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arXiv 2503.08577quant-phmath-phmath.MP
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酉群上热方程的基本解建立了$ε$-网与近似酉$t$-设计之间的改进关系

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

Oskar Słowik, Oliver Reardon-Smith, Adam Sawicki

备注：
35 pages, 1 figure
发表信息：
J. Phys. A: Math. Theor. 58 445301 (2025)
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AI总结：

本研究利用射影酉群上的热核构造狄拉克δ函数的多项式逼近，改进了$ε$-网与$δ$-近似酉$t$-设计之间的定量关系，大幅提升了相关$δ$上界，成果可应用于量子计算等多个领域。

摘要：

$ε$-网和酉（$δ$-近似）$t$-设计的概念在量子计算与量子信息领域十分重要且普遍存在。这两个概念密切相关，$t$、$δ$与$ε$之间的定量关系可应用于（非构造性的）无逆类Solovay-Kitaev定理以及随机量子电路等领域。近期研究的定量关系揭示了这两种构造之间的紧密联系：在参数选择合适的情况下，$ε$-网可充当酉$δ$-近似$t$-设计，反之亦然。在本工作中我们改进了这些结果，将$δ$-近似$t$-设计构成$ε$-网所需的$δ$上界从$δ\thickapprox \bigl(ε^{3/2}/d\bigr)^{d^2}$大幅提升至$δ\thickapprox \bigl(ε/d^{1/2}\bigr)^{d^2}$。我们通过利用射影酉群$\text{PU}(d) \cong \mathbf{U}(d)$上的热核构造狄拉克δ函数的多项式逼近来实现这一目标，我们研究了该热核的性质，其或可得到更广泛的应用。我们还概述了本研究结果在量子线路开销、量子复杂性以及黑洞物理学中的潜在应用。

英文摘要：

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.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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