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Fermionic Wave Functions from Neural-Network Constrained Hidden States
Javier Robledo Moreno, Giuseppe Carleo, Antoine Georges, James Stokes
TL;DR
Choosing an effective hidden-particle constraint is difficult because the space of possible functions is doubly exponential. The paper optimizes a neural-network-parametrized constraint together with Slater-determinant orbitals in an augmented fermionic space. The resulting family is universal and achieves accuracy competitive with state-of-the-art variational methods for the square-lattice Hubbard ground state.
Problem
Choosing an optimal hidden-particle constraint is unclear, and brute-force search is intractable because the number of possible constraint functions is doubly exponential.
Source: d. Benchmarks · p. 16Method
The method uses an augmented-space Slater determinant and optimizes its hidden-fermion constraint and orbital amplitudes, with the constraint parametrized by a neural network.
Source: §IV · p. 8Results
The ansatz is universal and achieves variational accuracy competitive with state-of-the-art methods for the square-lattice Hubbard ground state.
Source: §IV · p. 8Takeaways & Limitations
The ansatz does not rely on imposing symmetries, and the authors identify possible applications to systems without an underlying lattice.
Source: §IV · p. 8Brute-force search over constraint functions is intractable because their number is doubly exponential.
Source: d. Benchmarks · p. 16
Abstract
from arXiv · showhide
We introduce a systematically improvable family of variational wave functions for the simulation of strongly correlated fermionic systems. This family consists of Slater determinants in an augmented Hilbert space involving "hidden" additional fermionic degrees of freedom. These determinants are projected onto the physical Hilbert space through a constraint which is optimized, together with the single-particle orbitals, using a neural network parametrization. This construction draws inspiration from the success of hidden particle representations but overcomes the limitations associated with the mean-field treatment of the constraint often used in this context. Our construction provides an extremely expressive family of wave functions, which is proven to be universal. We apply this construction to the ground state properties of the Hubbard model on the square lattice, achieving levels of accuracy which are competitive with state-of-the-art variational methods.
I. BACKGROUND: STATES AND HAMILTONIAN
The paper studies interacting fermions using the Hubbard model and an augmented space of visible and hidden fermionic modes. A constraint connects the enlarged representation to physical states, while neural networks parametrize parts of the hidden-fermion determinant.
I. BACKGROUND: STATES AND HAMILTONIAN
- The method targets ground-state approximation for interacting fermionic Hamiltonians, illustrated with the Hubbard model on square and rectangular lattices with uniform hopping and onsite repulsion.More general geometries are left for future studies.Source: §I · p. 2
- The augmented-space picture uses a constraint to bring a correlated target state close to a Slater determinant, with visible-only determinants corresponding to physical Hartree-Fock states.The figure depicts the physical Hilbert space as a constrained subspace of the enlarged space.Source: Figure 1 · p. 2
- The visible modes correspond one-to-one with physical modes, and sampling imposes definite particle numbers for each spin species despite antisymmetry across species.The construction imposes full antisymmetry between spin species to increase the expressivity of the trial-wave-function family.Source: §I · p. 2
- For determinant amplitudes, visible configurations select rows of the orbital matrix, while a separate neural network maps visible-lattice occupancy to the hidden sub-matrix.The figure caption describes the visible-configuration row selection; the network input is the flattened visible occupancy.Source: Figure 2 · p. 3, §I · p. 3
A. States in the augmented Hilbert space: constraint function
The physical Fock space is embedded in a larger fermionic space by adding hidden modes and selecting representative hidden occupations for visible configurations. Optimizing this constraint alongside the augmented-space state produces a flexible family of correlated wave functions.
A. States in the augmented Hilbert space: constraint function
- The augmented Fock space adds hidden fermionic modes to the physical modes, while each visible configuration is paired with a selected hidden configuration.The hidden occupation is specified by a constraint function F(n) = ñ.Source: A. States in the augmented Hilbert… · p. 3
- The physical amplitude for visible occupancy n is the overlap of the trial state with an augmented basis state whose hidden occupancy is controlled by the constraint.This selects representative states spanning the physical Hilbert space within the enlarged space and removes unphysical states.Source: A. States in the augmented Hilbert… · p. 3
- Unlike earlier hidden-particle formulations with fixed physically motivated constraints, this work searches over the constraint function, yielding a highly flexible family.The number of possible constraint functions is doubly exponential.Source: A. States in the augmented Hilbert… · p. 3
- Geometrically, the constraint changes the physical subspace embedded in the enlarged space to bring the target correlated state close to the family of augmented-space Slater determinants.With no hidden modes, standard Slater determinants are recovered.Source: A. States in the augmented Hilbert… · p. 4
1. Generalities
The general construction takes an augmented-space Slater determinant with visible and hidden orbital components. Its hidden-particle configuration is determined from the visible configuration through a permutation-invariant mapping, preserving fermionic antisymmetry.
