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

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 $k+\mathrm{\mid }J\mathrm{\mid }$ and consists of $O(k\mathrm{\mid }J\mathrm{\mid })$ one- or two-qudit unitary gates when $\mathrm{\mid }J\mathrm{\mid }$ participants reconstruct $k$-qudit quantum secrets.