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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

TL;DR

This work advances the quantitative relationship between $\varepsilon$-nets and unitary $t$-designs by constructing polynomial approximations to the Dirac delta on the space of unitary channels via heat kernels on the projective unitary group $PU(d)$. By trimming the PU$(d)$ heat kernel to balanced polynomials and proving it approximates the Dirac delta with controllable $L^1$ and $L^2$ properties, the authors derive explicit bounds showing that a $\delta$-approximate $t$-design yields an $\varepsilon$-net with $t$ scaling like $t \gtrsim 32 d^{5/2} \log(d)/\varepsilon$ and with $\delta$ decaying roughly as $(\varepsilon/d^{1/2})^{d^2}$ up to polylogarithms. They further provide a polynomial approximation (trimmed heat kernel) whose properties enable both exact and approximate $t$-design constructions to produce efficient $\varepsilon$-nets, with significant implications for Solovay-Kitaev type overheads, quantum complexity, and information scrambling in black holes. The approach unifies harmonic analysis on compact Lie groups with quantum information tasks, offering a broadly applicable toolkit for analyzing finite gate sets and random circuits. Overall, the paper tightens the bridge between two central notions in quantum information while delivering practical bounds and techniques for circuit design and complexity questions.

Abstract

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.

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

TL;DR

This work advances the quantitative relationship between -nets and unitary -designs by constructing polynomial approximations to the Dirac delta on the space of unitary channels via heat kernels on the projective unitary group . By trimming the PU heat kernel to balanced polynomials and proving it approximates the Dirac delta with controllable and properties, the authors derive explicit bounds showing that a -approximate -design yields an -net with scaling like and with decaying roughly as up to polylogarithms. They further provide a polynomial approximation (trimmed heat kernel) whose properties enable both exact and approximate -design constructions to produce efficient -nets, with significant implications for Solovay-Kitaev type overheads, quantum complexity, and information scrambling in black holes. The approach unifies harmonic analysis on compact Lie groups with quantum information tasks, offering a broadly applicable toolkit for analyzing finite gate sets and random circuits. Overall, the paper tightens the bridge between two central notions in quantum information while delivering practical bounds and techniques for circuit design and complexity questions.

Abstract

The concepts of -nets and unitary (-approximate) -designs are important and ubiquitous across quantum computation and information. Both notions are closely related and the quantitative relations between , and 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 -nets functioning as unitary -approximate -designs and vice-versa, for appropriate choice of parameters. In this work we improve these results, significantly increasing the bound on the required for a -approximate -design to form an -net from to . We achieve this by constructing polynomial approximations to the Dirac delta using heat kernels on the projective unitary group , 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.

Paper Structure

This paper contains 14 sections, 20 theorems, 192 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main ideas
  3. Main results and applications
  4. The heat kernel on the projective unitary group
  5. Bounds for exact t-designs
  6. Bounds for delta-approximate t-designs
  7. Trimmed heat kernel as a polynomial approximation of Dirac delta
  8. Summary and future work
  9. Connecting the heat kernels on the projective and special unitary groups
  10. Bounding the polynomial approximation of the heat kernel
  11. Bounding the dominant term
  12. Bounding the correction term
  13. Bounding the L2-norm
  14. Well-definedness of the Poisson form of the heat kernel at non-regular points

Key Result

Lemma 1

Let $\varphi$ be a non-negative function on $\mathrm{SU}(d)$ Haar-normalised to 1. Fix $\epsilon > 0$ and consider a set $\tilde{B}_{P,\epsilon}$ defined by eqn:projective-epsilon-ball-in-sun, let its complement be $\tilde{B}_{P,\epsilon}^c$. Then

Figures (1)

  • Figure 1: Illustration of the distribution of the components of a heat kernel $H_P$ on $\mathrm{PU}(3)$, obtained via the averaging map applied to a heat kernel $H_S$ in Poisson form. Actual shapes and relative sizes are not depicted. The averaging takes place over $\Gamma$, which consists of three roots of unity, denoted by red, green and blue points in the central square region. The elements of $\Gamma$ act by shifting by the roots of unity along the dotted grey lines, corresponding to a torus. The heat kernel $H_S$ corresponds to the red peaks. Each repeated square region corresponds to the contribution from a different $k$-vector in the Poisson form, which lies on the grey dashed grid. Notice that only the central square region ($k=0$) corresponds to points in a group. However, the tails of the peaks from non-central square regions ($k \neq 0$) overlap with the central square region, contributing to the heat kernel. A ball $\tilde{B}_{P, \epsilon}$ corresponds to a sum of three balls in a central region, denoted by dotted lines. A ball $B_{\epsilon}$ corresponds to the red ball at the origin, and the grey region corresponds to its complement. Lemma \ref{['lem:intP']} states that the integral of $H_P$ over $\tilde{B}_{P, \epsilon}$ can be upper bounded by the integral of $H_S$ (proportional to the red component) over the grey region. This is outlined by the opacity of the blue and green peaks. Lemma \ref{['lem:tail-bound-smaller-than-dominant']} shows that this integral can be bounded by bounding the contribution from the central ($k=0$) red peak, which is obtained in Lemma \ref{['lem:I0_bound']}.

Theorems & Definitions (47)

  • Example 1: Heat equation on a circle and the Poisson summation formula
  • Remark 1
  • Example 2: Irreps of $\mathrm{SU}(2)$ and $\mathrm{SO}(3)$
  • Remark 2: Optimality of the trimming procedure
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Theorem 1
  • proof
  • ...and 37 more