Recently, a class of quantum secret sharing schemes called communication efficient quantum threshold secret sharing schemes (CE-QTS) was introduced. These schemes reduced the communication cost during secret recovery. In this paper, we introduce a general class of communication efficient quantum secret sharing schemes (CE-QSS) which include both threshold and non-threshold schemes. We propose a framework for constructing CE-QSS schemes to generalize the earlier construction of CE-QTS schemes which was based on the staircase codes. The main component in this framework is a class of quantum codes which we call the extended Calderbank-Shor-Steane codes. These extended CSS codes could have other applications. We derive a bound on communication cost for CE-QSS schemes. Finally, we provide a construction of CE-QSS schemes meeting this bound using the proposed framework.

In the above article, as one of the results, we proposed a construction for universal CE-QTS schemes. However, we recently found an error in this construction. We found that the Vandermonde matrix used for the encoding in this construction leads to a failure in secret recovery. Here we shortly describe why this error occurs and provide a rectification. By replacing the Vandermonde matrix with a Cauchy matrix, the secret recovery in the construction goes through. However, the construction now needs a higher field size. With this rectification, the construction is now correct. The remaining results in the article remain unaffected.

Coded caching is a technique for achieving increased throughput in cached networks during peak hours. Placement delivery arrays (PDAs) capture both placement and delivery scheme requirements in coded caching in a single array. Lifting is a method of constructing PDAs, where entries in a small base PDA are replaced with constituent PDAs that satisfy a property called Blackburn-compatibility. We propose two new constructions for Blackburn-compatible PDAs including a novel method for lifting Blackburn-compatible PDAs to obtain new sets of Blackburn-compatible PDAs. Both of these constructions improve upon previous tradeoffs between rate, memory and subpacketization. We generalize lifting constructions by defining partial Blackburn-compatibility between two PDAs w.r.t. a third PDA. This is a wider notion of Blackburn-compatibility making the original definition a special case. We show that some popular coded caching schemes can be defined as lifting constructions in terms of this extended notion.

Topological subsystem color codes (TSCCs) are an important class of topological subsystem codes that allow for syndrome measurement with only 2-body measurements. It is expected that such low complexity measurements can help in fault tolerance. While TSCCs have been studied over depolarizing noise model, their performance over the erasure channel has not been studied as much. Recently, we proposed erasure decoders for TSCCs and reported a threshold of 9.7%. In this paper, we continue our study of TSCCS over the erasure channel. We propose two erasure decoders for topological subsystem color codes. These decoders employ a mapping of the TSCCs to topological color codes (TCCs). In addition, these decoders use the technique of gauge fixing, where some of the gauge operators of the subsystem code are promoted to stabilizers. We perform gauge fixing using 4-body and 8-body gauge operators. With partial gauge fixing, we obtained a threshold of 17.7% on a TSCC derived from the square octagon lattice. Using an order maximal gauge fixing decoder we were able to improve the threshold to 44%. The performance of the order maximal gauge fixing decoder can be further improved to close to 50% in conjunction with an optimal erasure decoder for topolological color codes. We also study the correctability of erasures on the subsystem codes.

Coded caching is a technique where multicasting and coding opportunities are utilized to achieve better rate-memory tradeoff in cached networks. A crucial parameter in coded caching is subpacketization, which is the number of parts a file is to be split into for coding purposes. The Maddah-Ali-Niesen scheme has order-optimal rate, but the subpacketization is exponential in the number of users for certain memory regimes. In contrast, coded caching schemes designed using placement delivery arrays (PDAs) can have linear subpacketization with a penalty in rate. In this work, we propose several constructions of efficient PDAs through lifting, where a base PDA is expanded by replacing each entry by another PDA. By proposing and using the notion of Blackburn-compatibility of PDAs, we provide multiple lifting constructions with increasing coding gains. We compare the constructed coded caching schemes with other existing schemes for moderately high number of users and show that the proposed constructions are versatile and achieve a good rate-memory tradeoff at low subpacketizations.

