Qubit-Regularized Lattice Gauge Theories

• Qubit-regularized Hamiltonian lattice gauge theories is a finite-dimensional framework that replaces infinite-dimensional bosonic gauge-field spaces with qubits while retaining exact gauge invariance.
• It employs methods like quantum link models, monomer-dimer tensor networks, and plaquette encodings to map gauge dynamics onto spin chains and digital quantum circuits.
• The approach unveils continuum limits and universal phase structures, offering practical insights into confinement, deconfinement, and simulation strategies for quantum hardware.

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Qubit-regularized Hamiltonian lattice gauge theories are finite-dimensional Hamiltonian formulations of gauge theory in which the local bosonic gauge-field Hilbert space is replaced by a qubit- or qudit-encodable space while exact gauge invariance is retained. In this setting, the basic lattice dynamics is formulated directly in real Minkowski time, with evolution generated by $e^{-i\hat H t}$ rather than by a Euclidean path integral, and the microscopic gauge variables are represented either by quantum-link operators, by truncated irreducible-representation data, or by gauge-invariant plaquette variables. The program encompasses several distinct but related constructions—quantum link models and D-theory, monomer-dimer tensor-network formulations, plaquette-chain models with exact mappings to spin chains, large-$N_c$ plaquette encodings, and improved or renormalized effective Hamiltonians designed for quantum hardware—and seeks continuum gauge physics from strictly finite-dimensional microscopic degrees of freedom (Brower et al., 2020, Chandrasekharan, 8 Feb 2025).

1. Foundational reformulation of lattice gauge dynamics

The defining departure from traditional Wilson lattice gauge theory is twofold. First, one works in the Hamiltonian formalism with continuous physical time, so the basic dynamical object is the unitary evolution operator

$e^{-i\hat H t},$

not a Euclidean transfer matrix. Second, one replaces the infinite-dimensional local link Hilbert space by a finite register, typically a qubit, qutrit, or a small set of allowed irreducible representations. In the language of "Lattice Gauge Theory for a Quantum Computer" (Brower et al., 2020), this yields a Hamiltonian, Minkowski-time, qubit-native formulation in which gauge-invariant local kernels can be mapped directly to digital quantum circuits.

In the quantum-link and D-theory perspective associated with Brower, Chandrasekhar, and Wiese, gauge fields are represented by operator-valued links rather than by commuting group-valued c-numbers. For the non-Abelian case, the Hamiltonian is written in Kogut-Susskind-like form as

$$\hat H = \frac{g^2}{2} \sum_{x,\mu} \mathrm{Tr}\!\left[\hat E_L^2(x,\mu) + \hat E_R^2(x,\mu)\right] -\frac{1}{2g^2}\sum_{x,\mu,\nu}\mathrm{Tr}\!\left[\hat U_{\mu\nu}(x)\right] + \text{Quarks},$$

but the link operators are constructed from fermionic rishon operators and satisfy local noncommutativity, including $[\hat U,\hat U^\dagger]\neq 0$. The identity $E_R = U^\dagger E_L U$ does not hold in this finite-dimensional setting, although operators on different links still commute. The D-theory claim is that this microscopic deformation is an irrelevant UV effect in the continuum limit, so the regularized theory may remain in the same universality class after coarse graining (Brower et al., 2020).

A broader conceptual statement is made in "Qubit Regularization of Quantum Field Theories" (Chandrasekharan, 8 Feb 2025). There, qubit regularization is presented as a distinct regularization paradigm for quantum field theory: finite-dimensional local Hilbert spaces are not merely a computational truncation but the starting point of the microscopic definition. This position is historically linked to D-theory, where discrete variables in one higher dimension can generate the target continuum theory through dimensional reduction. A central question then becomes whether asymptotically free continuum behavior can emerge without sending the local Hilbert-space dimension to infinity.

