Implementing Quantum Secret Sharing on Current Hardware

TL;DR

This work tackles the practicality of quantum secret sharing (QSS) on current quantum hardware by implementing and benchmarking three QSS codes on IBM Brisbane. It develops explicit encoding/decoding circuits for the $((3,5))$ and $((5,7))$ qubit schemes and the $((2,3))$ qutrit scheme, and evaluates them using SWAP tests and entanglement fidelity, with comparisons between mid-circuit measurements (MCM) and delayed coherent measurements (DCM). The main findings show that $((3,5))$ and $((5,7))$ perform similarly in SWAP tests, the Steane code enhances entanglement fidelity, and the qutrit implementation lags due to deeper circuits; MCM generally outperforms DCM, and real hardware exhibits sizable gaps from simulations due to idle decoherence, cross-talk, and readout errors. The work provides practical guidance for near-term QSS deployment, highlighting the value of MCM, error-mitigation improvements, and potential benefits from native qudit support and scheduler-aware compilation.

Abstract

Quantum secret sharing is a cryptographic scheme that enables a secure storage and reconstruction of quantum information. While the theory of secret sharing is mature in its development, relatively few studies have explored the performance of quantum secret sharing on actual devices. In this work, we provide a pedagogical description of encoding and decoding circuits for different secret sharing codes, and we test their performance on IBM's 127-qubit Brisbane system. We evaluate the quality of implementation by performing a SWAP test between the decoded state and the ideal one, as well as by estimating how well the code preserves entanglement with a reference system. Results indicate that a ((3,5)) threshold secret sharing scheme performs slightly better overall than a ((5,7)) scheme based on the SWAP test, but is outperformed by the Steane Code scheme in regards to the entanglement fidelity. We also investigate one implementation of a ((2,3)) qutrit scheme and find that it performs the worst of all, which is expected due to the additional number of multi-qubit gate operations needed to encode and decode qutrits.

Figures (14)

• Figure 1: A general threshold QSS circuit implementation using an [[$n,m,d$]] stabilizer code. An arbitrary $m$-qubit secret $\ket{\psi}$ is encoded into $n$ qubits by a unitary $U$. A permutation $\Pi$ is performed to select an unauthorized set (erased set $E$) of $d-1$ qubits that get discarded. A fresh set of qubits is swapped in place of the latter qubit, and the encoding map is reversed. Finally, an error syndrome is obtained by measuring the $n-m$ qubits in the computational basis, and the appropriate correction $R_k$ is performed on the unmeasured qubits to recover the secret $\ket{\psi}$. Note that the error correction $R_k$ will depend on the permutation $\Pi$.
• Figure 2: (a) Initialization of arbitrary qubit quantum state. (b) Initialization of arbitrary qutrit quantum state using qubits.
• Figure 3: Quantum SWAP test circuit diagram.
• Figure 4: A circuit implementation of a five-qubit QSS protocol.
• Figure 5: A circuit implementation of a Seven-qubit (Steane) QSS protocol with two qubits being erased, $|E|=2$.