Pradeep Sarvepalli

A quantum secret sharing scheme is a cryptographic protocol by which a dealer can share a secret among a group of n players so that only certain subsets of players can recover the secret by collaboration. In this paper we propose communication efficient quantum threshold secret sharing schemes. They minimize the amount of quantum communication required to reconstruct the secret when more than the necessary number of players collaborate. They are based on a class of staircase codes proposed by Bitar and El Rouayheb. In a standard ((k,n)) quantum threshold scheme, any subset of k or more players can recover the secret. The quantum communication cost for reconstruction in such schemes is k qudits for each secret qudit. Using the proposed construction, any subset of d≥k players can also collaborate to recover the secret with a communication cost of d qudits for d-k+1 secret qudits. In other words, for the proposed schemes the quantum communication cost is only dd-k+1 qudits for every secret qudit. For d>k, proposed schemes are communication efficient with respect to standard schemes; and when d=2k-1, the quantum communication cost is reduced by a factor O(k). Further, when n=2k-1, the proposed schemes have optimal communication cost for secret reconstruction.

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.

Coded caching is a technique where we utilize multi-casting opportunities to reduce rate in cached networks. One limitation of coded caching schemes is that they reveal the demands of all users to their peers. In this work, we consider coded caching schemes that assure privacy for user demands with a particular focus on reducing subpacketization. For the 2-user, 2-file case, we present a new linear demand-private scheme with the lowest possible subpacketization. This is done by presenting the scheme explicitly and proving impossibility results under lower subpacketization. When only partial privacy is required, we show that subpacketization can be significantly reduced when there are a large number of files.

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 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.

The quantum erasure channel models phenomena such as loss or leakage of qubits. Using quantum codes, we can recover from such errors. In this paper, we are interested in studying the performance of two-dimensional color codes over the quantum erasure channel. Our approach makes use of the local equivalence between color codes and surface codes. We propose a variety of decoding algorithms for color codes over the erasure channel. First, we propose algorithms that decode by projecting the erasures on the color to surface codes. Then, instead of directly decoding on the color code or on the equivalent copies of surface codes, we decode jointly on the color code and the equivalent surface codes. We observe a threshold of 44.3% for the color code on the square octagonal lattice.

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.

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.

Twists are defects in the lattice which can be utilized to perform computations on encoded data. Twists have been studied in various classes of topological codes like qubit and qudit surface codes, qubit color codes, and qubit subsystem color codes. They are known to exhibit projective non-Abelian statistics, which is exploited to perform encoded gates. In this paper, we initiate the study of twists in qudit color codes over the odd prime alphabet. Specifically, we present a systematic construction of twists in qudit color codes that permute both charge and color of the excitations. We also present a mapping between generalized Pauli operators and strings in the lattice. Making use of the construction, we give protocols to implement generalized Clifford gates using charge-and-color-permuting twists.

In this paper, we study nonbinary toric codes and color codes. While the relationships between the qubit color codes and toric codes have been studied extensively from various perspectives, similar studies for qudit toric codes and color codes are missing. We show that just as in the binary case, qudit color codes can be mapped to qudit toric codes. Qudit color codes are known to have the capability for implementing the generalized Clifford group transversally. The proposed mapping of qudit color codes to toric codes would allow us to port the computational capabilities of the qudit color codes to qudit toric codes. Another application would be to decode qudit color codes via qudit toric codes. Along the way, we show an important structural result for higher-dimensional qudit toric codes. Specifically, we show that the toric codes over prime power dimension can be viewed as a direct sum of copies of codes over the prime dimension. This result simplifies the study of codes over prime power dimension by reducing them to the study of codes over the prime dimension. Furthermore, we provide explicit circuits for the transformation between qudit color codes and toric 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.