Fundamental solutions of heat equation on unitary groups establish an improved relation between $ε$-nets and approximate unitary $t$-designs

The concepts of $ε$-nets and unitary ($δ$-approximate) $t$-designs are important and ubiquitous across quantum computation and information. Both notions are closely related and the quantitative relations between $t$, $δ$ and $ε$ find applications in areas such as (non-constructive) inverse-free Solovay-Kitaev like theorems and random quantum circuits. In recent work, quantitative relations have revealed the close connection between the two constructions, with $ε$-nets functioning as unitary $δ$-approximate $t$-designs and vice-versa, for appropriate choice of parameters. In this work we improve these results, significantly increasing the bound on the $δ$ required for a $δ$-approximate $t$-design to form an $ε$-net from $δ\simeq \left(ε^{3/2}/d\right)^{d^2}$ to $δ\simeq \left(ε/d^{1/2}\right)^{d^2}$. We achieve this by constructing polynomial approximations to the Dirac delta using heat kernels on the projective unitary group $\mathrm{PU}(d) \cong\mathbf{U}(d)$, whose properties we studied and which may be applicable more broadly. We also outline the possible applications of our results in quantum circuit overheads, quantum complexity and black hole physics.