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

Abstract

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.

35 pages, 1 figure

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