Weak approximate unitary designs and applications to quantum encryption

TL;DR

This paper demonstrates that sampling a polynomial number of unitaries from an exact $t$-design yields an approximate $t$-design with positive probability, and applies this to construct a non-malleable quantum encryption scheme.

Contribution

It proves that polynomially many samples from an exact $t$-design form an approximate $t$-design and introduces a non-malleable quantum encryption scheme with similar security to the quantum one-time pad.

Findings

Sampling $d^t ext{poly}(t, ext{log} d, 1/ extpsilon)$ unitaries yields an $ extvarepsilon$-approximate $t$-design.

Constructs a non-malleable quantum encryption scheme with key size comparable to the quantum one-time pad.

Provides a partially derandomized approach to quantum encryption security.

Abstract

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 $d^t\mathrm{poly}(t,\log d, 1/\epsilon)$ unitaries from an exact $t$-design provides with positive probability an $\epsilon$-approximate $t$-design, if the error is measured in one-to-one norm distance of the corresponding $t$-twirling channels. As an application, we give a partially derandomized 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.