Verifiable hybrid secret sharing with few qubits

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 $\mathrm{\Omega }(logn)$ 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.

Figure 1

Lifting the secrecy of an $n$-node secret sharing scheme of a quantum state, i.e., increasing the value $p$ of nodes which gain no information about the secret state throughout the execution of the scheme. Here $t$ denotes the number of nodes that can perform arbitrary operations on their shares throughout the protocol, and hence corrupt the secret (active cheaters).

Figure 2

A sketch of a verifiable hybrid secret sharing (VHSS) protocol for $n=10$ nodes denoted $N_1,\cdots ,N_{10}$, with $n_q=7$ quantum ($•$) and $n_c=10$ classical ($▴$) shares. The quantum secret state $|\psi \rangle$ of the dealer is encrypted using a classical key $s$. The resulting encrypted state $\sigma$ and the key $s$ are then distributed by the dealer as quantum and classical shares respectively.

Figure 3

The encoding tree for (2,1,7)-VQSS protocol with seven nodes $N_1,\cdots ,N_7$, based on the Steane's ${\left[[7,1,3]\right]}_2$ code. The figure represents the encoding done in the sharing phase by each of the nodes.