Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2602.20661

Qudit stabiliser codes for $\mathbb{Z}_N$ lattice gauge theories with matter

Luca Spagnoli, Alessandro Roggero, Nathan Wiebe

quant-phSubmitted Feb 24, 2026• Announced Feb 25, 2026• 0 citations
arXivHTMLPDF

Abstract

In this work we extend the connection between Quantum Error Correction (QEC) and Lattice Gauge Theories (LGTs) by showing that a $\mathbb{Z}_N$ gauge theory with prime dimension $N$ coupled to dynamical matter can be expressed as a qudit stabilizer code. Using the stabilizer formalism we show how to formulate an exact mapping of the encoded $\mathbb{Z}_N$ gauge theory onto two different bosonic models, uncovering a logical duality generated by error correction itself. From this perspective, quantum error correction provides a unifying language to expose dual descriptions of lattice gauge theories. In addition, we generalize earlier $\mathbb{Z}_2$ constructions on qubits to $\mathbb{Z}_N$ on $N$-level qudits and demonstrate how universal fault-tolerant gates can be implemented via state injection between compatible qudit codes.

Related papers

Gauss law codes and vacuum codes from lattice gauge theories

Javier P. Lacambra, Aidan Chatwin-Davies, Masazumi Honda +1 more

We develop a comprehensive framework for constructing quantum error correcting codes (QECCs) from Abelian lattice gauge theories (LGTs) using quantum reference frames (QRFs) as a unifying formalism. We consider LGTs with arbitrary compact Abelian gauge groups supported on lattices in arbitrary numbers of spatial dimensions, and we work with both pure gauge theories and theories with couplings to bosonic and fermionic matter. The codes that we construct fall into two classes: First, Gauss law codes identify the code subspace with the full gauge-invariant sector of the theory. In models with matter coupled to gauge fields, these codes inherit a natural subsystem structure in which gauge-invariant Wilson loops and dressed matter excitations factorize the code space. Second, vacuum codes restrict the code subspace to the matter vacuum sector within the gauge-invariant subspace, yielding codes where errors correspond to gauge-invariant charge excitations rather than to violations of the Gauss law. Despite their distinct setup, we show that when the gauge group is finite, vacuum codes are unitarily equivalent to pure gauge theory Gauss law codes, and that when the group is continuous, this is only true upon a charge coarse-graining of the vacuum code. In all cases, QRFs provide a systematic apparatus for fully characterizing the codes' algebraic structures and correctable error sets. For clarity, we illustrate our general results in $\mathbb{Z}_2$-gauge theory, as well as in scalar and fermionic QED. These findings offer fundamental insights into the parallelism between quantum error correction and gauge theory and point toward practical advantages for simulating LGTs on noisy quantum devices.

quant-pharXiv:2604.06087•Apr 8, 2026•0 citations

Fault-tolerant simulation of Lattice Gauge Theories with gauge covariant codes

Luca Spagnoli, Alessandro Roggero, Nathan Wiebe

We show in this paper that a strong and easy connection exists between quantum error correction and Lattice Gauge Theories (LGT) by using the Gauge symmetry to construct an efficient error-correcting code for Abelian Z 2 LGTs. We identify the logical operations on this gauge covariant code and show that the corresponding Hamiltonian can be expressed in terms of these logical operations while preserving the locality of the interactions. Furthermore, we demonstrate that these substitutions actually give a new way of writing the LGT as an equivalent hardcore boson model. Finally we demonstrate a method to perform fault-tolerant time evolution of the Hamiltonian within the gauge covariant code using both product formulas and qubitization approaches. This opens up the possibility of inexpensive end to end dynamical simulations that save physical qubits by blurring the lines between simulation algorithms and quantum error correcting codes.

arXiv:2405.19293•Quantum•Jan 16, 2026•6 citations

Quantum Error Correction as Lattice Gauge Theory on the D4 Lattice — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: Quantum error correction (QEC) stabilizer codes, particularly CSS codes, map directly onto lattice gauge theory (LGT) with the B2 root system (D4 lattice) as the underlying geometric scaffold; gauge symmetry is not redundancy but an information-theoretic resource for fault tolerance. | MATH: Stabilizer group \( S \subset \mathcal{P}_n \) (Pauli group), CSS code condition \( H_X H_Z^T = 0 \) (mod 2); B2 root system: \( \{\pm e_1 \pm e_2, \pm e_1, \pm e_2\} \) → D4 lattice (checkerboard, kissing number 24); plaquette operators \( B_p = \prod_{\ell \in \partial p} \sigma_\ell^z \), vertex operators \( A_v = \prod_{\ell \ni v} \sigma_\ell^x \); toric code: \( |\psi\rangle = \sum_{g} |g\rangle \) with \( A_v|\psi\rangle = B_p|\psi\rangle = |\psi\rangle \). | CONNECTION: B2/D4 lattice is the root system of \( SO(8) \) — its Weyl group order 192, with Coxeter number 6. The ratio of distances between B2 root lengths (long:short = \( \sqrt{2}:1 \)) yields \( 1/\sqrt{2} \approx 0.707 \) Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

