Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Bounds in the Projective Unitary Group with Respect to Global Phase Invariant Metric

Bhanu Pratap Yadav, Mahdi Bayanifar, Olav Tirkkonen

TL;DR

The paper develops a geometric coding-theoretic framework on the projective unitary group ${\cal PU}_n$ under a global phase-invariant metric. It derives the volume of small metric balls, establishes Gilbert–Varshamov and Hamming bounds, and provides tight kissing-radius bounds to refine density and covering results. It also furnishes distortion-rate bounds for quantization and analyzes concrete PU$_n$ codebooks (projective Pauli, Clifford, diagonal Clifford hierarchy, and higher-level products), including numerical estimates for several semi-Clifford and higher-order codebooks, with simulations validating the theory. The results illuminate gate-approximation and circuit-design strategies in universal quantum computation, offering actionable bounds for codebook design and performance in quantum compilation tasks.

Abstract

We consider a global phase-invariant metric in the projective unitary group PUn, relevant for universal quantum computing. We obtain the volume and measure of small metric ball in PUn and derive the Gilbert-Varshamov and Hamming bounds in PUn. In addition, we provide upper and lower bounds for the kissing radius of the codebooks in PUn as a function of the minimum distance. Using the lower bound of the kissing radius, we find a tight Hamming bound. Also, we establish bounds on the distortion-rate function for quantizing a source uniformly distributed over PUn. As example codebooks in PUn, we consider the projective Pauli and Clifford groups, as well as the projective group of diagonal gates in the Clifford hierarchy, and find their minimum distances. For any code in PUn with given cardinality we provide a lower bound of covering radius. Also, we provide expected value of the covering radius of randomly distributed points on PUn, when cardinality of code is sufficiently large. We discuss codebooks at various stages of the projective Clifford + T and projective Clifford + S constructions in PU2, and obtain their minimum distance, distortion, and covering radius. Finally, we verify the analytical results by simulation.

Bounds in the Projective Unitary Group with Respect to Global Phase Invariant Metric

TL;DR

The paper develops a geometric coding-theoretic framework on the projective unitary group under a global phase-invariant metric. It derives the volume of small metric balls, establishes Gilbert–Varshamov and Hamming bounds, and provides tight kissing-radius bounds to refine density and covering results. It also furnishes distortion-rate bounds for quantization and analyzes concrete PU codebooks (projective Pauli, Clifford, diagonal Clifford hierarchy, and higher-level products), including numerical estimates for several semi-Clifford and higher-order codebooks, with simulations validating the theory. The results illuminate gate-approximation and circuit-design strategies in universal quantum computation, offering actionable bounds for codebook design and performance in quantum compilation tasks.

Abstract

We consider a global phase-invariant metric in the projective unitary group PUn, relevant for universal quantum computing. We obtain the volume and measure of small metric ball in PUn and derive the Gilbert-Varshamov and Hamming bounds in PUn. In addition, we provide upper and lower bounds for the kissing radius of the codebooks in PUn as a function of the minimum distance. Using the lower bound of the kissing radius, we find a tight Hamming bound. Also, we establish bounds on the distortion-rate function for quantizing a source uniformly distributed over PUn. As example codebooks in PUn, we consider the projective Pauli and Clifford groups, as well as the projective group of diagonal gates in the Clifford hierarchy, and find their minimum distances. For any code in PUn with given cardinality we provide a lower bound of covering radius. Also, we provide expected value of the covering radius of randomly distributed points on PUn, when cardinality of code is sufficiently large. We discuss codebooks at various stages of the projective Clifford + T and projective Clifford + S constructions in PU2, and obtain their minimum distance, distortion, and covering radius. Finally, we verify the analytical results by simulation.

Paper Structure

This paper contains 17 sections, 9 theorems, 75 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Projective Unitary Group
  4. Packing and Covering Problems
  5. Codebooks in the Projective Unitary Group
  6. Projective Pauli Group
  7. Projective Clifford Group
  8. Coodebooks from Higher Level of the Clifford Hierarchy
  9. Volume of Projective Unitary Group
  10. Minimum-distance Bounds on ${\cal PU}_n$
  11. Kissing Radius Bounds of Projective Unitary Group
  12. Minimum Distances of the Projective Unitary Codebooks
  13. Bounds on Codebook Distortion
  14. Distortion-Rate function in ${\cal PU}_n$
  15. Covering radius
  16. ...and 2 more sections

Key Result

Theorem 1

The volume of the ${\cal PU}_n$ is

Figures (7)

  • Figure 1: Theoretical and simulation results comparison of the measure of the ball in $\mathcal{PU}_2$, given by Corollary \ref{['Cor:Measurofball']}.
  • Figure 2: The upper bound $\bar{\varrho}$ and lower bound $\underline{\varrho}$ of kissing radius in $\mathcal{PU}_4$. These bounds are compared to simulated midpoints between two randomly generated codewords
  • Figure 3: Hamming bound \ref{['Eq:HammBound']} compared with tight Hamming bound \ref{['Cor:Lowerboundofkissingradius']} in $\mathcal{PU}_4$.
  • Figure 4: Comparison of minimum distances and theoretical bounds for different code families.
  • Figure 5: Minimum distances of $\textcolor{blue}{\tilde{\mathcal{C}}^l_{2,3}}$ , $\textcolor{blue}{\tilde{\mathcal{C}}^l_{2,4}}$ and $\tilde{\mathcal{C}}_{2,k}$ and comparison with GV \ref{['Eq:GVBound']} and Hamming \ref{['Eq:HammBound']} bounds in $\mathcal{PU}_2$.
  • ...and 2 more figures

Theorems & Definitions (18)

  • Theorem 1
  • proof
  • Corollary 1
  • proof
  • Lemma 1
  • proof
  • Theorem 2
  • proof
  • Corollary 2
  • proof
  • ...and 8 more