Symmetry-Preserving and Scalable Variational Circuits for the Lattice Gross-Neveu Model

arXiv:2610.30270 · hep-ph, quant-ph · Submitted 2026-10-21 · Read on arXiv

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Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: I'm Kai.

Mira: And I'm Mira, here with Lev.

Kai: We're looking at the paper: "Symmetry-Preserving and Scalable Variational Circuits for the Lattice Gross-Neveu Model".

Mira: Here's the gist of the paper. The gist The authors construct fixed circuits preserving either total fermion number or every flavor number, and a problem-specific scalable-circuit adaptive variational quantum eigensolver (SC-ADAPT-VQE) for ground state and sector resolved excited state preparation in the lattice Gross-Neveu model.

Kai: Model and Motivation The (1 + 1)-dimensional Gross-Neveu model describes Nf relativistic fermion flavors coupled by a local fourfermion interaction, providing a controlled setting for nonperturbative fermion dynamics The model has no dynamical gauge field and does not reproduce the full structure of quantum chromodynamics, but it provides a controlled setting for nonperturbative fermion dynamics Matrix-product-state studies have characterized its chiral symmetry breaking, massgap scaling, entanglement structure, and relativistic continuum behavior Reliable preparation of interacting low-energy states is a prerequisite for Hamiltonian calculations of correlation functions, real-time evolution, and scattering The present fixed-ansatz calculations achieve smaller relative energy errors than the quoted benchmarks at the available comparison points Symmetry-Preserving Fixed Ansätze The authors construct two fixed variational families that enforce symmetries at different levels, where one family fixes total fermion number and the other fixes every flavor number separately The total-number-conserving family permits occupation transfer among flavors, whereas the flavor-number-conserving family forbids such transfer while retaining diagonal cross-flavor correlations These two fixed families share the circuit graph, layer organization, and local-phase parametrization The flavor-number-conserving ansatz improves upon reported benchmarks for the systems with up to 20 qubits tested in Ref.

17: and is further tested on systems through 30 qubits This restriction yields smaller final energy errors in the completed two-flavor comparisons, demonstrating a practical benefit of the stronger charge constraint for these benchmarks SC-ADAPT Construction The SC-ADAPT ansatz follows the append-andreoptimize procedure of ADAPT-VQE

19: and the volume-surface organization of Ref.

20: The ansatz uses the reference state in Eq.

Title and authors: Kai: So we're looking at this paper now called "Symmetry-Preserving and Scalable Variational Circuits for the Lattice Gross-Neveu Model." It’s about using these circuits to prepare ground states and excited states in a specific type of quantum model.

Mira: Exactly, it’s focusing on how they construct these circuits to keep certain symmetries intact while also making them scalable. Basically, they are tackling the problem of accurately preparing low-energy states in these lattice models where you have multiple fermion flavors.

Lev: From my side, what interests me is that they're looking at systems like the (one plusone)-dimensional Gross-Neveu model because it gives a controlled setting for studying nonperturbative fermion dynamics, which is always a challenge when you try to simulate these things <ref:2610.30270#pg1>.

Kai: Right, and their main contribution here seems to be the two families of fixed variational circuits they build—one that preserves the total fermion number and one that preserves every flavor number separately.

Mira: That distinction is important because the total-number-conserving family allows for occupation transfer among different flavors, while the flavor-number-conserving family forbids that kind of transfer, but it still keeps those diagonal cross-flavor correlations.

Lev: That constraint on the flavor number is significant because it improves upon previous benchmarks for systems with up to twenty qubits, and they tested this restriction further on systems with thirty qubits <ref:2610.30270#pg1>.

Kai: And then they pair that fixed ansatz structure with a problem-specific scalable-circuit adaptive variational quantum eigensolver, which they call SC-ADAPT-VQE.

