Unitary t-designs are the bread and butter of quantum information theory and beyond. An important issue in practice is that of efficiently constructing good approximations of such unitary t-designs. Building on results by Aubrun (Comm. Math. Phys. 2009), we prove that sampling dtpoly(t, log d, 1/ε) unitaries from an exact t-design provides with positive probability an ε-approximate t-design, if the error is measured in one-to-one norm. As an application, we give a 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.

Taming Quantum Adversaries
Centrum Wiskunde & Informatica, Amsterdam (CWI), The Netherlands

Lancien, C, & Majenz, C. (2020). Weak approximate unitary designs and applications to quantum encryption. Quantum, 4. doi:10.22331/q-2020-08-28-313