We consider the setting of a distributed storage system where a single file is subdivided into smaller fragments of same size which are then replicated with a common replication factor across servers of identical cache size. An incoming file download request is sent to all the servers, and the download is completed whenever the request gathers all the fragments. At each server, we are interested in determining the set of fragments to be stored, and the sequence in which fragments should be accessed, such that the mean file download time for a request is minimized. We model the fragment download time as an exponential random variable independent and identically distributed for all fragments across all servers, and show that the mean file download time can be lower bounded in terms of the expected number of useful servers summed over all distinct fragment downloads. We present deterministic storage schemes that attempt to maximize the number of useful servers. We show that finding the optimal sequence of accessing the fragments is a Markov decision problem, whose complexity grows exponentially with the number of fragments. We propose heuristic algorithms that determine the sequence of access to the fragments which are empirically shown to perform well.

A$((k,n))$quantum threshold secret sharing (QTS) scheme is a quantum cryptographic protocol for sharing a quantum secret among$n$parties such that the secret can be recovered by any$k$or more parties while$k-1$or fewer parties have no information about the secret. Despite extensive research on these schemes, there has been very little study on optimizing the quantum communication cost during recovery. Recently, we initiated the study of communication efficient quantum threshold secret sharing (CE-QTS) schemes. These schemes reduce the communication complexity in QTS schemes by accessing$d>k$parties for recovery; here$d$is fixed ahead of encoding the secret. In contrast to the standard QTS schemes which require$k$qudits for recovering each qudit in the secret, these schemes have a lower communication cost of$\frac {d}{d-k+1}$. In this paper, we further develop the theory of communication efficient quantum threshold schemes. Here, we propose universal CE-QTS schemes which reduce the communication cost for all$d>k$simultaneously. We provide a framework based on ramp quantum secret sharing to construct CE-QTS and universal CE-QTS schemes. We give another construction for universal CE-QTS schemes based on Staircase codes. We derived a lower bound on communication complexity and show that our constructions are optimal. Finally, an information theoretic model is developed to analyse CE-QTS schemes and the lower bound on communication complexity is proved again using this model.

We consider a distributed storage system storing a single file, where the file is divided into equal sized fragments. The fragments are replicated with a common replication factor, and stored across servers with identical storage capacity. An incoming download request for this file is sent to all the servers, and it is considered serviced when all the unique fragments are downloaded. The download time for all fragments across all servers, is modeled as an independent and identically distributed (i.i.d.) random variable. The mean download time can be bounded in terms of the expected number of useful servers available after gathering each fragment. We find the mean number of useful servers after collecting each fragment, for a random storage scheme for replication codes. We show that the performance of the random storage for replication code achieves the upper bound for expected number of useful servers at every download asymptotically in number of servers for any storage capacity. Further, we show that the performance of this storage scheme is comparable to that of Maximum Distance Separable (MDS) coded storage.

Three dimensional (3D) toric codes are a class of stabilizer codes with local checks and come under the umbrella of topological codes. While decoding algorithms have been proposed for the 3D toric code on a cubic lattice, there have been very few studies on the decoding of 3D toric codes over arbitrary lattices. Color codes in 3D can be mapped to toric codes. However, the resulting toric codes are not defined on cubic lattice. They are arbitrary lattices with triangular faces. Decoding toric codes over an arbitrary lattice will help in studying the performance of color codes. Furthermore, gauge color codes can also be decoded via 3D toric codes. Motivated by this, we propose an efficient algorithm to decode 3D toric codes on arbitrary lattices (with and without boundaries).

Quantum secret sharing (QSS) is a cryptographic protocol in which a quantum secret is distributed among a number of parties where some subsets of the parties are able to recover the secret while some subsets are unable to recover the secret. In the standard ((k, n)) quantum threshold secret sharing scheme, any subset of k or more parties out of the total n parties can recover the secret while other subsets have no information about the secret. But recovery of the secret incurs a communication cost of at least k qudits for every qudit in the secret. Recently, a class of communication efficient QSS schemes were proposed which can improve this communication cost to $\frac{d}{{d - k + 1}}$ by contacting d ≥ k parties where d is fixed prior to the distribution of shares. In this paper, we propose a more general class of ((k, n)) quantum secret sharing schemes with low communication complexity. In these schemes the combiner can contact any d parties at the time of recovery where k ≤ d ≤ n. This is the first such class of universal communication efficient quantum threshold schemes.

Qubit loss is one of the forms of noise encountered in some quantum technologies. Such noise is modeled using the quantum erasure channel. Unlike the depolarizing noise, it is much more tractable, yet the performance of many quantum codes over the erasure channel has not been studied as extensively. In this paper, we study the performance of topological subsystem color codes (TSCCs) over the quantum erasure channel. It is the first such study of TSCCs over the erasure channel. We propose multiple decoding algorithms for TSCC and obtain the highest threshold of about 9.7% for the subsystem color code derived from the square octagon lattice.

