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arXiv:1911.09470 · Physical Review A

Verifiable hybrid secret sharing with few qubits

Victoria Lipińska, Gláucia Murta, Jérémy Ribeiro, Stephanie Wehner

Published Mar 20, 2020• 28 citations
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Abstract

We consider the task of sharing a secret quantum state in a quantum network in a verifiable way. We propose a protocol that achieves this task, while reducing the number of required qubits, as compared to the existing protocols. To achieve this, we combine classical encryption of the quantum secret with an existing verifiable quantum secret sharing scheme based on Calderbank-Shor-Steane quantum error correcting codes. In this way we obtain a verifiable hybrid secret sharing scheme for sharing qubits, which combines the benefits of quantum and classical schemes. Our scheme does not reveal any information to any group of less than half of the $n$ nodes participating in the protocol. Moreover, for sharing a one-qubit state each node needs a quantum memory to store $n$ single-qubit shares, and requires a workspace of at most $3n$ qubits in total to verify the quantum secret. Importantly, in our scheme an individual share is encoded in a single qubit, as opposed to previous schemes requiring $\mathrm{\ensuremath{\Omega}}(logn)$ qubits per share. Furthermore, we define a ramp verifiable hybrid scheme. We give explicit examples of various verifiable hybrid schemes based on existing quantum error correcting codes.

DOI
10.1103/physreva.101.032332

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We consider the task of sharing a secret quantum state in a quantum network in a verifiable way. We propose a protocol that achieves this task, while reducing the number of required qubits, as compared to the existing protocols. To achieve this, we combine classical encryption of the quantum secret with an existing verifiable quantum secret sharing scheme based on Calderbank-Shor-Steane quantum error correcting codes. In this way we obtain a verifiable hybrid secret sharing scheme for sharing qubits, which combines the benefits of quantum and classical schemes. Our scheme does not reveal any information to any group of less than half of the nnn nodes participating in the protocol. Moreover, for sharing a one-qubit state each node needs a quantum memory to store nnn single-qubit shares, and requires a workspace of at most 3n3n3n qubits in total to verify the quantum secret. Importantly, in our scheme an individual share is encoded in a single qubit, as opposed to previous schemes requiring \ensuremathΩ(logn)\mathrm{\ensuremath{\Omega}}(logn)\ensuremathΩ(logn) qubits per share. Furthermore, we define a ramp verifiable hybrid scheme. We give explicit examples of various verifiable hybrid schemes based on existing quantum error correcting codes.
Quantum secret-sharing and quantum error-correction schemes rely on multipartite decoding protocols, yet the nonlocal operations involved are challenging and sometimes infeasible. Here we construct a quantum secret-sharing protocol with a reduced number of quantum communication channels between the players. We introduce a scheme based on embedding a classical linear code into a quantum error-correcting code and then mapping the latter to a quantum secret-sharing protocol. In contrast to the Calderbank-Shor-Steane construction, we do not impose any restriction on the classical code; our protocol works with any arbitrary linear code. Our work paves the way towards the more general problem of simplifying the decoding of quantum error-correcting codes.
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We consider the task of secure multiparty distributed quantum computation on a quantum network. We propose a protocol based on quantum error correction which reduces the number of necessary qubits. That is, each of the nnn nodes in our protocol requires an operational workspace of n2+4n{n}^{2}+4nn2+4n qubits, as opposed to the previously shown $\mathrm{\ensuremath{\Omega}}\left(({n}^{3}+{n}^{2}{s}^{2})logn\right)$ qubits, where sss is a security parameter. Additionally, we reduce the communication complexity by a factor of O(n3log(n))O({n}^{3}log(n))O(n3log(n)) qubits per node compared to existing protocols. To achieve universal computation, we develop a distributed procedure for verifying magic states, which allows us to apply distributed gate teleportation and which may be of independent interest. We showcase our protocol in a small example for a seven-node network.
We present a general scheme for sharing quantum secrets, and an extension to sharing classical secrets, which contain all known quantum-secret-sharing schemes. In this framework, we show the equivalence of existence of both schemes, that is, the existence of a scheme sharing a quantum secret implies that the extended classical-secret-sharing scheme works, and vice versa. As a consequence of this, we find schemes sharing classical secrets for arbitrary access structures. We then clarify the relationship to quantum error correction and observe several restrictions thereby imposed, which for example indicates that for pure-state threshold schemes the share size qqq must scale with the number of players nnn as $q\ensuremath{\ge}\sqrt{n}$. These results also provide a way of searching for quantum-error-correcting codes.
Abstract This paper integrates dynamic participant management, verifiable secret reconstruction, and fault tolerance against noise and adversarial attacks. By leveraging graph states for scalable entanglement and a quantum error-correcting code ([ ⁣[n,1,d] ⁣][\![n,1,d]\!][[n,1,d]]), the protocol enables secure sharing of classical and quantum secrets among n participants, requiring a threshold k for reconstruction. A hybrid verification mechanism---combining randomly interleaved trap qubits and post-quantum hash functions---detects malicious behavior with probability 1−(34)m1 - \left(\frac{3}{4}\right)^m1−(43​)m for m traps, while dynamic graph-state reconfiguration permits seamless participant addition/removal. Formal security proofs demonstrate information-theoretic secrecy against collusion ( t < k participants), adaptive coherent attacks, and revoked dishonest participants, alongside tolerance for t=⌊(d−1)/2⌋t = \lfloor (d-1)/2 \rfloort=⌊(d−1)/2⌋ errors. The protocol’s composable security is established via the Universal Composability framework, and simulations on IBM Quantum achieve greater than 95\% secret recovery fidelity under depolarizing noise ((p = 0.01). Key innovations include linear resource scaling, stabilizer-based error correction, and compatibility with post-quantum cryptography. This work bridges theoretical security with practical feasibility, offering a robust framework for quantum-secure systems in dynamic, noisy environments.
We demonstrate a new construction for perfect quantum secret sharing (QSS) schemes based on imperfect “ramp” secret sharing combined with classical encryption, in which the individual parties' shares are split into quantum and classical components, allowing the former to be of lower dimension than the secret itself. We show that such schemes can be performed with smaller quantum components and lower overall quantum communication than required for existing methods. We further demonstrate that one may combine both imperfect quantum and imperfect classical secret sharing to produce an overall perfect QSS scheme, and that examples of such schemes (which we construct) can have the smallest quantum and classical share components possible for their access structures, something provably not achievable using perfect underlying schemes. Our construction has significant potential for being adapted to other QSS schemes based on stabilizer codes.