2. Gauge-invariant finite-dimensional encodings

A major structural development is the monomer-dimer tensor-network (MDTN) basis. In this formulation, the physical Hilbert space of a traditional $\mathrm{SU}(N)$ lattice gauge theory is organized by irreducible representations on links and sites together with local singlet-fusion labels. For each oriented link,

$${}^{\rm Trad}_\ell = \bigoplus_{\lambda_\ell} V_{\lambda_\ell}\otimes V_{\bar\lambda_\ell},$$

with basis states $\lvert D^{\lambda_\ell}_{ij}\rangle$, while matter at a site is expanded in basis states $\lvert \psi^{\lambda_s}_k\rangle$. Gauge invariance is imposed locally by projecting the site Hilbert space $N_c$0 onto its singlet subspace and labeling an orthonormal singlet basis by

$N_c$1

The resulting physical basis states are

$N_c$2

This basis is exact, orthonormal, and gauge invariant by construction, and it naturally supports qubit regularization by restricting the allowed set of link irreps (Chandrasekharan et al., 20 Feb 2025).

A particularly simple truncation is the anti-symmetric qubit regularization (ASQR), in which only anti-symmetric irreps are retained on each link. In the examples discussed, this means $N_c$3 for $N_c$4 and $N_c$5 for $N_c$6. Because the truncation is defined inside the gauge-invariant Hilbert space rather than at the level of unconstrained link variables, Gauss’s law remains exact and the pure-gauge sector can be made sign-problem free (Chandrasekharan, 26 Feb 2026, Chandrasekharan et al., 20 Feb 2025).

A different but complementary route is to build a larger effective link Hilbert space from many qubits. "Real-space blocking of qubit variables on parallel lattice gauge theory links for quantum simulation" (Shir et al., 2023) constructs an open ladder of $N_c$7 spin-$N_c$8 degrees of freedom with

$N_c$9

and uses a ferromagnetic ladder Hamiltonian to project onto the fully symmetric highest-spin multiplet. The projector

$e^{-i\hat H t},$0

defines an effective link of dimension $e^{-i\hat H t},$1, so $e^{-i\hat H t},$2 qubits emulate a single truncated $e^{-i\hat H t},$3 gauge link with electric cutoff $e^{-i\hat H t},$4.

A further gauge-invariant simplification appears in the large-$e^{-i\hat H t},$5 plaquette approach. By parameterizing the gauge-invariant Hilbert space in terms of plaquette loops rather than link variables, "Quantum Simulation of Large N Lattice Gauge Theories" shows that at leading order in $e^{-i\hat H t},$6 only single-plaquette loops survive, which permits a qutrit encoding

$e^{-i\hat H t},$7

or, after charge-conjugation projection, a single qubit per plaquette. For $e^{-i\hat H t},$8 this yields a constrained PXP-type Hamiltonian in the $e^{-i\hat H t},$9-even sector, though at $$\hat H = \frac{g^2}{2} \sum_{x,\mu} \mathrm{Tr}\!\left[\hat E_L^2(x,\mu) + \hat E_R^2(x,\mu)\right] -\frac{1}{2g^2}\sum_{x,\mu,\nu}\mathrm{Tr}\!\left[\hat U_{\mu\nu}(x)\right] + \text{Quarks},$$0 the leading approximation carries about $$\hat H = \frac{g^2}{2} \sum_{x,\mu} \mathrm{Tr}\!\left[\hat E_L^2(x,\mu) + \hat E_R^2(x,\mu)\right] -\frac{1}{2g^2}\sum_{x,\mu,\nu}\mathrm{Tr}\!\left[\hat U_{\mu\nu}(x)\right] + \text{Quarks},$$1 error in a $$\hat H = \frac{g^2}{2} \sum_{x,\mu} \mathrm{Tr}\!\left[\hat E_L^2(x,\mu) + \hat E_R^2(x,\mu)\right] -\frac{1}{2g^2}\sum_{x,\mu,\nu}\mathrm{Tr}\!\left[\hat U_{\mu\nu}(x)\right] + \text{Quarks},$$2 benchmark at $$\hat H = \frac{g^2}{2} \sum_{x,\mu} \mathrm{Tr}\!\left[\hat E_L^2(x,\mu) + \hat E_R^2(x,\mu)\right] -\frac{1}{2g^2}\sum_{x,\mu,\nu}\mathrm{Tr}\!\left[\hat U_{\mu\nu}(x)\right] + \text{Quarks},$$3, with only modest improvement from the first $$\hat H = \frac{g^2}{2} \sum_{x,\mu} \mathrm{Tr}\!\left[\hat E_L^2(x,\mu) + \hat E_R^2(x,\mu)\right] -\frac{1}{2g^2}\sum_{x,\mu,\nu}\mathrm{Tr}\!\left[\hat U_{\mu\nu}(x)\right] + \text{Quarks},$$4 corrections (Ciavarella et al., 2024).