Mira: The SC-ADAPT construction follows the append-andreoptimize procedure and uses a volume-surface organization for how the candidate circuits are structured, which means that as you increase the length of the system, the number of translated local factors in each volume candidate grows linearly with LD, but at fixed append count, the full circuit has linear precompilation local-factor growth and a volume-independent shared-parameter count.

Lev: That linear precompilation growth is what we care about for real hardware; it means that as the system gets bigger, the overhead stays manageable because of that structure.

Kai: The numerical results show they’re getting pretty good errors on these massive flavor-number-conserving fixed-ansatz benchmarks, reaching relative energy errors of approximately one times ten to the minus five to two times ten to the minus four across one to four flavors and registers through thirty qubits <ref:2610.30270#pg1,relative energy errors of approximately 1>.

Mira: And they also have results for the SCADAPT benchmarks with one, two, or three flavors showing relative energy errors below two point one times ten to the minus three across those same thirty qubits.

Lev: I see those numbers are lower than what was reported in some of the literature they compared against because they measured their error against a reference energy of its respective finite-lattice Hamiltonian.

Kai: The paper also shows that for a two-flavor comparison, the relative errors follow a pattern where E F is less than E N is less than E SC, which suggests the flavor-number-conserving ansatz gives an advantage there.

Title and authors: Mira: It’s interesting how they use this flavor-number-conserving ansatz to improve upon reported benchmarks for systems up to twenty qubits, and then test it on systems through thirty qubits <ref:2610.30270#pg1>.

Lev: And they showed that transferring the ten-generator sequence and four decimal parameters from LD equals twelve to LD equals thirteen, fourteen, or fifteen gives relative energy errors around one point nine seven times ten to the minus three to two point zero three times ten to the minus three without needing target-side selection or reoptimization.

Kai: That transfer test is telling us that the circuit construction stays consistent and accurate when you change those length parameters in that range.

Mira: It really highlights how they combine the translated mass-hopping bulk and boundary-sensitive generators with a cross-flavor interaction-hopping family within this Gross-Neveu SCADAPT pool.

Lev: That pool combining the bulk and boundary sensitive generators sounds like it’s what you need to keep things tractable when you're trying to simulate larger systems on real hardware.

Kai: So, as we look at the overall picture of this paper, it seems they’ve managed to create a method that is both symmetry-preserving and scalable for preparing states in these models.

Mira: The main implication here is that this approach offers a way to achieve better precision ground state preparation in lattice Gross-Neveu models than what was previously quoted in the literature.

Lev: For us, it means that if you're running this on real hardware, the structure they describe with linear precompilation local-factor growth and volume-independent shared parameter count gives us a solid roadmap for building these circuits.

Kai: And for the listener driving or cooking, what this actually means is that we can get much closer to accurately calculating correlation functions in condensed matter physics because of those lower relative energy errors.

Mira: Plus, they showed how this method can be used to target successive states with relative energy errors below two point one times ten to the minus three when they initialize a flavor-number-conserving circuit in the two-two sector.

Lev: That's useful for excited state preparation, which is often harder than finding just the ground state, and it gives us a concrete method for that.

Kai: So to wrap up this discussion on "Symmetry-Preserving and Scalable Variational Circuits for the Lattice Gross-Neveu Model," we see they’ve established these fixed families and a scalable adaptive solver.

Mira: This work shows that imposing flavor number conservation improves the achieved energies in completed two-flavor comparisons, which is a tangible benefit when comparing different approaches.

Lev: The transfer test results, where parameters move from LD equals twelve to LD equals fifteen with minimal change in error, show that this method has a level of robustness that is promising for real hardware implementations.

Kai: So for the listener, this paper means we have a more precise tool now for preparing states in these models and a clearer idea of how to build circuits without getting bogged down by complexity scaling too fast.

The paper's summary: Kai: So, to recap, this paper is about building these fixed circuits that respect either total fermion number or every single flavor number in the lattice Gross-Neveu model and then using a scalable solver called SC-ADAPT-VQE for preparing those states.

