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Physical Review A

Modified graph-state codes for single-node recovery in quantum distributed storage

Priya J. Nadkarni, Ankur Raina, Shayan Srinivasa Garani

Published Dec 30, 2020• 4 citations
Publisher

Abstract

Distributed quantum information stored over nodes within a quantum network are prone to node failures, motivating the need for coded quantum networks. We encode $k$ qubits of quantum information over an $n$-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 $n$, we design the network links to obtain an $(n\ensuremath{-}3)/n$ rate-optimal modified graph state code.

DOI
10.1103/physreva.102.062430

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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(n\ensuremath{-}2)/n(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(n\ensuremath{-}3)/n(n\ensuremath−3)/n rate-optimal modified graph state code.
Quantum graph states are highly entangled quantum states known to exhibit properties useful in applications such as quantum error correction, quantum computation, etc. We distribute quantum information across nodes, each having a qubit at its location, by using encoding techniques. We call it a quantum graph state code. By modeling the loss of a qubit due to failure of a node as a quantum erasure, we present a recovery mechanism for distributed quantum information using purification followed by an error correction procedure.
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.
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.
Graph states are generalized from qubits to collections of nnn qudits of arbitrary dimension DDD, and simple graphical methods are used to construct both additive and nonadditive, as well as degenerate and nondegenerate, quantum-error-correcting codes. Codes of distance 2 saturating the quantum Singleton bound for arbitrarily large nnn and DDD are constructed using simple graphs, except when nnn is odd and DDD is even. Computer searches have produced a number of codes with distances 3 and 4, some previously known and some new. The concept of a stabilizer is extended to general DDD, and shown to provide a dual representation of an additive graph code.
We investigate a class of quantum error-correcting codes to correct errors on both qubits and higher-state quantum systems represented as qudits. These codes arise from an original graph-theoretic representation of sets of quantum errors. In this framework, we represent the algebraic conditions for error correction in terms of edge avoidance between graphs providing a visual representation of the interplay between errors and error-correcting codes. Most importantly, this framework supports the development of quantum codes that correct against a predetermined set of errors, in contrast to current methods. A heuristic algorithm is presented, providing steps to develop codes that correct against an arbitrary noisy channel. We benchmark the correction capability of reflexive stabilizer codes for the case of single-qubit errors by comparison to existing stabilizer codes that are widely used. In addition, we present two instances of optimal encodings: an optimal encoding for fully correlated noise, which achieves a higher encoding rate than previously known, and a minimal encoding for single-qudit errors on a four-state system..