Construction	Finite local space	Structural idea
Quantum link / D-theory (Brower et al., 2020)	single qubit per link in minimal $$\hat H = \frac{g^2}{2} \sum_{x,\mu} \mathrm{Tr}\!\left[\hat E_L^2(x,\mu) + \hat E_R^2(x,\mu)\right] -\frac{1}{2g^2}\sum_{x,\mu,\nu}\mathrm{Tr}\!\left[\hat U_{\mu\nu}(x)\right] + \text{Quarks},$$5 truncation	operator-valued links and gauge-invariant Trotter kernels
MDTN / ASQR (Chandrasekharan et al., 20 Feb 2025)	$$\hat H = \frac{g^2}{2} \sum_{x,\mu} \mathrm{Tr}\!\left[\hat E_L^2(x,\mu) + \hat E_R^2(x,\mu)\right] -\frac{1}{2g^2}\sum_{x,\mu,\nu}\mathrm{Tr}\!\left[\hat U_{\mu\nu}(x)\right] + \text{Quarks},$$6 for $$\hat H = \frac{g^2}{2} \sum_{x,\mu} \mathrm{Tr}\!\left[\hat E_L^2(x,\mu) + \hat E_R^2(x,\mu)\right] -\frac{1}{2g^2}\sum_{x,\mu,\nu}\mathrm{Tr}\!\left[\hat U_{\mu\nu}(x)\right] + \text{Quarks},$$7; $$\hat H = \frac{g^2}{2} \sum_{x,\mu} \mathrm{Tr}\!\left[\hat E_L^2(x,\mu) + \hat E_R^2(x,\mu)\right] -\frac{1}{2g^2}\sum_{x,\mu,\nu}\mathrm{Tr}\!\left[\hat U_{\mu\nu}(x)\right] + \text{Quarks},$$8 for $$\hat H = \frac{g^2}{2} \sum_{x,\mu} \mathrm{Tr}\!\left[\hat E_L^2(x,\mu) + \hat E_R^2(x,\mu)\right] -\frac{1}{2g^2}\sum_{x,\mu,\nu}\mathrm{Tr}\!\left[\hat U_{\mu\nu}(x)\right] + \text{Quarks},$$9	gauge-invariant irrep basis with singlet labels $[\hat U,\hat U^\dagger]\neq 0$0
Real-space blocking (Shir et al., 2023)	$[\hat U,\hat U^\dagger]\neq 0$1 qubits $[\hat U,\hat U^\dagger]\neq 0$2 one spin-$[\hat U,\hat U^\dagger]\neq 0$3 link	symmetric ladder subspace realizes dimension $[\hat U,\hat U^\dagger]\neq 0$4
Large-$[\hat U,\hat U^\dagger]\neq 0$5 plaquette encoding (Ciavarella et al., 2024)	qutrit or qubit per plaquette	plaquette loops survive at leading order

3. Hamiltonians, constraints, and exact mappings

The simplest explicit quantum-link example is compact $[\hat U,\hat U^\dagger]\neq 0$6 gauge theory on a triangular lattice. For one link,

$[\hat U,\hat U^\dagger]\neq 0$7

with flux eigenstates $[\hat U,\hat U^\dagger]\neq 0$8. The minimal D-theory realization replaces this infinite-dimensional link space by a single qubit,

$[\hat U,\hat U^\dagger]\neq 0$9

so the link flux is restricted to $E_R = U^\dagger E_L U$0. On a triangular plaquette stacked along an extra direction $E_R = U^\dagger E_L U$1, the Hamiltonian contains an electric term, an inter-layer XY coupling, and a magnetic plaquette term, and exact gauge invariance is enforced by local generators

$E_R = U^\dagger E_L U$2

This permits Suzuki-Trotter decomposition into gauge-invariant local kernels: electric evolution uses $E_R = U^\dagger E_L U$3-type rotations, the XY term maps to two-qubit $E_R = U^\dagger E_L U$4 and $E_R = U^\dagger E_L U$5 interactions, and the plaquette term becomes a multi-qubit controlled circuit (Brower et al., 2020).

