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arXiv:2505.18840 · Quantum Information Processing

Measurement-free reconstruction circuit of quantum secrets in quantum secret sharing

Shogo Chiwaki, Ryutaroh Matsumoto

Published Sep 29, 2025• 1 citation
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Abstract

Abstract For a quantum secret sharing scheme built from a general quantum stabilizer code, no measurement-free circuit has been known for reconstructing its quantum secrets, except particular classes, such as one proposed by Cleve, Gottesman and Lo. We propose a measurement-free reconstruction circuit of quantum secrets in quantum secret sharing based on stabilizer codes. Our reconstruction circuit has width $$k+|J|$$ k + | J | and consists of O ( k | J |) one- or two-qudit unitary gates when | J | participants reconstruct k -qudit quantum secrets.

DOI
10.1007/s11128-025-04943-y

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Abstract For a quantum secret sharing scheme built from a general quantum stabilizer code, no measurement-free circuit has been known for reconstructing its quantum secrets, except particular classes, such as one proposed by Cleve, Gottesman and Lo. We propose a measurement-free reconstruction circuit of quantum secrets in quantum secret sharing based on stabilizer codes. Our reconstruction circuit has width k+∣J∣k+|J|k+∣J∣ k + | J | and consists of O ( k | J |) one- or two-qudit unitary gates when | J | participants reconstruct k -qudit quantum secrets.
Stabilizer-based quantum secret sharing has two methods to reconstruct a quantum secret: The erasure correcting procedure and the unitary procedure. It is known that the unitary procedure has a smaller circuit width. On the other hand, it is unknown which method has smaller depth and fewer circuit gates. In this letter, it is shown that the unitary procedure has smaller depth and fewer circuit gates than the erasure correcting procedure which follows a standard framework performing measurements and unitary operators according to the measurements outcomes, when the circuits are designed for quantum secret sharing using the [[5, 1, 3]] binary stabilizer code. The evaluation can be reversed if one discovers a better circuit for the erasure correcting procedure which does not follow the standard framework.
Helper quantum secret sharing is a form of secret sharing defined by its unique access structure. One special fixed party, called the helper, can work together with any other party to fully decode the secret. A blind helper is one who can provide this assistance while not holding any local information about the encoded secret. In this work, we analyze the general structure of QSS helper codes and present new code constructions. We fully characterize the structure of blind helper stabilizer codes and show that for the encoding of a single qubit, recovery is always possible using one-way local operations and classical communication (LOCC) from the helper to the targeted party. Furthermore, we demonstrate how such codes also allow for the helper to target larger subsets of parties by one-way LOCC, enabling them to be authorized to recover the secret. Finally, when each party only holds a qubit, we identify the general form of all helper codes (including non-stabilizer codes) and find that LOCC recovery is only possible in special cases.
A quantum secret sharing scheme is a cryptographic protocol by which a dealer can share a secret among a group of nnn players so that only certain subsets of players can recover the secret by collaboration. In this paper we propose communication efficient quantum threshold secret sharing schemes. They minimize the amount of quantum communication required to reconstruct the secret when more than the necessary number of players collaborate. They are based on a class of staircase codes proposed by Bitar and El Rouayheb. In a standard ((k,n))((k,n))((k,n)) quantum threshold scheme, any subset of kkk or more players can recover the secret. The quantum communication cost for reconstruction in such schemes is kkk qudits for each secret qudit. Using the proposed construction, any subset of $d\ensuremath{\ge}k$ players can also collaborate to recover the secret with a communication cost of ddd qudits for $d\ensuremath{-}k+1$ secret qudits. In other words, for the proposed schemes the quantum communication cost is only $\frac{d}{d\ensuremath{-}k+1}$ qudits for every secret qudit. For d>kd>kd>k, proposed schemes are communication efficient with respect to standard schemes; and when $d=2k\ensuremath{-}1$, the quantum communication cost is reduced by a factor O(k)O(k)O(k). Further, when $n=2k\ensuremath{-}1$, the proposed schemes have optimal communication cost for secret reconstruction.
A quantum secret-sharing (QSS) protocol consists of two main phases, called sharing and reconstruction. In the first phase, the dealer selects a secret, divides it into several shares, and sends each participant its share securely with a quantum channel. In the second phase, the participants run an interactive protocol in order to reconstruct the secret. If the participants can communicate via a broadcast channel, they can show their shares and learn the secrets simultaneously. So what happens if the channel is not simultaneous? In this paper, we propose a QSS protocol with cheaters by using partially and maximally entangled states. A secure and fair reconstruction mechanism is designed, in a way that each participant can learn or cannot learn the secret simultaneously.