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

The concepts of $\epsilon$-nets and unitary ($\delta$-approximate) $t$-designs are important and ubiquitous across quantum computation and information. Both notions are closely related and the quantitative relations between $t$, $\delta$ and $\epsilon$ 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 $\epsilon$-nets functioning as unitary $\delta$-approximate $t$-designs and vice-versa, for appropriate choice of parameters. In this work we improve these results, significantly increasing the bound on the $\delta$ required for a $\delta$-approximate $t$-design to form an $\epsilon$-net from $\delta \simeq \left(\epsilon^{3/2}/d\right)^{d^2}$ to $\delta \simeq \left(\epsilon/d^{1/2}\right)^{d^2}$. 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.