The plaquette-chain models provide non-Abelian examples with exact mappings to standard quantum spin chains. In the $E_R = U^\dagger E_L U$6 chain, the simplest qubit regularization keeps only singlet and doublet link irreps,

$E_R = U^\dagger E_L U$7

and the gauge-invariant Hamiltonian

$E_R = U^\dagger E_L U$8

splits into even and odd topological sectors. In the even sector, each plaquette has two states and the Hamiltonian maps exactly to a transverse-field Ising model in a uniform magnetic field,

$E_R = U^\dagger E_L U$9

while the odd sector differs only by the sign of the local field term (Siew et al., 11 Dec 2025).

The $\mathrm{SU}(N)$0 plaquette chain is formulated in the MDTN basis with link states

$\mathrm{SU}(N)$1

and Hamiltonian

$\mathrm{SU}(N)$2

The physical Hilbert space splits into three topological sectors, with $\mathrm{SU}(N)$3 as the charge-conjugation-invariant vacuum sector and $\mathrm{SU}(N)$4 exchanged by $\mathrm{SU}(N)$5. In a fixed sector, each plaquette has exactly three allowed states, so the model maps to the three-state quantum clock chain. In the vacuum sector,

$\mathrm{SU}(N)$6

with $\mathrm{SU}(N)$7 and $\mathrm{SU}(N)$8 (Siew et al., 1 Mar 2026).

These exact mappings matter because they transform gauge-theory continuum-limit questions into controlled questions about well-understood critical spin chains while retaining a direct gauge-theoretic interpretation of the excitations, sectors, and static-source observables.

4. Continuum limits and renormalization-group structures

A recurring issue in qubit regularization is whether asymptotically free or otherwise nontrivial continuum quantum field theories can emerge from strictly finite-dimensional microscopic Hilbert spaces. "Qubit Regularization of Quantum Field Theories" states that a common belief has been that asymptotically free QFTs require the local Hilbert-space dimension to grow without bound, but also points to two $\mathrm{SU}(N)$9-dimensional counterexamples in which asymptotic freedom appears within a strictly finite-dimensional local space through a novel RG flow. In that picture, the microscopic model flows to a decoupled fixed point, while the desired UV physics appears as a crossover over an intermediate range of scales rather than as the literal RG endpoint (Chandrasekharan, 8 Feb 2025).

The $${}^{\rm Trad}_\ell = \bigoplus_{\lambda_\ell} V_{\lambda_\ell}\otimes V_{\bar\lambda_\ell},$$0 plaquette chain realizes a more direct continuum-limit scenario. With $${}^{\rm Trad}_\ell = \bigoplus_{\lambda_\ell} V_{\lambda_\ell}\otimes V_{\bar\lambda_\ell},$$1, the critical point is at $${}^{\rm Trad}_\ell = \bigoplus_{\lambda_\ell} V_{\lambda_\ell}\otimes V_{\bar\lambda_\ell},$$2, where the continuum theory is the Ising conformal field theory of free massless Majorana fermions. The paper interprets this as asymptotic freedom in the toy gauge system. A nonzero $${}^{\rm Trad}_\ell = \bigoplus_{\lambda_\ell} V_{\lambda_\ell}\otimes V_{\bar\lambda_\ell},$$3 is a relevant perturbation that drives the model into the massive $${}^{\rm Trad}_\ell = \bigoplus_{\lambda_\ell} V_{\lambda_\ell}\otimes V_{\bar\lambda_\ell},$$4 field theory of Zamolodchikov. The finite-size step-scaling function gives the universal Ising value

$${}^{\rm Trad}_\ell = \bigoplus_{\lambda_\ell} V_{\lambda_\ell}\otimes V_{\bar\lambda_\ell},$$5

and the IR confining regime yields

$${}^{\rm Trad}_\ell = \bigoplus_{\lambda_\ell} V_{\lambda_\ell}\otimes V_{\bar\lambda_\ell},$$6

where $${}^{\rm Trad}_\ell = \bigoplus_{\lambda_\ell} V_{\lambda_\ell}\otimes V_{\bar\lambda_\ell},$$7 is the lightest glueball-like mass and $${}^{\rm Trad}_\ell = \bigoplus_{\lambda_\ell} V_{\lambda_\ell}\otimes V_{\bar\lambda_\ell},$$8 is the static string tension (Siew et al., 11 Dec 2025).