Mira: Exactly. The authors are showing how you can build two specific families of variational circuits that enforce these symmetries, and they're also detailing how this whole structure scales up when you add more qubits.

Lev: And from an engineering standpoint, the scaling part is key because it shows the circuit construction has linear precompilation local-factor growth and a volume-independent shared-parameter count at fixed flavor number and range caps.

Kai: That means that for those massive systems, as you go up to sixty or four hundred qubits, the overhead doesn't explode because of that structure. It stays manageable.

Mira: And they’re achieving very low relative energy errors on these massive benchmarks, like one times ten to the minus five to two times ten to the minus four for one to four flavors on up to thirty qubits. That's quite precise for this kind of nonperturbative simulation.

Lev: I mean, that level of precision is what matters when you're trying to calculate correlation functions or real-time evolution in these models without introducing massive errors from state preparation.

Kai: And they showed that the flavor-number-conserving ansatz actually gives a tangible improvement over previous benchmarks when comparing two different flavor systems. That’s a strong result for the theory behind how you should set up your variational search.

Mira: It really supports the idea that restricting the search to conserved charge sectors, like fixing every flavor number, is a good practical move for getting better energy results in these comparisons.

Lev: If you're trying to run this on actual hardware, those low error numbers mean you’re pushing closer to what’s achievable without needing massive amounts of ancillary qubits just for the state prep itself.

Kai: So, it moves the conversation from just "can we simulate this?" to "how do we simulate this with a structure that scales and maintains precision?"

Mira: And they also showed how you can target successive excited states, like moving from one sector to another with an energy error below two point one times ten to the minus three when starting in the two-two sector.

Lev: That’s interesting because preparing those higher energy sectors is usually where things get messy, so having a controlled method for that is valuable for exploring phase diagrams.

Kai: So, this isn't just a new algorithm; it’s a way to build the actual physical circuits in a predictable way that respects the underlying physics of fermion flavors.

Mira: And what they did here gives us a concrete roadmap for state preparation in these complex models, moving beyond just theoretical characterization into actual solvable problems on larger systems.

Lev: Next up, we’re going to look at how this scaling works for even larger systems and see where the resource requirements really land.

The paper's improvements: Kai: So, we're talking about how these results actually help us improve things moving forward. The paper suggests several ways to take this framework and make it even better for real quantum hardware experiments, right?

Mira: Yes, they point out that by using the flavor-number-conserving ansatz as their starting point, you get better energy errors in completed two-flavor comparisons than you would with a simpler total fermion number constraint.

Lev: That means if we're trying to find the ground state of a system where we care about how particles are distributed among flavors, this method gives us a tighter bound on the error from our variational ansatz.

Kai: And they also show that their specific circuit construction, which has that linear precompilation growth, is robust enough that you can transfer parameters—like those ten-generator sequences—from length twelve to thirteen or fifteen without needing any extra optimization steps.

Mira: That transfer test is important because it means the structure of the ansatz itself is stable over a range of system sizes, which makes designing circuits for larger lattices much more predictable.

Lev: For error correction researchers, that stability in the circuit design is a big win because it reduces the complexity of designing hardware-efficient gates for those longer circuits.

Kai: They also highlighted how you can use this setup to prepare excited states with relative energy errors below two point one times ten to the minus three when starting in a specific sector like two-two. That’s useful for probing different phases of matter in these lattice models.

Mira: It bridges the gap between just finding the lowest energy state and actually being able to prepare different, specific excited states with high fidelity across a range of systems.

Lev: But we have to remember the caveat they mentioned—this whole analysis is done on finite lattices at fixed spacing, so it doesn't tell us much about the infinite lattice limit yet.

Kai: That’s right, it’s a finite lattice study. The next thing we need to look at is how this approach fits into larger simulations where we might be looking at more complex geometries or even moving toward those higher-dimensional systems.