•Oct 3, 2026•0 citations

Gauging the Spacetime Code

Gideon Lee

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.

quant-pharXiv:2606.05664•Jun 5, 2026•0 citations

Quantum error correction with gauge symmetries

Abhishek Rajput, Alessandro Roggero, Nathan Wiebe

Abstract Quantum simulations of lattice gauge theories (LGTs) are often formulated on an enlarged Hilbert space containing both physical and unphysical sectors in order to retain a local Hamiltonian. We provide simple fault-tolerant procedures that exploit such redundancy by combining a phase flip error correction code with the Gauss’ law constraint to correct one-qubit errors for a $${{\mathbb{Z}}}_{2}$$ Z 2 or truncated U(1) LGT in 1+1 and 2+1 dimensions with a link flux cutoff of 1. Unlike existing work on detecting violations of Gauss’ law, our circuits are fault tolerant and the overall error correction scheme outperforms a naïve application of the [5, 1, 3] code. The constructions outlined can be extended to LGT systems with larger cutoffs and may be of use in understanding how to hybridize error correction and quantum simulation for LGTs in higher space-time dimensions and with different symmetry groups.

arXiv:2112.05186•npj Quantum Information•Apr 25, 2023•0 citations

Quantum Error Correction Codes for Truncated SU(2) Lattice Gauge Theories

Xiaojun Yao

We construct two quantum error correction codes for pure SU(2) lattice gauge theory in the electric basis truncated at the electric flux $j_{\rm max}=1/2$, which are applicable on quasi-1D plaquette chains, 2D honeycomb and 3D triamond and hyperhoneycomb lattices. The first code converts Gauss's law at each vertex into a stabilizer while the second only uses half of the vertices and is locally the carbon code. Both codes are able to correct single-qubit errors. The electric and magnetic terms in the SU(2) Hamiltonian are expressed in terms of logical gates in both codes. The logical-gate Hamiltonian in the first code exactly matches the spin Hamiltonian for gauge singlet states found in previous work.