Recent developments in the field of deep learning have motivated many researchers to apply these methods to problems in quantum information. Torlai and Melko first proposed a decoder for surface codes based on neural networks. Since then, many other researchers have applied neural networks to study a variety of problems in the context of decoding. An important development in this regard was due to Varsamopoulos et at. who proposed a two-step decoder using neural networks. Subsequent work of Maskara et at. used the same concept for decoding for various noise models. We propose a similar two-step neural decoder using inverse parity-check matrix for topological color codes. We show that it outperforms the state-of-the-art performance of non-neural decoders for independent Pauli errors noise model on a 2D hexagonal color code. Our final decoder achieves a threshold of 10%. Our result is comparable to the recent work on neural decoder for quantum error correction by Maskara et at. It appears that our decoder has advantages with respect to training cost and complexity of the network for higher distances when compared to that of Maskara et at.

Cubic codes were proposed by Haah as candidates for self-error correction in three dimensions (3D). While these codes are not self-correcting, Bravyi and Haah showed that they are partially self-correcting. In this paper we are interested in generalizations of the cubic code to prime alphabet. Kim initiated the study of such codes over prime alphabet. Haah also proposed a framework based on modules that enables the study of cubic codes over higher alphabet. However, there are many open questions remaining, especially those pertaining to the performance of nonbinary cubic codes. Building on Bravyi and Haah's decoder, we study the performance of nonbinary cubic codes over the quantum erasure and depolarizing channels. This is the first such study of nonbinary cubic codes.

Topological subsystem codes can combine the advantages of both topological codes and subsystem codes. In this paper we study two classes of topological subsystem codes with an emphasis on decoding them. First, we generalize the two step decoding algorithm of Suchara et al. to all topological subsystem color codes. Then, we propose a construction of subsystem surface codes and develop decoders for them generalizing the results of Bravyi et al. Our simulations for the topological subsystem color code derived from the square octagon lattice resulted in a noise threshold of 1.75%. This is comparable to the previous result of 1.95% by Bombin et al., who used a different algorithm.

In a recent work, Bombin, Duclos-Cianci, and Poulin showed that every local translationally invariant 2D topological stabilizer code is locally equivalent to a finite number of copies of Kitaev's toric code. For 2D color codes, Delfosse relaxed the constraint on translation invariance and mapped a 2D color code onto three surface codes. In this paper, we propose an alternate map based on linear algebra. We show that any 2D color code can be mapped onto exactly two copies of a related surface code. The surface code in our map is induced by the color code and easily derived from the color code. Furthermore, our map does not require any ancilla qubits for the surface codes.

The correspondence between linear codes and representable matroids is well known. But a similar correspondence between quantum codes and matroids is not known. We show that representable symplectic matroids over a finite field Fqcorrespond to Fq-linear quantum codes. This connection is straightforward but it does not appear to have been made earlier in literature. This correspondence is made through isotropic subspaces. We show that Calderbank-Shor-Steane (CSS) codes correspond to homogenous symplectic matroids while graph states, which figure so prominently in measurement based quantum computation, correspond to a special class of symplectic matroids, namely Lagrangian matroids. This association is useful in that it enables the study of symplectic matroids in terms of quantum codes and vice versa. Furthermore, it has application in the study of quantum secret sharing schemes.

Color codes are a class of topological codes that have come into prominence in the recent years. Like the surface codes the codespace defined by them can be associated to the degenerate ground state of a local Hamiltonian. In addition they can be designed to have an extended set of transversal encoded gates (compared to surface codes) increasing their appeal for fault tolerant quantum computation. In this paper we generalize the color codes to arbitrary prime power alphabet. We show that in the 2D case there exist color codes for which the nonbinary Clifford group can be implemented transversally.

In this paper we give new decoding algorithms for the generalized Shor codes and a class of subsystem codes due to Bacon and Casaccino. Our interest in these codes stems from the fact these codes can allow us to construct quantum codes from non-dual containing codes. In this paper we show how to decode these codes efficiently.

