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A Bravyi-König theorem for Floquet codes generated by locally conjugate instantaneous stabiliser groups

Jelena Mackeprang, Jonas Helsen

TL;DR

The paper addresses whether the Bravyi-König no-go constraint on fault-tolerant dynamics extends to Floquet codes built from locally conjugate instantaneous stabiliser groups. It develops a formal framework for Floquet codes, introducing a Floquet transition operator $K_{ ext{A,B}}$ and a constant-depth unitary $V_{ ext{A,B}}$ that implement code transitions, and it defines generalized logical unitaries that need not preserve the codespace at every step. The authors prove that the BK theorem extends to code-preserving unitaries in Floquet codes and further derive a canonical form for generalized unitaries, showing BK also holds for these under locality and $ au= ext{O}(1)$ assumptions. The result clarifies fundamental limits for fault-tolerant operations in dynamical/spacetime codes and informs the design of Floquet-code-based quantum memories and processors, highlighting how measurements and local transitions constrain achievable logical gates.

Abstract

The Bravyi-König (BK) theorem is an important no-go theorem for the dynamics of topological stabiliser quantum error correcting codes. It states that any logical operation on a $D$-dimensional topological stabiliser code that can be implemented by a short-depth circuit acts on the codespace as an element of the $D$-th level of the Clifford hierarchy. In recent years, a new type of quantum error correcting codes based on Pauli stabilisers, dubbed Floquet codes, has been introduced. In Floquet codes, syndrome measurements are arranged such that they dynamically generate a codespace at each time step. Here, we show that the BK theorem holds for a definition of Floquet codes based on locally conjugate stabiliser groups. Moreover, we introduce and define a class of generalised unitaries in Floquet codes that need not preserve the codespace at each time step, but that combined with the measurements constitute a valid logical operation. We derive a canonical form of these generalised unitaries and show that the BK theorem holds for them too.

A Bravyi-König theorem for Floquet codes generated by locally conjugate instantaneous stabiliser groups

TL;DR

The paper addresses whether the Bravyi-König no-go constraint on fault-tolerant dynamics extends to Floquet codes built from locally conjugate instantaneous stabiliser groups. It develops a formal framework for Floquet codes, introducing a Floquet transition operator and a constant-depth unitary that implement code transitions, and it defines generalized logical unitaries that need not preserve the codespace at every step. The authors prove that the BK theorem extends to code-preserving unitaries in Floquet codes and further derive a canonical form for generalized unitaries, showing BK also holds for these under locality and assumptions. The result clarifies fundamental limits for fault-tolerant operations in dynamical/spacetime codes and informs the design of Floquet-code-based quantum memories and processors, highlighting how measurements and local transitions constrain achievable logical gates.

Abstract

The Bravyi-König (BK) theorem is an important no-go theorem for the dynamics of topological stabiliser quantum error correcting codes. It states that any logical operation on a -dimensional topological stabiliser code that can be implemented by a short-depth circuit acts on the codespace as an element of the -th level of the Clifford hierarchy. In recent years, a new type of quantum error correcting codes based on Pauli stabilisers, dubbed Floquet codes, has been introduced. In Floquet codes, syndrome measurements are arranged such that they dynamically generate a codespace at each time step. Here, we show that the BK theorem holds for a definition of Floquet codes based on locally conjugate stabiliser groups. Moreover, we introduce and define a class of generalised unitaries in Floquet codes that need not preserve the codespace at each time step, but that combined with the measurements constitute a valid logical operation. We derive a canonical form of these generalised unitaries and show that the BK theorem holds for them too.
Paper Structure (20 sections, 20 theorems, 104 equations)

This paper contains 20 sections, 20 theorems, 104 equations.

Table of Contents

  1. Introduction
  2. Preservation of information
  3. Reversible pairs of stabiliser groups
  4. Preservation of information
  5. Logical effect of the Floquet projections
  6. Floquet transition operator
  7. Transformation of logical operations
  8. Floquet codes defined by locally conjugate stabiliser groups
  9. The BK theorem for code-preserving unitaries
  10. Logical operations beyond code preservation
  11. Error detect-ability and self correcting errors
  12. Error detectability
  13. Self-correction of errors
  14. Preservation of logical information and logical equivalence
  15. Canonical form of generalised logical unitaries in Floquet transitions
  16. ...and 5 more sections

Key Result

Theorem 1

[Bravyi-König theorem for generalised logical unitaries (informal)] Let $\mathcal{A}_0 \rightarrow \mathcal{A}_1 \rightarrow \ldots \rightarrow \mathcal{A}_{{\tau}_1}$ be a dynamical code, where each $\mathcal{A}_t$ is a $D$-dimensional topological stabiliser code. At each time step one applies a co

Theorems & Definitions (44)

  • Definition 1: Generalised logical unitary in dynamical code (informal)
  • Theorem 1
  • Definition 2: Reversible pair/Conjugate stabiliser groups
  • Proposition 1
  • Corollary 1
  • Lemma 1
  • proof
  • Proposition 2: Shared logical operators
  • Definition 3: The transitions operator $K_{\mathcal{A},\mathcal{B}}$
  • Lemma 2
  • ...and 34 more