The $${}^{\rm Trad}_\ell = \bigoplus_{\lambda_\ell} V_{\lambda_\ell}\otimes V_{\bar\lambda_\ell},$$9 plaquette chain has an analogous but distinct structure. The critical point

$\lvert D^{\lambda_\ell}_{ij}\rangle$0

maps to the $\lvert D^{\lambda_\ell}_{ij}\rangle$1 parafermion conformal field theory with central charge

$\lvert D^{\lambda_\ell}_{ij}\rangle$2

A magnetic perturbation $\lvert D^{\lambda_\ell}_{ij}\rangle$3 with thermal perturbation $\lvert D^{\lambda_\ell}_{ij}\rangle$4 drives the system into a massive, non-integrable infrared theory. The correlation length scales as

$\lvert D^{\lambda_\ell}_{ij}\rangle$5

so finite-size scaling is controlled by $\lvert D^{\lambda_\ell}_{ij}\rangle$6. In the massive IR regime, the lightest $\lvert D^{\lambda_\ell}_{ij}\rangle$7-even and $\lvert D^{\lambda_\ell}_{ij}\rangle$8-odd excitations are interpreted as quasi-one-dimensional glueballs, with continuum estimates

$\lvert D^{\lambda_\ell}_{ij}\rangle$9

At the UV fixed point these two lightest states are degenerate, $\lvert \psi^{\lambda_s}_k\rangle$0 (Siew et al., 1 Mar 2026).

Taken together, these results indicate that qubit-regularized Hamiltonian gauge theories admit more than one continuum mechanism. One route is tuning to a known critical spin-chain CFT followed by a relevant deformation to a massive relativistic IR theory. Another is the crossover scenario advocated in the broader qubit-regularization program. This suggests that finite-dimensional regularization does not, by itself, preclude continuum gauge dynamics.

5. Confinement, deconfinement, and universal phase structure

The phase structure of qubit-regularized gauge theories can be studied directly in the truncated, gauge-invariant Hilbert space. In the MDTN-based pure-gauge models, the Hamiltonian

$\lvert \psi^{\lambda_s}_k\rangle$1

penalizes non-singlet links through

$\lvert \psi^{\lambda_s}_k\rangle$2

At $\lvert \psi^{\lambda_s}_k\rangle$3 the ground state has all links in the singlet representation, and static charges are connected by a flux string with linearly increasing energy cost. At large $\lvert \psi^{\lambda_s}_k\rangle$4, plaquette fluctuations dominate and the flux can spread, producing a deconfined regime. In the pure-gauge sector these models are sign-problem free and can be investigated by classical Monte Carlo (Chandrasekharan, 26 Feb 2026).

The natural finite-temperature observable is the susceptibility

$\lvert \psi^{\lambda_s}_k\rangle$5

where $\lvert \psi^{\lambda_s}_k\rangle$6 inserts heavy static sources. In the confined phase, $\lvert \psi^{\lambda_s}_k\rangle$7 saturates in the thermodynamic limit; in the deconfined phase it scales as $\lvert \psi^{\lambda_s}_k\rangle$8. If the transition is second order, the finite-size scaling form is

$\lvert \psi^{\lambda_s}_k\rangle$9

The qubit-regularized $N_c$00 and $N_c$01 models reproduce the expected Svetitsky-Yaffe universality classes: in two spatial dimensions, $N_c$02 follows 2D Ising and $N_c$03 follows the 3-state Potts model; in three spatial dimensions, $N_c$04 follows 3D Ising and $N_c$05 is first order, as in the conventional theory (Chandrasekharan, 26 Feb 2026).

The MDTN finite-temperature study gives explicit critical estimates. For $N_c$06 in $N_c$07, the transition is consistent with 2D Ising exponents

$N_c$08

For $N_c$09 in $N_c$10, it is consistent with 3-state Potts exponents

$N_c$11

For $N_c$12 in $N_c$13, the transition is continuous with

$N_c$14

while $N_c$15 in $N_c$16 is first order (Chandrasekharan et al., 20 Feb 2025).

The same MDTN framework also supplies a zero-temperature indicator of deconfinement in one dimension. In the $N_c$17 plaquette chain with static sources separated by distance $N_c$18, the ground-state energy behaves as

$N_c$19

and at large $N_c$20,

$N_c$21

Thus the string tension is continuously tunable and vanishes as $N_c$22, which the paper identifies with a deconfined quantum critical point (Chandrasekharan et al., 20 Feb 2025).

