The multiple assignment scheme is to assign one or more shares to single participant so that any kind of access structure can be realized by classical secret sharing schemes. We propose its quantum version including ramp secret sharing schemes. Then we propose an integer optimization approach to minimize the average share size.

A secret sharing scheme allows a secret to be shared among a set of participants, P, such that only authorized subsets of P can recover the secret, but any unauthorized subset can not recover the secret. It can be used to protect important secret data, such as cryptographic keys, from being lost or destroyed without accidental or malicious exposure. In this paper, we consider secret sharing schemes based on interpolating polynomials. We show that, by simply increasing the number of shares held by each participant, there is a multiple assignment scheme for any monotone access structure such that cheating can be detected with very high probability by any honest participant even the cheaters form a coalition in order to deceive him.