  In the standard model of quantum secret sharing, typically, one is interested
in minimal authorized sets for the reconstruction of the secret. In such a
setting, reconstruction requires the communication of all the shares of the
corresponding authorized set. If we allow for non-minimal authorized sets, then
we can trade off the size of the authorized sets with the amount of
communication required for reconstruction. Based on the staircase codes,
proposed by Bitar and El Rouayheb, we propose a class of quantum threshold
secret sharing schemes that are also communication efficient. We call them
$((k,2k-1,d))$ communication efficient quantum secret sharing schemes where
$k\leq d\leq2k-1$. Using the proposed construction, we can recover a secret of
$d-k+1$ qudits by communicating $d$ qudits whereas using the standard
$((k,2k-1))$ quantum secret sharing requires $k(d-k+1)$ qudits to be
communicated. In other words, to share a secret of one qudit, the standard
quantum secret sharing requires $k$ qudits whereas the proposed schemes
communicate only $\frac{d}{d-k+1}$ qudits per qudit in the communication
complexity. Proposed schemes can reduce communication overheads by a factor
$O(k)$ with respect to standard schemes, when $d$ equals $2k-1$. Further, we
show that our schemes have optimal communication cost for secret
reconstruction.