quant-pharXiv:2511.13721•Physical review. D/Physical review. D.•Nov 18, 2025•1 citation
Qudit stabiliser codes for ZN\mathbb{Z}_NZN​ lattice gauge theories with matter
In this work we extend the connection between Quantum Error Correction (QEC) and Lattice Gauge Theories (LGTs) by showing that a ZN\mathbb{Z}_NZN​ gauge theory with prime dimension NNN coupled to dynamical matter can be expressed as a qudit stabilizer code. Using the stabilizer formalism we show how to formulate an exact mapping of the encoded ZN\mathbb{Z}_NZN​ gauge theory onto two different bosonic models, uncovering a logical duality generated by error correction itself. From this perspective, quantum error correction provides a unifying language to expose dual descriptions of lattice gauge theories. In addition, we generalize earlier Z2\mathbb{Z}_2Z2​ constructions on qubits to ZN\mathbb{Z}_NZN​ on NNN-level qudits and demonstrate how universal fault-tolerant gates can be implemented via state injection between compatible qudit codes.
We develop a comprehensive framework for constructing quantum error correcting codes (QECCs) from Abelian lattice gauge theories (LGTs) using quantum reference frames (QRFs) as a unifying formalism. We consider LGTs with arbitrary compact Abelian gauge groups supported on lattices in arbitrary numbers of spatial dimensions, and we work with both pure gauge theories and theories with couplings to bosonic and fermionic matter. The codes that we construct fall into two classes: First, Gauss law codes identify the code subspace with the full gauge-invariant sector of the theory. In models with matter coupled to gauge fields, these codes inherit a natural subsystem structure in which gauge-invariant Wilson loops and dressed matter excitations factorize the code space. Second, vacuum codes restrict the code subspace to the matter vacuum sector within the gauge-invariant subspace, yielding codes where errors correspond to gauge-invariant charge excitations rather than to violations of the Gauss law. Despite their distinct setup, we show that when the gauge group is finite, vacuum codes are unitarily equivalent to pure gauge theory Gauss law codes, and that when the group is continuous, this is only true upon a charge coarse-graining of the vacuum code. In all cases, QRFs provide a systematic apparatus for fully characterizing the codes' algebraic structures and correctable error sets. For clarity, we illustrate our general results in Z2\mathbb{Z}_2Z2​-gauge theory, as well as in scalar and fermionic QED. These findings offer fundamental insights into the parallelism between quantum error correction and gauge theory and point toward practical advantages for simulating LGTs on noisy quantum devices.
We show in this paper that a strong and easy connection exists between quantum error correction and Lattice Gauge Theories (LGT) by using the Gauge symmetry to construct an efficient error-correcting code for Abelian Z 2 LGTs. We identify the logical operations on this gauge covariant code and show that the corresponding Hamiltonian can be expressed in terms of these logical operations while preserving the locality of the interactions. Furthermore, we demonstrate that these substitutions actually give a new way of writing the LGT as an equivalent hardcore boson model. Finally we demonstrate a method to perform fault-tolerant time evolution of the Hamiltonian within the gauge covariant code using both product formulas and qubitization approaches. This opens up the possibility of inexpensive end to end dynamical simulations that save physical qubits by blurring the lines between simulation algorithms and quantum error correcting codes.
FINDING: Quantum error correction (QEC) stabilizer codes, particularly CSS codes, map directly onto lattice gauge theory (LGT) with the B2 root system (D4 lattice) as the underlying geometric scaffold; gauge symmetry is not redundancy but an information-theoretic resource for fault tolerance. | MATH: Stabilizer group S⊂Pn S \subset \mathcal{P}_n S⊂Pn​ (Pauli group), CSS code condition HXHZT=0 H_X H_Z^T = 0 HX​HZT​=0 (mod 2); B2 root system: {±e1±e2,±e1,±e2} \{\pm e_1 \pm e_2, \pm e_1, \pm e_2\} {±e1​±e2​,±e1​,±e2​} → D4 lattice (checkerboard, kissing number 24); plaquette operators Bp=∏ℓ∈∂pσℓz B_p = \prod_{\ell \in \partial p} \sigma_\ell^z Bp​=∏ℓ∈∂p​σℓz​, vertex operators Av=∏ℓ∋vσℓx A_v = \prod_{\ell \ni v} \sigma_\ell^x Av​=∏ℓ∋v​σℓx​; toric code: ∣ψ⟩=∑g∣g⟩ |\psi\rangle = \sum_{g} |g\rangle ∣ψ⟩=∑g​∣g⟩ with Av∣ψ⟩=Bp∣ψ⟩=∣ψ⟩ A_v|\psi\rangle = B_p|\psi\rangle = |\psi\rangle Av​∣ψ⟩=Bp​∣ψ⟩=∣ψ⟩. | CONNECTION: B2/D4 lattice is the root system of SO(8) SO(8) SO(8) — its Weyl group order 192, with Coxeter number 6. The ratio of distances between B2 root lengths (long:short = 2:1 \sqrt{2}:1 2​:1) yields 1/2≈0.707 1/\sqrt{2} \approx 0.707 1/2​≈0.707 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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.
Abstract Quantum simulations of lattice gauge theories (LGTs) are often formulated on an enlarged Hilbert space containing both physical and unphysical sectors in order to retain a local Hamiltonian. We provide simple fault-tolerant procedures that exploit such redundancy by combining a phase flip error correction code with the Gauss’ law constraint to correct one-qubit errors for a Z2{{\mathbb{Z}}}_{2}Z2​ Z 2 or truncated U(1) LGT in 1+1 and 2+1 dimensions with a link flux cutoff of 1. Unlike existing work on detecting violations of Gauss’ law, our circuits are fault tolerant and the overall error correction scheme outperforms a naïve application of the [5, 1, 3] code. The constructions outlined can be extended to LGT systems with larger cutoffs and may be of use in understanding how to hybridize error correction and quantum simulation for LGTs in higher space-time dimensions and with different symmetry groups.
We construct two quantum error correction codes for pure SU(2) lattice gauge theory in the electric basis truncated at the electric flux jmax=1/2j_{\rm max}=1/2jmax​=1/2, which are applicable on quasi-1D plaquette chains, 2D honeycomb and 3D triamond and hyperhoneycomb lattices. The first code converts Gauss's law at each vertex into a stabilizer while the second only uses half of the vertices and is locally the carbon code. Both codes are able to correct single-qubit errors. The electric and magnetic terms in the SU(2) Hamiltonian are expressed in terms of logical gates in both codes. The logical-gate Hamiltonian in the first code exactly matches the spin Hamiltonian for gauge singlet states found in previous work.