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Transversal gates of the ((3,3,2)) qutrit code and local symmetries of the absolutely maximally entangled state of four qutrits

Ian Tan

TL;DR

The paper uncovers a fundamental link between entanglement structure and quantum error correction by proving a bijection between LU orbits of perfect tensors and LU orbits of certain $((n-1,D,n/2))_D$ codes. Focusing on the four-qutrit case, it shows that the unique $((3,3,2))_3$ code $\mathcal{C}$ corresponds to the unique AME state $|\Phi\rangle$ up to LU, and that the transversal gate group coincides with the Weyl group of $\mathcal{C}$ within Vinberg's graded Lie algebra framework. The work embeds the 3-qutrit system into the exceptional Lie algebra $\mathfrak{e}_6$ via a $\mathbb{Z}_3$-grading, identifying the Cartan subspace with $\mathcal{C}$ and linking stabilizers, transversal gates, and local symmetries. Using Kempf–Ness theory and Vinberg's theory, it derives generators for both transversal gates and the local symmetry group $S(|\Phi|)$, and computes their finite orders, revealing a tight algebraic structure governing LU-equivalence of AME states and associated codes. Collectively, these results advance the geometric invariant theory perspective on quantum information and illuminate how group-theoretic symmetries dictate fault-tolerant operations and state reachability.

Abstract

We provide a proof that there exists a bijection between local unitary (LU) orbits of absolutely maximally entangled (AME) states in $(\mathbb{C}^D)^{\otimes n}$ where $n$ is even, also known as perfect tensors, and LU orbits of $((n-1,D,n/2))_D$ quantum error correcting codes. Thus, by a result of Rather et al. (2023), the AME state of 4 qutrits and the pure $((3,3,2))_3$ qutrit code $\mathcal{C}$ are both unique up to the action of the LU group. We further explore the connection between the 4-qutrit AME state and the code $\mathcal{C}$ by showing that the group of transversal gates of $\mathcal{C}$ and the group of local symmetries of the AME state are closely related. Taking advantage of results from Vinberg's theory of graded Lie algebras, we find generators of both of these groups.

Transversal gates of the ((3,3,2)) qutrit code and local symmetries of the absolutely maximally entangled state of four qutrits

TL;DR

The paper uncovers a fundamental link between entanglement structure and quantum error correction by proving a bijection between LU orbits of perfect tensors and LU orbits of certain codes. Focusing on the four-qutrit case, it shows that the unique code corresponds to the unique AME state up to LU, and that the transversal gate group coincides with the Weyl group of within Vinberg's graded Lie algebra framework. The work embeds the 3-qutrit system into the exceptional Lie algebra via a -grading, identifying the Cartan subspace with and linking stabilizers, transversal gates, and local symmetries. Using Kempf–Ness theory and Vinberg's theory, it derives generators for both transversal gates and the local symmetry group , and computes their finite orders, revealing a tight algebraic structure governing LU-equivalence of AME states and associated codes. Collectively, these results advance the geometric invariant theory perspective on quantum information and illuminate how group-theoretic symmetries dictate fault-tolerant operations and state reachability.

Abstract

We provide a proof that there exists a bijection between local unitary (LU) orbits of absolutely maximally entangled (AME) states in where is even, also known as perfect tensors, and LU orbits of quantum error correcting codes. Thus, by a result of Rather et al. (2023), the AME state of 4 qutrits and the pure qutrit code are both unique up to the action of the LU group. We further explore the connection between the 4-qutrit AME state and the code by showing that the group of transversal gates of and the group of local symmetries of the AME state are closely related. Taking advantage of results from Vinberg's theory of graded Lie algebras, we find generators of both of these groups.
Paper Structure (23 sections, 23 theorems, 44 equations, 2 figures)

This paper contains 23 sections, 23 theorems, 44 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Overview
  3. Some important groups
  4. Quantum error correcting codes and uniform states
  5. Quantum error correcting codes
  6. Stabilizer codes
  7. Transversal gates
  8. Uniform states
  9. The correspondence between perfect tensors and MDS codes
  10. New codes from old ones
  11. Correspondence of LU orbits
  12. Connections to prior literature
  13. The 4-qutrit AME state and its associated code
  14. Geometric invariant theory and applications
  15. Critical points
  16. ...and 8 more sections

Key Result

theorem 2.2

If $\mathcal{C}$ is a code with parameters $((n,K,d))_D$, then $\log_D K\leq n-2(d-1)$.

Figures (2)

  • Figure 1: A visualization of the vector graph $L_3$ associated to $W(\mathcal{C})$. The vectors $e_i$ are defined in \ref{['eq:vectors']}. There is no edge between two vectors if and only if they are orthogonal.
  • Figure 2: Generators of the local symmetry group $S(\ket{\Phi})$. We let $\omega=e^{2\pi \mathrm i/3}$.

Theorems & Definitions (50)

  • Definition 2.1
  • theorem 2.2: Quantum Singleton bound
  • proof
  • theorem 2.3
  • proof
  • Definition 2.4
  • theorem 2.5
  • proof
  • theorem 3.1
  • proof
  • ...and 40 more