We develop a mathematical framework for the quantum simulation of lattice gauge theories using gauge-invariant packaged quantum states \cite{Ma2017,Ma2025}. In this formalism, every single excitation transforms as a complete \textbf{irreducible representation (irrep)} of the local gauge group, preventing any appearance of fractional or partial \textbf{internal quantum numbers (IQNs)}. Multi-particle excitations can form nontrivial packaged entangled states that are also gauge invariant, thereby forbidding partial or fractional IQNs. In other words, all IQNs of such packaged entangled states remain inseparably entangled. This ``packaging principle'' ensures that physical states remain confined to the correct gauge sector and excludes partial charges or colors, even when multiple excitations are entangled. We illustrate this approach for $\mathrm{U}(1)$, $\mathrm{SU}(2)$, and $\mathrm{SU}(3)$ lattice gauge theories, discussing explicit constructions, Trotterized Hamiltonian evolution, and gauge-invariant measurements on a quantum simulator. We also outline how packaged states can mitigate gauge-violating errors and serve as natural building blocks for gauge-invariant coding schemes, while noting that standard quantum error correction is still required against typical local noise that respects gauge symmetry.
We develop a mathematical framework for the quantum simulation of lattice gauge theories using gauge-invariant packaged quantum states \cite{Ma2017,Ma2025}. In this formalism, every single excitation transforms as a complete \textbf{irreducible representation (irrep)} of the local gauge group, preventing any appearance of fractional or partial \textbf{internal quantum numbers (IQNs)}. Multi-particle excitations can form nontrivial packaged entangled states that are also gauge invariant, thereby forbidding partial or fractional IQNs. In other words, all IQNs of such packaged entangled states remain inseparably entangled. This ``packaging principle'' ensures that physical states remain confined to the correct gauge sector and excludes partial charges or colors, even when multiple excitations are entangled. We illustrate this approach for U(1), SU(2), and SU(3) lattice gauge theories, discussing explicit constructions, Trotterized Hamiltonian evolution, and gauge-invariant measurements on a quantum simulator. We also outline how packaged states can mitigate gauge-violating errors and serve as natural building blocks for gauge-invariant coding schemes, while noting that standard quantum error correction is still required against typical local noise that respects gauge symmetry.
Symmetry packaging is the phenomenon whereby, upon particle creation, all the internal quantum numbers (IQNs) become locked into a single irreducible representation (irrep) block of the gauge group, as required by locality and gauge invariance. The resulting packaged quantum states exhibit characteristic symmetry constraints and entanglement patterns. We develop a group-theoretic framework to describe the symmetry packaging for a variety of concrete symmetries and to classify the corresponding packaged states:(1) We prove that for any finite or compact group G, there exist G-associated packaged subspaces, in which every vector is automatically a packaged state. In particular, in multi-particle systems, any nontrivial representation of G induces inseparable packaged entanglement that locks together all IQNs.(2) We apply this framework to symmetry packaging in finite groups (cyclic group ZN, charge conjugation C, fermion parity, parity P, time reversal T, and dihedral groups), compact groups (U(1), SU(N), SU(2), and SU(3)), p-form symmetries, and hybrid symmetries. In each case, gauge invariance and superselection rules forbid the factorization of the resulting states. We illustrate how Bell-type packaged entangled states, color confinement, and hybrid gauge-invariant configurations all arise naturally. These results yield a complete classification of packaged quantum states.(3) Finally, we extend the packaging principle to incorporate full spacetime symmetry and hybrid systems of local, global, and Lorentz/Poincaré charges.Our approach unifies tools from group theory, gauge theory, and topological classification. These results may be useful for potential applications in high energy physics, quantum field theory, and quantum technologies.
We demonstrate that quantum field excitations can generate packaged entangled states, in which all internal quantum numbers (IQNs) (e.g., electric charge, flavor, and color) are inseparably entangled and constrained to irreducible representation (irrep) blocks. This is a consequence of local gauge invariance and superselection rules. The confinement restricts the net gauge charge to a single superselection sector, thereby excluding cross-sector superpositions but allowing entanglement within one sector. We establish theorems that: \textbf{(1)} Explain how these packaged entangled states naturally arise from quantum field excitations, \textbf{(2)} Show how they remain gauge invariant or transform covariantly within a fixed net-charge sector, and \textbf{(3)} Illustrate how external degrees of freedom (DOFs) (e.g., spin or momentum) can combine with packaged internal charges to yield gauge-invariant entanglement. Finally, we show that spin or momentum measurements on these hybrid states induce a collapse of the internal entanglement.
