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A theory of quantum error correction for permutation-invariant codes

Yingkai Ouyang, Gavin K. Brennen

TL;DR

This work develops a general theory of quantum error correction for permutation-invariant (PI) codes by exploiting the representation theory of the symmetric group. It presents a two-stage decoding framework: Stage 1 projects noisy states onto irreducible representations via total angular momentum measurements to identify a standard Young tableau, and Stage 2 recovers the state to the codespace using either an inverse quantum Schur transform with amplitude rebalancing or a teleportation-based protocol, with a simpler deletion-error route. The theory shows that a PI code of distance $d$ can correct all errors of Kraus weight up to $t$ when $d\ge 2t+1$, and offers near-term, hardware-efficient decoding using geometric phase gates that reduces reliance on individual qubit addressing. The framework also extends to bosonic-mode-assisted QEC, enabling $J^2$ and modular $J^z$ measurements, state synthesis in the Dicke subspace, and teleportation-based recovery, providing a practical path toward implementing QEC in platforms where permutation symmetry and collective operations are natural.

Abstract

We present for the first time a general theory of error correction for permutation invariant (PI) codes. Using representation theory of the symmetric group we construct efficient algorithms that can correct any correctible error on any PI code. These algorithms involve measurements of total angular momentum, quantum Schur transforms or logical state teleportations, and geometric phase gates. For erasure errors, or more generally deletion errors, on certain PI codes, we give a simpler quantum error correction algorithm.

A theory of quantum error correction for permutation-invariant codes

TL;DR

This work develops a general theory of quantum error correction for permutation-invariant (PI) codes by exploiting the representation theory of the symmetric group. It presents a two-stage decoding framework: Stage 1 projects noisy states onto irreducible representations via total angular momentum measurements to identify a standard Young tableau, and Stage 2 recovers the state to the codespace using either an inverse quantum Schur transform with amplitude rebalancing or a teleportation-based protocol, with a simpler deletion-error route. The theory shows that a PI code of distance can correct all errors of Kraus weight up to when , and offers near-term, hardware-efficient decoding using geometric phase gates that reduces reliance on individual qubit addressing. The framework also extends to bosonic-mode-assisted QEC, enabling and modular measurements, state synthesis in the Dicke subspace, and teleportation-based recovery, providing a practical path toward implementing QEC in platforms where permutation symmetry and collective operations are natural.

Abstract

We present for the first time a general theory of error correction for permutation invariant (PI) codes. Using representation theory of the symmetric group we construct efficient algorithms that can correct any correctible error on any PI code. These algorithms involve measurements of total angular momentum, quantum Schur transforms or logical state teleportations, and geometric phase gates. For erasure errors, or more generally deletion errors, on certain PI codes, we give a simpler quantum error correction algorithm.
Paper Structure (18 sections, 1 theorem, 57 equations, 6 figures)

This paper contains 18 sections, 1 theorem, 57 equations, 6 figures.

Table of Contents

  1. Introduction
  2. PI codes
  3. Correcting t single qubit errors
  4. Young diagrams and Young tableau
  5. Symmetrizing lemma
  6. Syndrome extraction by measuring total angular momentum
  7. Amplitude rebalancing
  8. Further projections and recovery
  9. Teleportation based QEC
  10. Simpler correction for deletion errors
  11. Bosonic mode assisted QEC
  12. Measurement of J2
  13. Modular measurement of Jz.
  14. State preparation of PI codes
  15. Operations for teleportation
  16. ...and 3 more sections

Key Result

Lemma 1

Let $\mathcal{C}$ be any $N$-qubit PI code of distance $d$. Let $\mathcal{N}$ be any quantum channel with Kraus operators $K$ of weight at most $t$. Then if $d \ge 2t+1$, the channels $\mathcal{N}$ and $\mathcal{S} \circ \mathcal{N}$ are both correctible with respect to $\mathcal{C}$.

Figures (6)

  • Figure 1: A gnu state. Illustration of a gnu state $|+_{g,n,u,s}\rangle =(|0_{g,n,u,s}\rangle + |1_{g,n,u,s}\rangle)/\sqrt 2$ where $g=21$, $n=2\lfloor g/2\rfloor+1$, $u=1+1/n$, and $s = g$ so that the number of qubits is $N=gn+2g=483$. Here, the horizontal axis depicts the weights $w$ of the Dicke states, and the vertical axis depicts the values of amplitude $\langle D^{gnu+s}_w|+_{g,n,u,s}\rangle$. The colors red and blue correspond to the Dicke state amplitudes of $|0_{g,n,u,s}\rangle$ and $|1_{g,n,u,s}\rangle$ respectively, and which are related by a global bit flip.
  • Figure 2: Measurement of the total angular momentum of nested consecutive sets of qubits. This measurement collapses the state onto an SYT. When total spin $j_k$ increases, we add a numbered box to the first row. Otherwise, $j_k$ decreases, and we add a numbered box to the second row. The 'plus' and 'minus' signs depicted when mapped to '0' and '1' respectively, give a binary string that is the Young-Yamanouchi representation of the SYT.
  • Figure 3: Decoupling. (Left) A Bratelli diagram illustrating how a given SYT $\textsf{T}$, here the one appearing in Fig. \ref{['fig:cgs']}, can be visualized as a sequential process of addition of elementary spin$-1/2$ qubits. The kink in the middle represents an event where coupling spin number $3$ to the joint $j=1$ space of the first two reduced the total spin angular momentum to $j=1/2$. (Right) An inverse Clebsch-Gordan transformation decouples the spin 5/2 state into three spin states, one with spin 1, one with spin 1/2 and another with spin 2.
  • Figure 4: QEC after obtaining the SYT syndrome. An error occurs on a seven-qubit PI code PoR04kubischta2023not, and a SYT syndrome measurement obtains the SYT depicted on the upper left. This algorithm proceeds using this SYT syndrome information. The classical input of the left side of the circuit depicts the Young-Yamanouchi representation ${\bf x}$ of this SYT. We permute the qubits so that the qubit that correspond to the component in ${\bf x}$ equal to 1 is rearranged as the last qubit. We apply Subroutine A, wherein we apply an inverse Clebsch-Gordan transformations followed by a computational basis measurement of the decoupled qubit and subsequently, a measurement of total angular momentum. We repeat Subroutine A until we obtain a projection onto maximal angular momentum space. Each pair of red lines depicts one bit of classical information, and each black depicts one qubit of quantum information. After successful projection onto the symmetric space, we perform the amplitude rebalancing algorithm to rebalance the deformation of the amplitudes of the logical codewords that happened during the course of performing Subroutine A.
  • Figure 5: Teleportation. Schematic for the teleportation of a state $\rho$ that is a T-code to a state $\rho_L$ in a shifted gnu code, with the help of (1) an ancilla logical state $|+_L\rangle$ in the shifted gnu code, (2) a logical controlled-NOT gate $\rm{C}_{\textsf{A}}{\rm X}_{\textsf{B}}$ between code A and code B, (3) a modulo measurement, and (4) a conditional $X_{\rm schur}$ operation depending on the logical-$Z$ basis measurement outcome.
  • ...and 1 more figures

Theorems & Definitions (2)

  • Lemma 1: Symmetrizing lemma
  • proof