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

• URL: http://arxiv.org/abs/2503.08577v1
• Date: Tue, 11 Mar 2025 16:10:45 GMT
• Title: Fundamental solutions of heat equation on unitary groups establish an improved relation between $ε$-nets and approximate unitary $t$-designs
• Authors: Oskar Słowik, Oliver Reardon-Smith, Adam Sawicki,
• Abstract summary: The concepts of $epsilon$-nets and unitary $delta$-approximate $t$-designs are important and ubiquitous across quantum computation and information.<n>We improve the bound on the $delta$ required for a $epsilon$-net from $delta simeq left(epsilon3/2/dright)d2$ to form an $epsilon$-net from $delta simeq left(epsilon/d1/2right)d2$
• Score: 1.3654846342364308
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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