A correspondence between quantum error correcting codes and quantum reference frames

TL;DR

The paper establishes a foundational correspondence between quantum error correcting codes (QECCs) and gauge theories through quantum reference frames (QRFs) within the perspective-neutral framework. It develops a detailed dictionary linking QECC constructs (code space, error sets, decodings) to QRF structures (perspective-neutral space, internal frames, dressing) and demonstrates this with Pauli stabilizer codes, notably deriving a one-to-one relation between maximal correctable error sets and tensor factorizations that separate redundant frame data from logical information. A novel error duality based on Pontryagin duality links electric Pauli errors to magnetic gauge-fixing errors, enriching the understanding of error architecture and recoverability. The work applies these ideas to surface codes, showing how lattice gauge theory structures map onto stabilizer codes and how dual representations illuminate dual excitations and frame fields. The results pave the way for new code designs tailored to gauge theories and enhance the theoretical understanding of information encoding in gauge-invariant contexts, with potential practical impact on quantum simulations of gauge theories and quantum gravity.

Abstract

In a gauge theory, a collection of kinematical degrees of freedom is used to redundantly describe a smaller amount of gauge-invariant information. In a quantum error correcting code (QECC), a collection of computational degrees of freedom that make up a device's physical layer is used to redundantly encode a smaller amount of logical information. We elaborate this parallel in terms of quantum reference frames (QRFs), which are a universal toolkit for dealing with symmetries in quantum systems and which define the gauge theory analog of encodings. The result is a precise dictionary between QECCs and QRF setups within the perspective-neutral framework for gauge systems. Concepts from QECCs like error sets and correctability translate to novel insights into the informational architecture of gauge theories. Conversely, the dictionary provides a systematic procedure for constructing symmetry-based QECCs and characterizing their error correcting properties. In this initial work, we scrutinize the dictionary between Pauli stabilizer codes and their corresponding QRF setups. We show that there is a one-to-one correspondence between maximal correctable error sets and tensor factorizations splitting system from error-generated QRF degrees of freedom. Relative to this split, errors corrupt only redundant frame data, leading to a novel characterization of correctability. When passed through the dictionary, standard Pauli errors behave as electric excitations that are dual, via Pontryagin duality, to magnetic excitations related to gauge-fixing. This gives rise to a new class of correctable errors and a systematic error duality. We illustrate our findings in surface codes, which themselves connect quantum error correction with gauge systems. Our exploratory investigations pave the way for foundational applications to gauge theories and for eventual practical applications to quantum simulation.

Figures (8)

• Figure 1: The Hilbert spaces in quantum error correction and the maps among them.
• Figure 2: The Hilbert spaces in the perspective-neutral framework for QRFs and the maps among them.
• Figure 3: Surface code on a rectangular lattice of size $L \times H$; it has $(L+1)\times(H+1)$ vertical edges and $L\times H$ horizontal ones, supporting $2LH+L+H+1$ physical qubits in total. Illustrated in red and green are plaquettes and vertex operators, as well as non-trivial string operators which capture invariant data about the one logical qubit stored in the code subspace.
• Figure 4: In the single-defect sector, the choice of reference frame is specified by a choice of path $\gamma_v$ for every vertex $v$, and a choice of dual path $\gamma_f'$ for every face $f$. An error creating a defect at location $v$ (resp. $f$) is dressed by $R_{\{v\}, \emptyset} = S^Z (\gamma_v)$ (resp. $R_{\emptyset , \{f\}} = S^X (\gamma_f')$) to form a gauge-invariant (or, equivalently, a logical) operator.
• Figure 5: Example of a spanning forest $T$ and a dual spanning forest $T'$ (we have defined $T'$ as a subset of $\mathcal{E}$, but it is here represented as a subset of dual edges to facilitate visualization). Here $T$ and $T'$ have two connected components each. For any $v \in \mathcal{V}$ (resp. $f\in \mathcal{F}$), there is a unique simple path $\gamma_v \in T$ (resp. dual path $\gamma_f' \in T'$) which connects $v$ (resp. $f$) to one of the rough (resp. smooth) boundaries. For convenience, the root-edges of $T$ and $T'$ are represented as dotted lines. Note that the edge in the top-right corner does not belong to either $T$ or $T'$.

Theorems & Definitions (87)

• Example 2.1: Three-qubit repetition code
• Example 2.2: QRFs in the three-qubit repetition code
• Lemma 4.1: Equality of the code and perspective-neutral projectors
• proof
• Lemma 4.2: Existence of a faithful representation of $G$ on a subset of physical qubits
• proof
• Lemma 4.3: Existence of a seed state for local QRFs
• proof
• Theorem 4.4: Local QRFs for stabilizer codes
• proof