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A family of permutationally invariant quantum codes

Arda Aydin, Max A. Alekseyev, Alexander Barg

TL;DR

This work develops a comprehensive framework for permutation-invariant quantum codes and introduces a new family ${\mathcal Q}_{g,m,\delta,\epsilon}$ that encodes one logical qubit into $n=2gm+\delta+1$ physical qubits. It proves general Knill–Laflamme-type conditions (C1–C4) for correcting $t$ Pauli errors by linking them to $2t$-deletion erasures, and specialized results show the same codes can correct $t$ deletions, $t$ Pauli errors, and, under suitable parameters, $t$ amplitude-damping errors. The authors demonstrate shorter code lengths than prior explicit permutationally invariant codes and provide examples with transversal gate capabilities, including $T$ and $\sqrt{T}$ implementations in certain instances. Collectively, the results advance constructive, scalable, permutation-invariant quantum error-correcting codes with practical error-model robustness and links to ground-state spin systems.

Abstract

We construct a new family of permutationally invariant codes that correct $t$ Pauli errors for any $t\ge 1$. We also show that codes in the new family correct quantum deletion errors as well as spontaneous decay errors. Our construction contains some of the previously known permutationally invariant quantum codes as particular cases, which also admit transversal gates. In many cases, the codes in the new family are shorter than the best previously known explicit permutationally invariant codes for Pauli errors and deletions. Furthermore, our new code family includes a new $((4,2,2))$ optimal single-deletion-correcting code. As a separate result, we generalize the conditions for permutationally invariant codes to correct $t$ Pauli errors from the previously known results for $t=1$ to any number of errors. For small $t$, these conditions can be used to construct new examples of codes by computer.

A family of permutationally invariant quantum codes

TL;DR

This work develops a comprehensive framework for permutation-invariant quantum codes and introduces a new family that encodes one logical qubit into physical qubits. It proves general Knill–Laflamme-type conditions (C1–C4) for correcting Pauli errors by linking them to -deletion erasures, and specialized results show the same codes can correct deletions, Pauli errors, and, under suitable parameters, amplitude-damping errors. The authors demonstrate shorter code lengths than prior explicit permutationally invariant codes and provide examples with transversal gate capabilities, including and implementations in certain instances. Collectively, the results advance constructive, scalable, permutation-invariant quantum error-correcting codes with practical error-model robustness and links to ground-state spin systems.

Abstract

We construct a new family of permutationally invariant codes that correct Pauli errors for any . We also show that codes in the new family correct quantum deletion errors as well as spontaneous decay errors. Our construction contains some of the previously known permutationally invariant quantum codes as particular cases, which also admit transversal gates. In many cases, the codes in the new family are shorter than the best previously known explicit permutationally invariant codes for Pauli errors and deletions. Furthermore, our new code family includes a new optimal single-deletion-correcting code. As a separate result, we generalize the conditions for permutationally invariant codes to correct Pauli errors from the previously known results for to any number of errors. For small , these conditions can be used to construct new examples of codes by computer.
Paper Structure (14 sections, 23 theorems, 137 equations)

This paper contains 14 sections, 23 theorems, 137 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Kraus Operators and the Knill--Laflamme conditions
  4. Quantum Deletion Channel
  5. Error correction conditions for permutation-invariant codes
  6. A new family of permutation-invariant Codes
  7. Deletion Correction Property
  8. Transversality
  9. Spontaneous Decay Errors
  10. Basics of the amplitude damping channel
  11. Generalization of the Pollatsek--Ruskai Conditions
  12. Concluding remarks
  13. Acknowledgment
  14. Proofs of combinatorial lemmas from Sec. \ref{['sec:new family']}

Key Result

Theorem 2.1

Let ${\mathscr C}$ be a quantum code with an orthonormal basis $\lvert{\boldsymbol c}_0\rangle, \lvert{\boldsymbol c}_1\rangle,\ldots, \lvert{\boldsymbol c}_{k-1}\rangle$, and let ${\EuScript A}$ be a quantum channel with Kraus operators ${\boldsymbol A}_i$. There exists a quantum recovery operator for some constants $g_{ab} \in {\mathbb C}$.

Theorems & Definitions (46)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Example 1
  • Theorem 2.1: Knill--Laflamme conditions
  • Definition 3.1
  • Lemma 3.1: ouyangEquivalence, Lemma 3.1
  • Lemma 3.2: ouyangEquivalence, Lemma 4.2
  • Definition 3.2
  • Example 2
  • ...and 36 more