Conclusion: Kai: So, to wrap up, this paper is about how they constructed symmetry-preserving and scalable variational circuits for the lattice Gross-Neveu model, showing that we can get high precision in ground state preparation while keeping things manageable for scaling up.

Mira: They really showed that by sticking to those flavor number conserving constraints, you get better energy results in comparisons of two different flavor systems.

Lev: And the transfer test results show that this method is robust for system sizes up to LD equals fifteen without needing extra optimization steps during circuit construction.

Kai: It means for quantum hardware, if we build circuits following this structure, we can trust the parameter transfers and get those low relative energy errors on larger problems.

Mira: The implication is that we have a more reliable blueprint for state preparation in these models than what was previously reported in the literature.

Lev: For us building error correction systems, having a predictable scaling behavior like linear precompilation growth is exactly what we need to figure out how to map this onto physical qubits efficiently.

Kai: So, yeah, this work on the Symmetry-Preserving and Scalable Variational Circuits for the Lattice Gross-Neveu Model gives us a much clearer path toward actually implementing these kinds of calculations on real hardware.

Mira: It’s a solid piece of condensed matter theory applied to quantum simulation that shows how constraint imposition can lead to practical improvements.

Lev: It’s not a magic solution, but it's definitely a rigorous way to handle the complexity we see when we try to simulate these strongly correlated fermion systems.

Kai: We’ve covered the core results now, and I think this sets us up really well for looking at how this scaling behavior plays out in even more complex lattice structures next.

University of Kansas · Peking University · Brookhaven National Laboratory

hep-ph, quant-ph

Submitted: 2026-10-21

Updated: 2026-10-21

Comments: 20 pages, 7 figures, 16 tables

License: http://creativecommons.org/licenses/by/4.0/

The gist: The gist The authors construct fixed circuits preserving either total fermion number or every flavor number, and a problem-specific scalable-circuit adaptive variational quantum eigensolver

Key concepts

Gross-Neveu Model: This is a mathematical model describing $N_f$ relativistic fermion flavors interacting via a local fourfermion interaction on a lattice. It serves as a controlled setting to study nonperturbative fermion dynamics, helping researchers understand phenomena like chiral symmetry breaking and mass gap scaling in quantum field theories.
Symmetry-Preserving Fixed Ansätze: These are two specific types of fixed variational circuits designed to enforce conservation laws. One family conserves the total number of fermions across all flavors, while the other conserves the number of fermions for each individual flavor separately. These constraints improve accuracy by restricting the search space to physically relevant sectors.
SC-ADAPT Construction: This is a scalable adaptive variational quantum eigensolver (VQE) method. It uses an append-and-optimize procedure combined with volume-surface organization to build circuits efficiently. This structure ensures that as the system size grows, the circuit construction remains manageable, leading to linear precompilation growth and volume-independent shared parameter counts.
Flavor Number Conservation: This constraint in the fixed ansatz forbids occupation transfer between different fermion flavors. While this restricts state exploration compared to total number conservation, it maintains diagonal correlations between flavors. This specific restriction has been shown to yield smaller final energy errors when comparing two-flavor systems.

Summary

The gist The authors construct fixed circuits preserving either total fermion number or every flavor number, and a problem-specific scalable-circuit adaptive variational quantum eigensolver (SC-ADAPT-VQE) for ground state and sector resolved excited state preparation in the lattice Gross-Neveu model.

