Weak approximate unitary designs and applications to quantum encryption

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 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.

Quantum Private Broadcasting

“…However, it has been shown that -approximate unitary t-designs on n qubits can be efficiently constructed with local random circuits that are polynomial in n, t and log(1/ ) [BHH16]. In this work, we use the construction of an -approximate unitary t-design from [LM20], where they prove an upper bound for when the unitaries are sampled from an exact tdesign (Theorem 3.1). They show that when at most C(td) t (t log d) 6 / 2 unitaries are sampled from a t-design for some constant C, then this is an -approximate unitary t-design with probability at least 1 2 .…”

“…The proof of Theorem 6.6 follows similarly to [LM20], with altered bounds due to ρ being in the symmetric subspace. To see this, we need the following result based from [Aub09] Lemma 5, now adjusted so that ρ ∈ D(Sym(d t )) and U ⊗t i is being applied instead of simply U i .…”

“…Using Schur-Weyl duality and Schur's Lemma [FH91] similarly to [LM20], one can write the following for ρ ∈ D(H ⊗t d ):…”

“…The indistinguishability of ciphertexts for our t-QPB encryption scheme is based on the definitions from [LM20], which compares the encryption scheme with that of a 'state replacement channel' σ . For a fixed σ ∈ D(H ⊗t d ), this is defined as…”

“…When t = 1, Definition 3.2 corresponds to the conventional information theoretic encryption [LM20], where there is no restriction in the input space.…”