Quantum Private Information Retrieval from MDS-coded and Colluding Servers

Abstract

In quantum private information retrieval (QPIR), a user retrieves a classical file from multiple servers by downloading quantum systems without revealing the identity of the file. The QPIR capacity is the maximal achievable ratio of the retrieved file size to the total download size. In this paper, the capacity of QPIR from MDS-coded and colluding servers is studied. Two classes of QPIR, called stabilizer QPIR and dimension squared QPIR induced from classical strongly linear PIR are defined, and the related QPIR capacities are derived. For the non-colluding case, the general QPIR capacity is derived when the number of files goes to infinity. The capacities of symmetric and non-symmetric QPIR with coded and colluding servers are proved to coincide, being double to their classical counterparts. A general statement on the converse bound for QPIR with coded and colluding servers is derived showing that the capacities of stabilizer QPIR and dimension squared QPIR induced from any class of PIR are upper bounded by twice the classical capacity of the respective PIR class. The proposed capacity-achieving scheme combines the star-product scheme by Freij-Hollanti et al. and the stabilizer QPIR scheme by Song et al. by employing (weakly) self-dual Reed-Solomon codes. I. INTRODUCTIONWith the amount of data stored in distributed storage systems steadily increasing, the demand for user privacy has surged in recent years. One notion that has received considerable attention is private information retrieval (PIR), where the Partial results have been accepted to ISIT 2021 [1]. C. Hollanti and M.

The Capacity of Classical Summation over a Quantum MAC with Arbitrarily Replicated Inputs

“…The QMAC model, with batch processing, also allows arbitrarily large DSC gains for certain partial functions (see Appendix A). On the other hand, the largest observed DSC gain for exact computation of total functions thus far is only 2 in the SMP setting [24] with or without batch processing [18,24,33,34]. To the best of our knowledge, DSC gains larger than 2 for exact computations of total functions, while unlikely, have not been formally ruled out.…”

“…In order to quantify the utility of multiparty quantum entanglements, the key figure of merit in the Σ-QMAC is the multiplicative gain in capacity that is enabled by entanglement, relative to the corresponding unentangled setting. In the literature such a gain is known as distributed superdense coding gain [18,[33][34][35][36][37][42][43][44][45][46]. Quantifying the utility of multiparty entanglements by characterizing the distributed superdense coding (DSC) gain in the Σ-QMAC is the immediate focus of this work.…”

“…Connection to N -sum Box: The Σ-QMAC is also immediately related to the N -sum box [33], which is a black box abstraction of stabilizer based linear computations. The DSC gains in many previously studied settings, including recent applications in quantum private information retrieval (QPIR), can be realized through the N -sum box abstraction [18,[34][35][36][37][43][44][45][46].…”

“…Indeed the foremost challenge in pursuing this direction is to identify formulations that are both insightful from a multiparty quantum entanglement perspective and also information theoretically tractable. Our choice of the Σ-QMAC setting draws inspiration from the following observations -1) the many-to-one (multiple access (MAC)) setting is among the most tractable in network information theory, 2) the capacity of finite field linear computations in noiseless settings, while still open in general, has seen much progress in the network coding literature, in particular elegant solutions have been found for the case of scalar linear computations (i.e., sum computations) [25][26][27][28][29], and 3) the stabilizer formalism [30][31][32] from quantum error correction lends itself nicely to linear black-box abstractions for the quantum multiple access channel (QMAC) [33], and has been instrumental to recent advances in quantum private information retrieval (QPIR) [33][34][35][36][37][38] that implicitly involve optimal specialized linear computations. The Σ-QMAC setting is described next.…”

Capacity of Quantum Symmetric Private Information Retrieval With Collusion of All But One of Servers

“…Following the conference version of this work, the paper [13] proposed a QSPIR protocol for coded and colluding servers which works for any [n, k]-MDS code and secure against t-collusion with t = n − k. Their protocol is an extension of the protocol of this paper by combination with the classical PIR protocol [18], and it achieves better rates than the classical counterparts [18], [24]. The paper [29] improved the capacity result of this paper for any number of colluding servers: For any 1 < t < n, the t-private QSPIR capacity is min{1, 2(n − t)/n}.…”