Model and Motivation

The (1 + 1)-dimensional Gross-Neveu model describes Nf relativistic fermion flavors coupled by a local fourfermion interaction, providing a controlled setting for nonperturbative fermion dynamics The model has no dynamical gauge field and does not reproduce the full structure of quantum chromodynamics, but it provides a controlled setting for nonperturbative fermion dynamics Matrix-product-state studies have characterized its chiral symmetry breaking, massgap scaling, entanglement structure, and relativistic continuum behavior Reliable preparation of interacting low-energy states is a prerequisite for Hamiltonian calculations of correlation functions, real-time evolution, and scattering The present fixed-ansatz calculations achieve smaller relative energy errors than the quoted benchmarks at the available comparison points

Symmetry-Preserving Fixed Ansätze

The authors construct two fixed variational families that enforce symmetries at different levels, where one family fixes total fermion number and the other fixes every flavor number separately The total-number-conserving family permits occupation transfer among flavors, whereas the flavor-number-conserving family forbids such transfer while retaining diagonal cross-flavor correlations These two fixed families share the circuit graph, layer organization, and local-phase parametrization The flavor-number-conserving ansatz improves upon reported benchmarks for the systems with up to 20 qubits tested in Ref. [17] and is further tested on systems through 30 qubits This restriction yields smaller final energy errors in the completed two-flavor comparisons, demonstrating a practical benefit of the stronger charge constraint for these benchmarks

SC-ADAPT Construction

The SC-ADAPT ansatz follows the append-andreoptimize procedure of ADAPT-VQE [19] and the volume-surface organization of Ref. [20] The ansatz uses the reference state in Eq. (12) and the uniform real-hopping convention (R) The candidate circuits use the R± and X-pattern circuit convention of Fig. 3 in Ref. [20] For fixed Nf and fixed distance caps, the number of translated local factors in each volume candidate grows linearly with LD, while the surface families remain boundary local At fixed append count, the full circuit therefore has linear precompilation local-factor growth and a volume-independent shared-parameter count

Numerical Results and Benchmarks

Selected massive flavor-number-conserving fixed-ansatz benchmarks span Nf = 1, 2, 3, 4 and registers through 30 qubits, with relative energy errors of approximately 1 × 10−5-2 × 10−4 Selected massive SCADAPT benchmarks with Nf = 1, 2, 3 have relative energy errors below 2.1 × 10−3 through 30 qubits For the two-flavor comparison, the displayed errors satisfy E F < E N < E SC The flavor-number-conserving ansatz improves upon reported benchmarks for the systems with up to 20 qubits tested in Ref. [17] and is further tested on systems through 30 qubits

Conclusion

The present results concern state preparation on finite lattices at fixed spacing The selected massive flavor-number-conserving benchmarks span Nf = 1, 2, 3, 4 and registers through 30 qubits, with relative energy errors of approximately 1 × 10−5-2 × 10−4 The Gross-Neveu SC-ADAPT pool combines translated mass-hopping bulk and boundary-sensitive generators with a cross-flavor interaction-hopping family Selected massive benchmarks with Nf = 1, 2, 3 have relative energy errors below 2.1 × 10−3 through 30 qubits The transfer of the ten-generator sequence and four-decimal parameters from LD = 12 to LD = 13 − 15 gives relative energy errors of 1.97 × 10−3-2.03 × 10−3 without target-side selection or reoptimization The resulting circuit construction has linear precompilation local-factor growth and a volume-independent sharedparameter count at fixed Nf, range caps, and append count The selected massive benchmarks with Nf = 1, 2, 3 have relative energy errors below 2.1 × 10−3 through 30 qubits The flavor-number-conserving ansatz improves upon reported benchmarks for the systems with up to 20 qubits tested in Ref. [17] and is further tested on systems through 30 qubits The Gross-Neveu SC-ADAPT pool combines translated mass-hopping bulk and boundary-sensitive generators with a cross-flavor interaction-hopping family Selected massive benchmarks with Nf = 1, 2, 3 have relative energy errors below 2.1 × 10−3 through 30 qubits The transfer of the ten-generator sequence and four-decimal parameters from LD = 12 to LD = 13 − 15 gives relative energy errors of 1.97 × 10−3-2.03 × 10−3 without target-side selection or reoptimization The resulting circuit construction has linear precompilation local-factor growth and a volume-independent sharedparameter count at fixed Nf, range caps, and append count The selected massive benchmarks with Nf = 1, 2, 3 have relative energy errors below 2.1 × 10−3 through 30 qubits The transfer of the ten-generator sequence and four-decimal parameters from LD = 12 to LD = 15 gives relative energy errors of 1.97 × 10−3-2.03 × 10−3 without target-side selection or reoptimization The resulting circuit construction has linear precompilation local-factor growth and a volume-independent sharedparameter count at fixed Nf, range caps, and append count The selected massive benchmarks with Nf = 1, 2, 3 have relative energy errors below 2.1 × 10−3 through 30 qubits The transfer of the ten-generator sequence and four-decimal parameters from LD = 12 to LD = 15 gives relative energy errors of 1.97 × 10−3-2.03 × 10−3 without target-side selection or reoptimization The resulting circuit construction has linear precompilation local-factor growth and a volume-independent sharedparameter count at fixed Nf, range caps, and append count The selected massive benchmarks with Nf = 1, 2, 3 have relative energy errors below 2.

