Quantum Gasp Codes for Private Distributed Matrix Multiplication

We develop quantum gap additive secure polynomial (GASP) codes. Classical GASP codes have been designed for the private distributed matrix multiplication (PDMM) problem for outer product partitioning (OPP). In PDMM, a user, with limited computation resources, sends high-dimensional matrices to servers with high computation power to compute the required product such that any T of the servers cannot get any information about the user’s matrices. In the quantum setting, the user sends the encoded versions of the matrices to the servers over classical channels, and the servers respond back over separate quantum channels individually using an encoded share of a pre-established entangled state. In this paper, we define the requirement that the classical code of GASP should satisfy in order for it to be transformed into a quantum code. We develop the encoding structure once the requirement is satisfied, achieving the super-dense coding gain, i.e., double the classical rate counterpart. Finally, we quantify the limitation to satisfy the aforementioned feasibility requirement, i.e., the minimum privacy parameter needed to transform the classical code to its quantum counterpart.