Good Gottesman-Kitaev-Preskill Codes from the NTRU Cryptosystem

Since their first experimental demonstrations [3, 5] bosonic Quantum-Error Correction with the Gottesman- Kitaev-Preskill (GKP) code [6] has seen a rapid increase of interest in experimental and theoretical investigations and has become a framework of interest in large scale quantum computing using photonics [1, 2] and supercon- ducting platforms [7]. While much research has been dedicated to obtain effective qubits from single-mode systems that are to be integrated into larger qubit-based networks, we argue that this approach is only scratching the surface of the potential quantum computing with the GKP code offers when compared to the more general, lattice theoretic perspective on these code [4, 9, 10]. To demonstrate this, we construct random good GKP codes derived from a cryptographic attack on the NTRU cryptosystem [8] and investigate the decoding problem of these codes. The presented scheme offers an example of a trapdoor decodable quantum error correcting code and is a first step towards cryptographic protocols built on the decoding problem of GKP codes which we believe can have wide application for secure quantum communication and cloud-based quantum computing.

 

[1]  Bartolucci et. al. Fusion-based quantum computation, 2021.

[2]  J. E. Bourassa et al. Quantum, 5:392, 2021.

[3]  P. Campagne-Ibarcq et al., Nature, 584:368–372, 2020.

[4]  Jonathan Conrad, Jens Eisert, and Francesco Arzani. Quantum, 6:648, feb 2022.

[5]  C. Fluehmann et al. Nature, 566:513–517, 2019.

[6]  D. Gottesman, A. Kitaev, and J. Preskill. Phys. Rev. A, 64:012310, 2001.

[7]  Arne L. Grimsmo and Shruti Puri. PRX Quantum, 2:020101, Jun 2021.

[8]  Jeffrey Hoffstein, Jill Pipher, and Joseph H. Silverman. In Algorithmic Number Theory, Lecture Notes in Computer Science, page 267–288, Berlin, Heidelberg, 1998. Springer.

[9]  Baptiste Royer, Shraddha Singh, and S.M. Girvin.  PRX Quantum, 3(1), mar 2022.

[10]  Frank Schmidt and Peter van Loock. Physical Review A, 105(4), apr 2022.