Exponential lower bounds for low-degree strategies in position-based quantum cryptography

Abstract

Non-local quantum computation (NLQC) consists on the implementation of a bipartite unitary $U_{AB}$ by two cooperating distant players by means of a two-round protocol with a single intermediate round of simultaneous communication. It is a major open question to know whether there exists a unitary $U_{AB}$ whose implementation in NLQC requires extra quantum systems with dimensions ${\rm exp}(n^{Ω(1)})$, where $n$ is the dimension of systems $A$ and $B$. This type of exponential lower bound is essential for security of position-based quantum cryptography, since the capabilities of the adversaries are precisely described by NLQC. It has also been connected to many other problems, such as the optimality of universal quantum simulators, or the expected properties of holographic quantum gravity. Currently the best lower bounds are only sublinear in $n$. In this paper we consider the family of diagonal ${\pm 1}$ valued unitaries $U_{\varepsilon}$, indexed by an element of the Boolean hypercube $\varepsilon \in \{\pm 1\}^{n^2}$, i.e. $U_{\varepsilon}|i\rangle_A|j\rangle_B =\varepsilon_{ij} |i\rangle_A|j\rangle_B$. We show that if $U_{\varepsilon}$ can be implemented in NLQC with constant accuracy for all $\varepsilon$, and the dependency on $\varepsilon$ in the first round of the associated strategy is a polynomial of degree $O(n^{\frac{1}{4}-δ})$, then the implementation of at least one $U_{\varepsilon}$ in NLQC requires resources scaling as $\exp\left(Ω\left(\frac{n^{2δ}}{(\log n)^2}\right)\right)$. The proof is based on geometric properties of particular Banach spaces, namely their type constants, together with random estimates of Boolean functions valued on them.