage (with shared-randomness assistance) problem is introduced. A set of constraints is identified for the classical problem, such that classical linear code constructions translate to quantum storage codes, and the converse bounds for the two settings utilize similar insights. In particular, the capacity characterizations for the classical and quantum settings are shown to be identical in all cases where the capacity is settled.
We consider (n,k)(n,k)(n,k) MDS-coded distributed storage over Fq\mathbb{F}_qFq​ with per-node storage ααα symbols. For the oblivious update problem, where a single message symbol changes and neither helpers nor the stale node know which, the classical lower bound is αklog⁡2qαk \log_2 qαklog2​q bits. We prove that when the kkk contacted helpers share prior quantum entanglement, the update bandwidth is ⌈α/2⌉⋅klog⁡2q\lceil α/2 \rceil \cdot k \log_2 q⌈α/2⌉⋅klog2​q bits-equivalent, a factor approaching 2 reduction. For α=2α= 2α=2, a [[k,k−2]]q[[k, k-2]]_q[[k,k−2]]q​ CSS code achieves bandwidth klog⁡2qk \log_2 qklog2​q with one qudit per helper. For general ααα, a [[⌈α/2⌉k,⌈α/2⌉k−α]]q[[\lceil α/2 \rceil k, \lceil α/2 \rceil k - α]]_q[[⌈α/2⌉k,⌈α/2⌉k−α]]q​ CSS code achieves the bound with ⌈α/2⌉\lceil α/2 \rceil⌈α/2⌉ qudits per helper. The matching converse uses the superdense coding bound: the stale node holds all transmitted qudits and hence the entangled partners, so each helper's channel supports at most D2D^2D2 distinguishable signals for dimension DDD. The result holds for all (n,k)(n,k)(n,k) pairs with sufficiently large prime qqq.
A distributed quantum storage code maps a quantum message to N storage nodes, of arbitrary specified sizes, such that the stored message is robust to an arbitrary specified set of erasure patterns. The sizes of the storage nodes, and erasure patterns may not be homogeneous. The capacity of distributed quantum storage is the maximum feasible size of the quantum message (relative to the sizes of the storage nodes), when the scaling of the size of the message and all storage nodes by the same scaling factor is allowed. Representing the decoding sets as hyperedges in a storage graph, the capacity is characterized for various graphs, including MDS graph, wheel graph, Fano graph, and intersection graph. The achievability is related via quantum CSS codes to a classical secure storage problem. Remarkably, our coding schemes utilize non-trivial alignment structures to ensure recovery and security in the corresponding classical secure storage problem, which leads to similarly non-trivial quantum codes. The converse is based on quantum information inequalities, e.g., strong sub-additivity and weak monotonicity of quantum entropy, tailored to the topology of the storage graphs.
This work investigates the use of quantum resources in distributed storage systems. Consider an (n,k,d)(n,k,d)(n,k,d) distributed storage system in which a file is stored across nnn nodes such that any kkk nodes suffice to reconstruct the file. When a node fails, any ddd helper nodes transmit information to a newcomer to rebuild the system. In contrast to the classical repair, where helper nodes transmit classical bits, we allow them to send classical information over quantum channels to the newcomer. The newcomer then generates its storage by performing appropriate measurements on the received quantum states. In this setting, we fully characterize the fundamental tradeoff between storage and repair bandwidth (total communication cost). Compared to classical systems, the optimal storage--bandwidth tradeoff can be significantly improved with the enhancement of quantum entanglement shared only among the surviving nodes, particularly at the minimum-storage regenerating point. Remarkably, we show that when d≥2k−2d \geq 2k-2d≥2k−2, there exists an operating point at which \textit{both storage and repair bandwidth are simultaneously minimized}. This phenomenon breaks the tradeoff in the classical setting and reveals a fundamentally new regime enabled by quantum communication.