Fundamental solutions of heat equation on unitary groups establish an improved relation between $ε$-nets and approximate unitary $t$-designs
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917048579260416 |
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| author | Słowik, Oskar Reardon-Smith, Oliver Sawicki, Adam |
| author_facet | Słowik, Oskar Reardon-Smith, Oliver Sawicki, Adam |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_08577 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Fundamental solutions of heat equation on unitary groups establish an improved relation between $ε$-nets and approximate unitary $t$-designs Słowik, Oskar Reardon-Smith, Oliver Sawicki, Adam Quantum Physics Mathematical Physics 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. |
| title | Fundamental solutions of heat equation on unitary groups establish an improved relation between $ε$-nets and approximate unitary $t$-designs |
| topic | Quantum Physics Mathematical Physics |
| url | https://arxiv.org/abs/2503.08577 |