Recently, quantum error-correcting codes were proposed that capitalize on the fact that many physical error models lead to a significant asymmetry between the probabilities for bit flip and phase flip errors. An example for a channel which exhibits such asymmetry is the combined amplitude damping and dephasing channel, where the probabilities of bit flips and phase flips can be related to relaxation and dephasing time, respectively. We give systematic constructions of asymmetric quantum stabilizer codes that exploit this asymmetry. Our approach is based on a CSS construction that combines BCH and finite geometry LDPC codes.

Recently, operator quantum error-correcting codes have been proposed to unify and generalize decoherence free subspaces, noiseless subsystems, and quantum error-correcting codes. This correspondence introduces a natural construction of such codes in terms of Clifford codes, an elegant generalization of stabilizer codes due to Knill. Character-theoretic methods are used to derive a simple method to construct operator quantum error-correcting codes from any classical additive code over a finite field, which obviates the need for self-orthogonal codes.

Convolutional stabilizer codes promise to make quantum communication more reliable with attractive online encoding and decoding algorithms. This paper introduces a new approach to convolutional stabilizer codes based on direct limit constructions. A quantum Singleton bound for pure convolutional stabilizer codes is given. A familiy of quantum convolutional codes is derived from generalized Reed-Solomon codes. These codes are shown to be optimal with respect to the (quantum) Singleton bound.

Duadic group algebra codes are a generalization of quadratic residue codes. This paper addresses an open problem raised by Zhu concerning the existence of duadic group algebra codes. These codes can be used to construct degenerate quantum stabilizer codes that have the nice feature that many errors of small weight do not need error correction.

Classical Bose–Chaudhuri–Hocquenghem (BCH) codes that contain their (Euclidean or Hermitian) dual codes can be used to construct quantum stabilizer codes; this correspondence studies the properties of such codes. It is shown that a BCH code of length$n$can contain its dual code only if its designed distance$\delta =O(\sqrt {n})$, and the converse is proved in the case of narrow-sense codes. Furthermore, the dimension of narrow-sense BCH codes with small design distance is completely determined, and – consequently – the bounds on their minimum distance are improved. These results make it possible to determine the parameters of quantum BCH codes in terms of their design parameters.

Good quantum codes, such as quantum MDS codes, are typically nondegenerate, meaning that errors of small weight require active error-correction, which is -unfortunately- itself prone to errors. In this paper, examples of degenerate quantum codes are constructed that alleviate this problem in that they allow some errors of small weight that do not require active error correction. In particular, two new families of [[n, 1, ges radicn]]qdegenerate quantum codes are derived from classical duadic codes

An attractive feature of BCH codes is that one can infer valuable information from their design parameters (length, size of the finite field, and designed distance), such as bounds on the minimum distance and dimension of the code. In this paper, it is shown that one can also deduce from the design parameters whether or not a primitive, narrow-sense BCH contains its Euclidean or Hermitian dual code. This information is invaluable in the construction of quantum BCH codes. A new proof is provided for the dimension of BCH codes with small designed distance, and simple bounds on the minimum distance of such codes and their duals are derived as a consequence. These results allow us to derive the parameters of two families of primitive quantum BCH codes as a function of their design parameters

One formidable difficulty in quantum communication and computation is to protect information-carrying quantum states against undesired interactions with the environment. To address this difficulty, many good quantum error-correcting codes have been derived as binary stabilizer codes. Fault-tolerant quantum computation prompted the study of nonbinary quantum codes, but the theory of such codes is not as advanced as that of binary quantum codes. This paper describes the basic theory of stabilizer codes over finite fields. The relation between stabilizer codes and general quantum codes is clarified by introducing a Galois theory for these objects. A characterization of nonbinary stabilizer codes over$bf F_q$in terms of classical codes over$ bf F_q^2$is provided that generalizes the well-known notion of additive codes over$ bf F_4$of the binary case. This paper also derives lower and upper bounds on the minimum distance of stabilizer codes, gives several code constructions, and derives numerous families of stabilizer codes, including quantum Hamming codes, quadratic residue codes, quantum Melas codes, quantum Bose–Chaudhuri–Hocquenghem (BCH) codes, and quantum character codes. The puncturing theory by Rains is generalized to additive codes that are not necessarily pure. Bounds on the maximal length of maximum distance separable stabilizer codes are given. A discussion of open problems concludes this paper.

We construct nonbinary quantum codes from classical generalized Reed-Muller codes and derive the conditions under which these quantum codes can be punctured. We provide a partial answer to a question raised by Grassl, Beth and Rotteler on the existence of q-ary quantum MDS codes of length n with q les n les q2- 1