Unitary designs play an important role in quantum information theory and beyond. Applications include the ubiquitous decoupling technique in quantum information theory, quantum encryption, randomized benchmarking and more.
Researchers have studied a weaker form of approximate unitary designs motivated by operations on isolated quantum systems. They showed that approximate unitary designs of much smaller size exist in this restricted setting compared to the general setting.
They exemplified the usefulness of such objects by describing a construction of a quantum encryption scheme that, in addition to ensuring confidentiality of a message akin to the quantum one-time pad, protects the message against tampering by attackers without (or with small) quantum memory.
As an application, they have provided a partially de-randomized construction of a quantum encryption scheme that has roughly the same key size and security as the quantum one-time pad, but possesses the additional property of being non-malleable against adversaries without quantum side information.