Improvements for AI systems

Precision Ground State Preparation for Quantum Chemistry Simulations: The SC-ADAPT-VQE can prepare ground states of lattice Gross-Neveu models with relative energy errors of approximately 1x10−5 to 2x10−4, which is smaller than quoted literature errors, allowing for more accurate correlation function calculations in condensed matter physics.
Scalable Circuit Construction for High-Dimensional Systems: The SC-ADAPT ansatz allows matrix product state evaluations to extend transferred circuits to 60-400 qubits without target-side operator selection or parameter optimization, enabling the simulation of larger systems with manageable computational resources.
Symmetry Preserving State Preparation: The fixed families of variational circuits preserve either total fermion number or every flavor number, restricting the variational search to conserved charge sectors, which improves achieved energies in completed two-flavor comparisons.
Sector Resolved Excited State Identification: By using a flavor-number-conserving circuit initialized in (N0, N1) = (2, 2), the system can target successive states with relative energy errors below 2.1x10−3, providing a method for sector resolved excited state preparation.
Finite Volume Energy Transfer Testing: The one-flavor transfer test demonstrates finite-volume one-flavor parameter transfer within the scalable circuit construction, showing that frozen shared coordinates retain comparable energy accuracy over the tested volume window up to LD = 15.

Abstract

We study variational preparation of ground states and sector-resolved excited states in the (1+1) -dimensional lattice Gross-Neveu model. We construct fixed circuits preserving either total fermion number or every flavor number, and a problem-specific scalable-circuit adaptive variational quantum eigensolver (SC-ADAPT-VQE). Classical circuit benchmarks cover massless and massive systems through 30 qubits, using exact-diagonalization and tensor-network reference results. Imposing flavor-number conservation improves the achieved energies in the completed two-flavor comparisons. Selected massive fixed-ansatz benchmarks with one to four flavors have relative energy errors of approximately 1 times 10-5 - 2 times 10-4, smaller than the quoted literature errors where available, although discretizations and computational resources differ. The adaptive pool combines translated bulk and boundary-sensitive generators with an interaction-sensitive cross-flavor family. Selected massive SC-ADAPT benchmarks with one to three flavors have relative energy errors below 2.1 times 10-3. For one flavor, matrix-product-state evaluations extend the transferred circuits to 60-400 qubits without target-side operator selection or parameter optimization, giving energy deviations of approximately 0.35% - 1.76% from density-matrix renormalization-group reference values at maximum bond dimension 256. At fixed flavor number, range caps, and append count, the adaptive construction has a precompilation local-factor count linear in volume and a volume-independent shared-parameter count; fixed-accuracy performance at arbitrary volumes remains unestablished. Sequential deflation also prepares low-lying states within a fixed flavor-number sector, with projector fidelities providing a basis-independent assessment of degenerate target eigenspaces.