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A Search for High-Threshold Qutrit Magic State Distillation Routines

Shiroman Prakash, Rishabh Singhal

TL;DR

The paper investigates whether contextuality suffices for universal quantum computation by examining thresholds for distilling the qutrit strange state. It develops a weight-enumerator framework that links magic-state distillation performance to stabilizer-code enumerators, with a simplified form for the strange state enabling large-scale searches up to $[[23,1]]_3$. The authors find over 600 $23$-qutrit CSS codes that distill the strange state with cubic noise suppression, but none surpass the $[[11,1,5]]_3$ Golay code’s threshold $\epsilon_* \approx 0.387$, suggesting that high thresholds are rare even in large code families. The results illuminate how distillation behavior scales with code size and symmetry and highlight the potential genericity of distillation in large codes while underscoring the difficulty of beating the Golay benchmark.

Abstract

Determining the best attainable threshold for qudit magic state distillation is directly related to the question of whether or not contextuality is sufficient for universal quantum computation. We show that the performance of a qudit correcting code for magic state distillation is captured by its complete weight enumerator. For the qutrit strange state -- a maximally magic non-stabilizer state -- the performance of a code is captured by its simple weight enumerator. This result allows us to carry out an extensive search for high-threshold magic state distillation routines for the strange state. Our search covers all $[[n,1]]_3$ qutrit stabilizer codes with a complete set of transversal Clifford gates for $n\leq 23$, and all $[[n,1]]_3$ stabilizer codes with a transversal $H^2$ gate with $n \leq 9$ qudits. For $n=23$, we find over 600 CSS codes that can distill the qutrit strange state with cubic noise suppression. While none of these codes surpass the threshold of the 11-qutrit Golay code, their existence suggests that, for large codes, the ability to distill the qutrit strange state is somewhat generic.

A Search for High-Threshold Qutrit Magic State Distillation Routines

TL;DR

The paper investigates whether contextuality suffices for universal quantum computation by examining thresholds for distilling the qutrit strange state. It develops a weight-enumerator framework that links magic-state distillation performance to stabilizer-code enumerators, with a simplified form for the strange state enabling large-scale searches up to . The authors find over 600 -qutrit CSS codes that distill the strange state with cubic noise suppression, but none surpass the Golay code’s threshold , suggesting that high thresholds are rare even in large code families. The results illuminate how distillation behavior scales with code size and symmetry and highlight the potential genericity of distillation in large codes while underscoring the difficulty of beating the Golay benchmark.

Abstract

Determining the best attainable threshold for qudit magic state distillation is directly related to the question of whether or not contextuality is sufficient for universal quantum computation. We show that the performance of a qudit correcting code for magic state distillation is captured by its complete weight enumerator. For the qutrit strange state -- a maximally magic non-stabilizer state -- the performance of a code is captured by its simple weight enumerator. This result allows us to carry out an extensive search for high-threshold magic state distillation routines for the strange state. Our search covers all qutrit stabilizer codes with a complete set of transversal Clifford gates for , and all stabilizer codes with a transversal gate with qudits. For , we find over 600 CSS codes that can distill the qutrit strange state with cubic noise suppression. While none of these codes surpass the threshold of the 11-qutrit Golay code, their existence suggests that, for large codes, the ability to distill the qutrit strange state is somewhat generic.
Paper Structure (19 sections, 6 theorems, 89 equations, 5 figures)

This paper contains 19 sections, 6 theorems, 89 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Heisenberg-Weyl operators and the Clifford group
  4. Discrete Wigner functions
  5. Qudit stabilizer codes
  6. Weight enumerators
  7. Distillation and weight enumerators
  8. A general formulation in terms of complete weight enumerators
  9. The strange state and simple weight enumerators
  10. Conditions for magic state distillation
  11. Search for distillation routines
  12. Narrowing the search space via symmetry
  13. Stabilizer codes with trivial syndrome
  14. Stabilizer codes with a complete set of transversal Clifford gates
  15. Discussion
  16. ...and 4 more sections

Key Result

Theorem 1

Let $\hat{\rho}$ be a single qudit mixed state described by the Wigner function $W(\hat{\rho}; \alpha, \beta)$ and let $\mathcal{S}$ be an $[[n,k]]_p$ stabilizer code with trivial syndrome.

Figures (5)

  • Figure 1: A histogram of all the thresholds that arise from 23-qubit CSS codes that are able to distill the strange state. We found a total of 646 codes, and the highest threshold was $\epsilon_* = 0.318$.
  • Figure 2: The distillation performance of the $[[23,1,5]]_3$ code defined via equation \ref{['23-code']}.
  • Figure 3: A scatter plot of the success probability (logarithmic scale) and threshold for the 646 CSS codes that distill the strange state.
  • Figure 4: The distillation performance for the $[[13,1,4]]_3$ code of SharmaGarani2024. For small $\epsilon_{\rm in}$, we find $\epsilon_{\rm out} \to 3/2$, which, from Equation \ref{['mixed']}, corresponds to a mixture of two states orthogonal to the strange state.
  • Figure 5: The distillation performance for the $[[29,1,7]]_3$ code of SharmaGarani2024. The threshold for distillation is zero.

Theorems & Definitions (12)

  • Theorem 1
  • proof : Proof of part (a)
  • proof : Proof of part (b):
  • Corollary 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • ...and 2 more