Resource Reduction in Multiparty Quantum Secret Sharing of both Classical and Quantum Information under Noisy Scenario

TL;DR

The paper tackles the vulnerability of multiparty quantum secret sharing to realistic noise and develops a resource-efficient quantum error correction approach. By applying a simplified Shor-based repetition code that encodes each single-qubit share into 3 copies, it reduces the standard $9$-qubit overhead to $3$ and yields lower average error rates across bit-flip, phase-flip, and amplitude-damping noise. The authors demonstrate, through analytic expressions and simulations, that this 3-qubit scheme outperforms both the conventional Shor code and other small codes in the considered settings, and they extend the approach to the SSQI protocol built on QSSCM. The work provides a practical route to robust, real-world deployment of multiparty QSSCM/SSQI and offers insights applicable to other single-qubit-based quantum protocols.

Abstract

Quantum secret sharing (QSS) enables secure distribution of information among multiple parties but remains vulnerable to noise. We analyze the effects of bit-flip, phase-flip, and amplitude damping noise on the multiparty QSS for classical message (QSSCM) and secret sharing of quantum information (SSQI) protocols proposed by Zhang et al. (Phys. Rev. A, 71:044301, 2005). To scale down these effects, we introduce an efficient quantum error correction (QEC) scheme based on a simplified version of Shor's code. Leveraging the specific structure of the QSS protocols, we reduce the qubit overhead from the standard 9 of Shor's code to as few as 3 while still achieving lower average error rates than existing QEC methods. Thus, our approach can also be adopted for other single-qubit-based quantum protocols. Simulations demonstrate that our approach significantly enhances the protocols' resilience, improving their practicality for real-world deployment.

Figures (6)

• Figure 1: Error on secret $e^a_{1,g}$ as function \ref{['eq:gen_damp']} of damping strengths $\gamma_A, \gamma_B$ and $\gamma_C$ for channels from Alice to Charlie, from Bob to Charlie and from Charlie to Alice, respectively. Observe that $\gamma_A$ and $\gamma_C$ affect similarly, while $\gamma_B$ effects differently.
• Figure 2: The plots of the error $e_1^g$ as a function \ref{['eq:gen_flip_error']} of the qubit error probability $p$ are shown for $n = 3, 4, 5,$ and $6$. For protocols involving an even number of channels, a higher error probability increases the likelihood of an even number of flips, which can paradoxically lead to a reduction in the overall error. However, this does not happen for odd number of channels, leading the error on secret to $1$, as qubit error probability reaches to $1$.
• Figure 3: Error on reconstructed secret is plotted against channel error probability $p$ with and without repetition code. The plot is generated from analytic equations \ref{['eq:gen_flip_error']} and \ref{['eq:QEC_gen_error']} and the simulated results. It shows that if $p<0.5$, the repetition code can reduce the error in the reconstructed secret.
• Figure 4: Error on reconstructed secret is plotted against damping probability $\gamma$ with and without repetition code. The plot is generated from analytic equations \ref{['eq:single_damp']} and \ref{['eq:QEC_damp']} and the simulated results. It shows that the repetition code can reduce the error in the reconstructed secret for all values of the damping strength.
• Figure 5: Plots show the simulated errors on reconstructed secret against Pauli (bit-flip or phase-flip) error $p$. Observe that repetition code performs better than existing $\llbracket5,1,3\rrbracket$ perfect QEC laflamme1996perfect. Even the five-qubit codes perform worse than the no encoding scenario. The reason is for five-qubit code, all the five qubits are going through the error, making the error very high compared to one-qubit error in no encoding scenario, and the QEC fails to recover it. Although for the repetition code, three qubits are going through the error, the QEC restricts it below the no encoding threshold.