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Intrinsic Quantum Codes: One Code To Rule Them All

Eric Kubischta, Ian Teixeira

TL;DR

The paper reframes quantum error correction in intrinsic, representation-theoretic terms: a code is a subspace $\mathcal{I}$ of a representation $V$ of a symmetry group $G$, with errors organized by isotypic sectors of $\mathscr{L}(V)$. The central Schur-Bootstrap result shows that any $G$-equivariant embedding into a physical space $\mathcal{H}$ preserves the Knill-Laflamme conditions and yields a covariant extrinsic code $\mathcal{C}=\mathcal{U}(\mathcal{I})$ with the same error protection, plus transversality for the induced logical group $\mathsf{G}$. Defining intrinsic distance $d$ via the smallest tensor power in which irreps appear, the authors prove that a code of distance $d$ in the intrinsic setting yields a corresponding extrinsic code of distance $d$ under $G$-equivariant realizations. They illustrate this with a minimal SU(2) intrinsic code $\{\{ {\mathbf{5}^{\prime}} ,2,2\}\}$, which generates a family of extrinsic codes across qubit, qudit, bosonic, and rotor platforms all covariant under $\mathsf{G}=S_3$, and reveals moduli spaces like $\mathbb{C}P^{m-1}$ and a depth notion that refines conventional distance. This framework promises hardware-agnostic design of covariant quantum codes and systematic discovery of new codes across diverse physical implementations.

Abstract

Drawing on the analogy between intrinsic and extrinsic geometry, we define an intrinsic quantum code as a subspace of a group representation. We show how this abstract code rules over any physical (extrinsic) realization by dictating its error-correction properties. By finding one intrinsic code, we simultaneously uncover properties of all its realizations. This approach brings diverse codes into a unified framework and binds them through a single underlying symmetry.

Intrinsic Quantum Codes: One Code To Rule Them All

TL;DR

The paper reframes quantum error correction in intrinsic, representation-theoretic terms: a code is a subspace of a representation of a symmetry group , with errors organized by isotypic sectors of . The central Schur-Bootstrap result shows that any -equivariant embedding into a physical space preserves the Knill-Laflamme conditions and yields a covariant extrinsic code with the same error protection, plus transversality for the induced logical group . Defining intrinsic distance via the smallest tensor power in which irreps appear, the authors prove that a code of distance in the intrinsic setting yields a corresponding extrinsic code of distance under -equivariant realizations. They illustrate this with a minimal SU(2) intrinsic code , which generates a family of extrinsic codes across qubit, qudit, bosonic, and rotor platforms all covariant under , and reveals moduli spaces like and a depth notion that refines conventional distance. This framework promises hardware-agnostic design of covariant quantum codes and systematic discovery of new codes across diverse physical implementations.

Abstract

Drawing on the analogy between intrinsic and extrinsic geometry, we define an intrinsic quantum code as a subspace of a group representation. We show how this abstract code rules over any physical (extrinsic) realization by dictating its error-correction properties. By finding one intrinsic code, we simultaneously uncover properties of all its realizations. This approach brings diverse codes into a unified framework and binds them through a single underlying symmetry.

Paper Structure

This paper contains 13 sections, 2 theorems, 43 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main Result
  3. Application
  4. Conclusion
  5. Parameterizing 6 qubit realizations of the $\{\mkern-4mu\{ {\mathbf{5}^{ \prime} } ,2,2 \}\mkern-4mu\}$ code
  6. Perturbative Order and Depth
  7. Bosonic Codes (Detailed Analysis)
  8. Rigid Rotor Codes (Detailed Analysis)
  9. Molecular Codes (Detailed Analysis)
  10. Buckyball Codes
  11. Landau Level Codes (Detailed Analysis )
  12. Intrinsic $\{\mkern-4mu\{ {\mathbf{14}^{ \prime} } ,2,3 \}\mkern-4mu\}$ Code
  13. Related Work

Key Result

Theorem 1

If $\mathcal{I} \subset V$ satisfies the Knill-Laflamme condition for all errors in some isotypic subspace of $\mathscr{L}(V)$, then any physical realization $\mathcal{U}(\mathcal{I})=\mathcal{C} \subset \mathcal{H}$ will satisfy the Knill-Laflamme condition for all errors in the corresponding isoty

Figures (1)

  • Figure 1: "One Code to rule them all, One Code to find them, One Code to bring them all, and in the Symmetry bind them," adapted from J.R.R. Tolkien, The Fellowship of the Ring (1954) JRR. Figure generated with ChatGPT-5 ChatGPT.

Theorems & Definitions (2)

  • Theorem : Schur-Bootstrap
  • Corollary : Multi-qudit Schur--Bootstrap