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.
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.
Recent advances in quantum error-correction (QEC) have shown that it is often beneficial to understand fault-tolerance as a dynamical process, a circuit with redundant measurements that help correct errors, rather than as a static code equipped with a syndrome extraction circuit. Spacetime codes have emerged as a natural framework to understand error correction at the circuit level while leveraging the traditional QEC toolbox. Here, we introduce a framework based on chain complexes and chain maps to model spacetime codes and transformations between them. We show that stabilizer codes, quantum circuits, and decoding problems can all be described using chain complexes, and that the equivalence of two spacetime codes can be characterized by specific maps between chain complexes, the fault-tolerant maps, that preserve the number of encoded qubits, fault distance, and minimum-weight decoding problem. As an application of this framework, we extend the foliated cluster state construction from stabilizer codes to any spacetime code, showing that any Clifford circuit can be transformed into a measurement-based protocol with the same fault-tolerant properties. To this protocol, we associate a chain complex which encodes the underlying decoding problem, generalizing previous cluster state complex constructions. Our method enables the construction of cluster states from non-CSS, subsystem, and Floquet codes, as well as from logical Clifford operations on a given code.
Typical stabilizer codes aim to solve the general problem of fault-tolerance without regard for the structure of a specific system. By incorporating a broader representation-theoretic perspective, we provide a generalized framework that allows the code designer to take this structure into account. For any representation of a finite group, we produce a quantum code with a code space invariant under the group action, providing passive error mitigation against errors belonging to the image of the representation. Furthermore, errors outside this scope are detected and diagnosed by performing a projective measurement onto the isotypic components corresponding to irreducible representations of the chosen group, effectively generalizing syndrome extraction to symmetry-resolved quantum measurements. We show that all stabilizer codes are a special case of this construction, including qudit stabilizer codes, and show that there is a natural one logical qubit code associated to the dihedral group. Thus we provide a unifying framework for existing codes while simultaneously facilitating symmetry-aware codes tailored to specific systems.
We develop the procedures of gauging and ungauging, reveal their operational meaning and propose their generalization in a systematic manner within the framework of quantum error-correcting codes. We demonstrate with an example of the subsystem Bacon-Shor code that the ungauging procedure can result in models with unusual symmetry operators constrained to live on lower-dimensional structures. We apply our formalism to the three-dimensional gauge color code (GCC) and show that its codeword space is equivalent to the Hilbert space of six copies of Z2 lattice gauge theory with 1-form symmetries. We find that three different stabilizer Hamiltonians associated with the GCC correspond to distinct thermal symmetry-protected topological (SPT) phases in the presence of the stabilizer symmetries of the GCC. One of the considered Hamiltonians describes the Raussendorf-Bravyi-Harrington model, which is universal for measurement-based quantum computation at non-zero temperature. We also propose a general procedure of creating D-dimensional SPT Hamiltonians from (D+1)-dimensional CSS stabilizer Hamiltonians by exploiting a relation between gapped domain walls and transversal logical gates. As a result, we find an explicit two-dimensional realization of a non-trivial fracton SPT phase protected by fractal-like symmetries.
In recent years, the spacetime code has arisen as a candidate for a unifying view of fault tolerance in space and time. On the other hand, the recent study of dynamical phases has increasingly turned its attention to fault tolerance as a notion of a dynamically stable process. In this work, I explore one pathway between the two, achieved by gauging the spacetime code. This gives rise to a lattice gauge theory that inherits the elements of fault tolerance associated with a circuit, with Gauss laws corresponding to equivalence relations between configurations of spacetime errors and Wilson loops corresponding to detectors. The obtained gauge theory finds a surprisingly wide array of applications, from quantum error correction to condensed matter physics, and even learning theory: (1) It contains in its description foliated computation, and hence gives rise to one version of a gauge theory for measurement-based quantum computation. (2) For a class of topologically ordered mixed states, it gives us a gauge-theoretic language to describe the classical memory associated with the state. (3) The gauge-invariant observables of the theory which describe detectors also coincide with the learnable degrees of freedom of circuit Pauli noise.
We ask what is the general framework for a quantum error correcting code that is defined by a sequence of measurements. Recently, there has been much interest in Floquet codes and space-time codes. In this work, we define and study the distance of a dynamical code. This is a subtle concept and difficult to determine: At any given time, the system will be in a subspace which forms a quantum error-correcting code with a given distance, but the full error correction capability of that code may not be available due to the schedule of measurements associated with the code. We address this challenge by developing an algorithm that tracks information we have learned about the error syndromes through the protocol and put that together to determine the distance of a dynamical code, in a non-fault-tolerant context. We use the tools developed for the algorithm to analyze the initialization and masking properties of a generic Floquet code. Further, we look at properties of dynamical codes under the constraint of geometric locality with a view to understand whether the fundamental limitations on logical gates and code parameters imposed by geometric locality for traditional codes can be surpassed in the dynamical paradigm. We find that codes with a limited number of long range connectivity will not allow non-Clifford gates to be implemented with finite depth circuits in the 2D setting.
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.