Fundamental solutions of the heat equation on unitary groups establish an improved relation between <i>ε</i> -nets and approximate unitary <i>t</i> -designs

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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>δ</mml:mi> <mml:mo>≃</mml:mo> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>ϵ</mml:mi> <mml:mrow> <mml:mn>3</mml:mn> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>d</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow> <mml:msup> <mml:mi>d</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>δ</mml:mi> <mml:mo>≃</mml:mo> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>ϵ</mml:mi> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:msup> <mml:mi>d</mml:mi> <mml:mrow> <mml:mn>1</mml:mn> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow> <mml:msup> <mml:mi>d</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> . We achieve this by constructing polynomial approximations to the Dirac delta using heat kernels on the projective unitary group <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mi>PU</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>d</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>≅</mml:mo> <mml:mrow> <mml:mi mathvariant="bold">U</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>d</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> , 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.