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arXiv:2412.15317

A correspondence between quantum error correcting codes and quantum reference frames

Sylvain Carrozza, Aidan Chatwin-Davies, Philipp A. Höhn, Fabio M. Mele

Published Dec 19, 2024• 0 citations
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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.

DOI
10.48550/arxiv.2412.15317

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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.
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Gauge theories and quantum error-correcting codes share the same underlying structure: both use constraints to identify a specific subspace of the full Hilbert space. In quantum error correction, these constraints are known as stabilizers, while in gauge theories they correspond to Gauss law. In this work, we consider a family of discrete Abelian lattice gauge theories described by a ZN\mathbb{Z}_{N}ZN​ gauge group with NNN an arbitrary power of two. In this setting, we find a set of stabilizers for the gauge-invariant subspace which is an alternative to the Gauss operators, and we call them binary Gauss stabilizers. We use this alternative stabilizer group to build practical error-correcting codes exploiting the gauge symmetries of the system without the addition of extra qubits. The applications of our finding are not limited to error correction though. We also provide a new strategy of gauge fixing to remove the redundancies based on our alternative stabilizer, which might provide advantages with respect to already-existing approaches such as the axial gauge. Our results provide new tools to study lattice gauge theories and their quantum simulation, and opens directions for future work at the interface of lattice gauge theory and quantum information.
Dynamical stabilizer codes (DSCs) have recently emerged as a powerful generalization of static stabilizer codes for quantum error correction, replacing a fixed stabilizer group with a sequence of non-commuting measurements. This dynamical structure unlocks new possibilities for fault tolerance but also introduces new challenges, as errors must now be tracked across both space and time. In this work, we provide a physical and topological understanding of DSCs by establishing a correspondence between qudit Pauli measurements and non-invertible symmetries in 4+1-dimensional 2-form gauge theories. Sequences of measurements in a DSC are mapped to a fusion of the operators implementing these non-invertible symmetries. We show that the error detectors of a DSC correspond to endable surface operators in the gauge theory, whose endpoints define line operators, and that detectable errors are precisely those surface operators that braid non-trivially with these lines. Finally, we demonstrate how this framework naturally recovers the spacetime stabilizer code associated with a DSC.
We generalize the notion of quantum reference frames (QRFs) to cases where the frame does not necessarily correspond to a tensor factor subsystem, but to a covariant quantum instrument. This unlocks a variety of physical applications: "frames of labeling" for indistinguishable particles, suggesting explanations for the symmetrization postulate and the absence of parastatistics, and yielding a transparent description of the entanglement of bosons and fermions; and relational clocks reproducing the Schrödinger equation exactly even when all subsystems are interacting or when there are frequency superselection sectors. Our work generalizes the perspective-neutral approach to QRFs pioneered by Höhn and co-authors, which we reconstruct from a simple operational scenario. We give a resource-theoretic grounding of this framework, and show how the notion of completely covariant operations explains the relevance of the charge-zero sector and the pure-state transformation behavior across perspectives. This also suggests operational clarifications of some aspects of constraint quantization, e.g. of the meaning of constraint equations such as C∣ψ⟩=0C|ψ\rangle=0C∣ψ⟩=0. Some of our results, such as our generalization of relationalization maps to instruments, apply more broadly to other QRF frameworks too, and they contribute to bridging the gap between operational quantum information theory and the internal QRF research program.
Employing internal quantum systems as reference frames is a crucial concept in quantum gravity, gauge theories, and quantum foundations whenever external relata are unavailable. In this work, we give a comprehensive and self-contained treatment of such quantum reference frames (QRFs) for the case when the underlying configuration space is a finite Abelian group, significantly extending our previous work [M. Krumm, P. A. Höhn, and M. P. Müller, Quantum 5, 530 (2021)]. The simplicity of this setup admits a fully rigorous quantum information–theoretic analysis, while maintaining sufficient structure for exploring many of the conceptual and structural questions also pertinent to more complicated setups. We exploit this to derive several important structures of constraint quantization with quantum information–theoretic methods and to reveal the relation between different approaches to QRF covariance. In particular, we characterize the “physical Hilbert space”—the arena of the “perspective-neutral” approach—as the maximal subspace that admits frame-independent descriptions of purifications of states. We then demonstrate the kinematical equivalence and, surprising, dynamical inequivalence of the “perspective-neutral” and the “alignability” approach to QRFs. While the former admits unitaries generating transitions between arbitrary subsystem relations, the latter, remarkably, admits no such dynamics when requiring symmetry-preservation. We illustrate these findings by example of interacting discrete particles, including how dynamics can be described “relative to one of the subystems.”