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arXiv:2511.13721 · Physical review. D/Physical review. D.

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

Xiaojun Yao

quant-phhep-lathep-phhep-thSubmitted Nov 17, 2025• Announced Nov 18, 2025• 1 citation• Revised Jun 11, 2026 (v2)
v1 · Nov 17, 2025v2 · Jun 11, 2026
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Abstract

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.

Journal reference
Phys. Rev. D 113, 114512 (2026)
DOI
10.1103/5yg7-6mph
Comments
12 pages, 10 figures; v2: added more details regarding code II

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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 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.
Arbitrary-Distance Quantum Error Correction with Gauss's Law for Z2\mathbb Z_2Z2​ Lattice Gauge Theory
It has previously been shown by Rajput, Roggero, and Wiebe that Z2\mathbb Z_2Z2​ Gauss's law constraints can be used to build efficient quantum error-correcting codes (QECCs) that are robust against arbitrary single-qubit errors. In this work, we generalize the construction to be robust against arbitrary ttt-qubit errors, where ttt is any positive integer. This includes a derivation of the optimal Gauss's law code within the considered family by minimizing the number of physical qubits required for a given code distance. Finally, we compare our codes against other efficient QECCs on metrics such as the number of physical qubits, the locality of the encoded Hamiltonian, and the logical error rate in the code capacity setting. Compared to using a domain-agnostic code for every lattice degree of freedom, we find that the Gauss's law code primarily excels at reducing the locality of the encoded Hamiltonian. Moreover, the physical qubit overhead is also reduced for t≤3t \le 3t≤3 (distance d≤7d \le 7d≤7).
We present a protocol for actively suppressing Gauss law violations in quantum simulations of SU(2) lattice gauge theory. Mid-circuit measurements extract a syndrome (J,M,N)(J,M,N)(J,M,N) characterising the gauge-violation sector at each vertex by resolving both the total angular momentum and the magnetic quantum numbers of the violation through a group quantum Fourier transform. A syndrome-conditional recovery operation maps the state back to the gauge-invariant subspace, and the procedure is iterated as a sweep over vertices in a process we call gauge cooling. We prove that every single-qubit Pauli error at a coordination-four vertex with four spin-1/21/21/2 edges is detected by the gauge syndrome, and we show that the Knill--Laflamme conditions fail for syndrome-based recovery alone whenever the singlet multiplicity exceeds one. The residual physical-subspace errors carry a structured Pauli decomposition with vanishing YYY component, which suggests compatibility with concatenation by a CSS stabilizer code. We demonstrate the protocol on a single-plaquette simulation of the Kogut--Susskind Hamiltonian truncated to the spin-1/21/21/2 representation under depolarising and amplitude damping noise, and we observe that gauge cooling restores approximate gauge invariance and improves fidelity at noise rates representative of current superconducting hardware.
Abstract Lattice gauge theory should be able to address significant new scientific questions when implemented on quantum computers. In practice, error-mitigation techniques have already allowed encouraging progress on small lattices. In this work we focus on a truncated version of SU(2) gauge theory, which is a familiar non-Abelian step toward quantum chromodynamics. First, we demonstrate effective error mitigation for imaginary time evolution on a lattice having two square plaquettes, obtaining the ground state using an IBM quantum computer and observing that this would have been impossible without error mitigation. Then we propose the triamond lattice as an expedient approach to lattice gauge theories in three spatial dimensions and we derive the Hamiltonian. Finally, error-mitigated imaginary time evolution is applied to the three-dimensional triamond unit cell, and its ground state is obtained from an IBM quantum computer. Future work will want to relax the truncation on the gauge fields, and the triamond lattice is increasingly valuable for such studies.
Gauss's law-based quantum error correction (GLQEC) offers a promising approach to reducing qubit overhead in lattice gauge theory simulations by leveraging built-in symmetries. For applications of GLQEC to 1+1D lattice quantum electrodynamics (QED), we identify two significant trade-offs. First, we prove via dimension-counting arguments that GLQEC requires periodic electric fields, thereby constraining the design space for lattice QED simulations. Second, we numerically compare GLQEC with a universal quantum error correction (UQEC) code, specifically the d=3d=3d=3 bitflip repetition code, and find that while GLQEC can achieve lower logical error rates in single-round error correction, it exhibits faster decoherence to the steady-state mixed ensemble under multiple rounds. The mixing speed penalty is manifest in observables of interest for both memory experiments and Hamiltonian evolution. We identify a mixing speed threshold, pth=0.277(2)p_{th}=0.277(2)pth​=0.277(2), above which using GLQEC exhibits even faster decoherence than without error correction. Our results highlight fundamental limitations of symmetry-based error correction schemes and inform corresponding constraints on formulations of lattice gauge theories compatible with error-robust quantum simulation techniques.