A common misconception is that finite-dimensional truncation necessarily destroys universal gauge-theory physics. The finite-temperature evidence does not support that view: within the studied classes of models, the universal confinement-deconfinement behavior matches that of conventional $N_c$23 lattice gauge theory.

6. Digital quantum simulation, hardware mappings, and improvement strategies

Because the Hamiltonian formulation is already unitary and local, qubit-regularized theories are especially suited to digital quantum simulation. In the triangular-lattice $N_c$24 quantum-link model, real-time amplitudes such as

$N_c$25

are approximated by gauge-invariant Suzuki-Trotter kernels. The electric kernel uses $N_c$26-type rotations, the XY kernel uses entangling gates for $N_c$27 and $N_c$28 interactions, and the plaquette kernel becomes a controlled multi-qubit circuit. For larger lattices, commuting plaquette sets can be scheduled by coloring plaquettes on the dual hexagonal lattice (Brower et al., 2020).

A distinct line of work adapts Symanzik improvement to Hamiltonian simulation. "Improved Hamiltonians for Quantum Simulations" constructs

$N_c$29

with magnetic terms

$N_c$30

and kinetic terms

$N_c$31

At tree level the coefficients

$N_c$32

push the leading classical discretization error from $N_c$33 to $N_c$34. The expected resource implication is a reduction of about $N_c$35 qubits at fixed physical accuracy. For $N_c$36 gauge theory, the rectangle gate implemented on ibm_perth required 12 CNOTs and 20 additional one-qubit gates, and a two-plaquette improved Trotter step required at least 28 CNOTs and 40 one-qubit gates. With Pauli twirling, the average reported rectangle fidelity was approximately

$N_c$37

while for $N_c$38 without twirling the fidelity was approximately

$N_c$39

The paper therefore presents improvement as a qubit-saving strategy whose current limitation is hardware noise rather than formal implementability (Carena et al., 2022).

A complementary improvement program addresses gauge-field truncation rather than lattice-spacing artifacts. "Quantum Simulation of Lattice QCD with Improved Hamiltonians" derives low-dimensional effective Hamiltonians for truncated $N_c$40 Kogut-Susskind theories using Schrieffer-Wolff perturbation theory and SRG/IMSRG flows. In $N_c$41 dimensions, even the zero-electric-field truncation can reproduce gaps, baryon masses, and vacuum electric observables once virtual flux sectors are integrated out into effective couplings. The real-time meson simulation on IBM Perth used 5 qubits instead of 18 qubits for the full gauge-theory Hamiltonian, with short-time dynamics matching exact evolution well (Ciavarella, 2023).

Hardware-level demonstrations of qubit-regularized gauge dynamics already exist in both Abelian and non-Abelian settings. "A qubit model for U(1) lattice gauge theory" reduces the independent gauge links to a minimum using Gauss’s law and shows that as few as two qubits per gauge link are useful for strong-coupling real-time propagation and real-time collisions in two spatial dimensions, with a three-dimensional extension also developed (Lewis et al., 2019). "Investigating how to simulate lattice gauge theories on a quantum computer" studies a truncated Hamiltonian $N_c$42 theory on both the D-Wave quantum annealer and IBM gate-based devices, using exact Gauss-law-respecting basis generation, Kitaev-Feynman clock states for annealer-based time evolution, and Suzuki-Trotter circuits for gate-based evolution; the reported outcome includes the first observation on quantum hardware of a traveling excitation in $N_c$43 lattice gauge theory (Mendicelli, 2023).

The large-$N_c$44 plaquette approach and the MDTN/ASQR constructions indicate that the most consequential simplification may be to encode the physical Hilbert space directly rather than to encode unconstrained link variables and then impose Gauss’s law afterward. Whether higher-dimensional non-Abelian qubit-regularized models possess second-order quantum critical points whose continuum limits are ordinary Yang-Mills theory, or instead different interacting gauge theories, remains an open question, but the existing results already establish exact gauge invariance, controlled continuum limits in quasi-one-dimensional settings, correct finite-temperature universality classes, and explicit real-time circuit constructions on qubit hardware (Chandrasekharan, 26 Feb 2026).