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-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.
Symmetry packaging is the phenomenon whereby, upon particle creation, all the internal quantum numbers (IQNs) become locked into a single irreducible representation (irrep) block of the gauge group, as required by locality and gauge invariance. The resulting packaged quantum states exhibit characteristic symmetry constraints and entanglement patterns. We develop a group-theoretic framework to describe the symmetry packaging for a variety of concrete symmetries and to classify the corresponding packaged states: \textbf{(1)} We prove that for any finite or compact group G, there exist G-associated packaged subspaces, in which every vector is automatically a packaged state. In particular, in multi-particle systems, any nontrivial representation of G induces inseparable packaged entanglement that locks together all IQNs. \textbf{(2)} We apply this framework to symmetry packaging in finite groups (cyclic group ZN, charge conjugation C, fermion parity, parity P, time reversal T, and dihedral groups), compact groups (U(1), SU(N), SU(2), and SU(3)), p-form symmetries, and hybrid symmetries. In each case, gauge invariance and superselection rules forbid the factorization of the resulting states. We illustrate how Bell-type packaged entangled states, color confinement, and hybrid gauge-invariant configurations all arise naturally. These results yield a complete classification of packaged quantum states. \textbf{(3)} Finally, we extend the packaging principle to incorporate full spacetime symmetry and hybrid systems of local, global, and Lorentz/Poincaré charges. Our approach unifies tools from group theory, gauge theory, and topological classification. These results may be useful for potential applications in high energy physics, quantum field theory, and quantum technologies.
Quantum simulation of non-Abelian gauge theories requires careful handling of gauge redundancy. We address this challenge by presenting universal principles for treating gauge symmetry that apply to any quantum simulation approach, clarifying that physical states need not be represented solely by gauge singlets. Both singlet and non-singlet representations are valid, with distinct practical trade-offs, which we elucidate using analogies to BRST quantization. We demonstrate these principles within a complete quantum simulation framework based on the orbifold lattice, which enables explicit and efficient circuit constructions relevant to real-world QCD. For singlet-based approaches, we introduce a Haar-averaging projection implemented via linear combinations of unitaries, and analyze its cost and truncation errors. We also introduce an efficient simulation protocol with an additional term to the Hamiltonian that eliminates non-singlet states from the low-energy spectrum. Beyond the singlet-approach, we show how non-singlet approaches can yield gauge-invariant observables through wave packets and string excitations. This non-singlet approach is proven to be both universal and efficient. Working in temporal gauge, we provide explicit mappings of lattice Yang-Mills dynamics to Pauli-string Hamiltonians suitable for Trotterization. Classical simulations of small systems validate convergence criteria and quantify truncation and Trotter errors, showing concrete resource estimates and scalable circuit recipes for SU(N) gauge theories. Our framework provides both conceptual clarity and practical tools toward quantum advantage in simulating non-Abelian gauge theories.
Packaged quantum states are gauge-invariant states in which all internal quantum numbers (IQNs) form an inseparable block. This feature gives rise to novel packaged entanglements that encompass all IQNs, which is important both for fundamental physics and for quantum technology. Here we develop a framework for gauge-invariant quantum information processing based on packaged quantum states. We propose the necessary and sufficient conditions for a valid packaged superposition state of a single particle and multi-particle. We then present the details of constructing gauge-invariant packaged qubits (or qudits), packaged gates, and packaged circuits (which commute with the total charge operator). These serve as alternative foundation for gauge-invariant quantum information science. We then adapt conventional quantum error-correction codes, quantum algorithms, and quantum communication protocols to the (d×D)-dimensional hybrid-packaged subspace. This high-dimensional hybrid-packaged subspace is flexible for pruning and scaling to match available physics systems. Thus, packaged quantum information processing becomes feasible and testable. Our results show that the gauge-invariant packaged quantum states may provide a possible route toward robust, fault-tolerant, and secure quantum technologies.