1. Generalities
- The augmented Slater determinant is built from Ntot orbital functions, with ˜N = Ntot − N hidden fermions added beyond the N visible fermions.The orbitals are assembled into a matrix whose columns correspond to the orbital functions.Source: §1 · p. 4
- The orbital matrix separates amplitudes on visible and hidden modes into four blocks, distinguishing visible and hidden orbitals evaluated in each sector.These blocks are φv, χv, φh, and χh.Source: §1 · p. 4
- Because the Slater determinant has fixed total particle number, the constraint maps visible configuration x to hidden configuration x̃ for amplitude evaluation.The constraint function must be invariant under permutations of the visible configuration to respect Fermi statistics.Source: §1 · p. 4
2. Universality and connection to other wave function ans¨atze
The hidden-fermion determinant family is universal on a lattice and connects to established wave-function forms. These links include backflow, Jastrow-type states, and configuration-interaction expansions.
2. Universality and connection to other wave function ans¨atze
- The ansatz is universal on the lattice because its determinant can represent a lookup table of amplitudes for an arbitrary target state.A construction with ˜N ≥ 1 may require the number of hidden modes to grow combinatorially with the number of physical modes.Source: §2 · p. 4
- The construction resembles backflow, but only orbitals restricted to hidden states have multi-particle position dependence.In regular backflow, orbitals depend on the coordinates of all particles.Source: §2 · p. 4
- With ˜N = 1 and χv = φh = 0, the amplitudes reduce to a visible Slater determinant times a symmetric function, including Gutzwiller and Jastrow factors.The symmetric factor can also be a generalized neural-network Jastrow factor.Source: §2 · p. 4
- A Laplace expansion of the determinant yields a configuration-interaction wave function containing single through ˜N-tuple excitations when the specified Hartree-Fock and virtual orbitals are used.The virtual orbitals are chosen as the first ˜N virtual orbitals.Source: §2 · p. 4
3. Parametrized constraint function, practical implementation
The constraint is represented through neural networks acting on visible occupations, allowing its parameters to be optimized together with the enlarged-space orbitals. The approach targets flexible constraint functions while keeping amplitude and gradient evaluation only a constant-factor more resource-intensive than an N-fermion determinant.
3. Parametrized constraint function, practical implementation
- Neural networks take visible occupation numbers as input to represent the permutation-invariant hidden sub-matrix, with a separate network for each row.The networks are multilayer perceptrons with hyperbolic-tangent activations.Source: §3 · p. 5
- The network parameters and visible-orbital matrices are jointly optimized, and sufficiently large networks can represent arbitrary constraint functions.The paper describes the network parametrization as a way to make choosing the optimal constraint tractable.Source: §3 · p. 5
- Amplitude and gradient evaluation is the only VMC step requiring more resources than an N-fermion determinant, and the increase is by a constant factor.The enlarged determinant and its derivatives are evaluated using LU factorization.Source: §3 · p. 5
C. Methods
The variational parameters are optimized by stochastic reconfiguration to minimize the energy expectation. Monte Carlo sampling estimates the expectations and gradients.
C. Methods
- The visible-orbital matrices and neural-network weights and biases are jointly optimized using stochastic reconfiguration.The objective is the variational state's Hamiltonian expectation value.Source: C. Methods · p. 5
- A new trial state is optimized from scratch for every Hamiltonian parameter choice.Source: C. Methods · p. 5
- Expectation values and objective gradients are computed by Markov chain Monte Carlo sampling configurations with probability proportional to |ψθ(x)|2.The implementation uses NetKet and Jax automatic differentiation.Source: C. Methods · p. 5
III. NUMERICAL EXPERIMENTS
The experiments benchmark square-lattice energies against exact diagonalization and study larger rectangular systems with complementary reference methods. They also examine stripe orders at one-eighth hole doping.
III. NUMERICAL EXPERIMENTS
- On 4 × 4 square lattices at n = 1/2 and n = 5/8, accuracy is measured by variational-energy error relative to the exact-diagonalization ground-state energy.Source: §III · p. 5
- For 4 × L lattices at n = 7/8, the study compares open-long-side energies with DMRG and examines competing stripe orders.Periodic boundary conditions are used along the short side.Source: §III · p. 5
- For periodic long-side boundaries, larger systems are compared with Slater-Jastrow and neural-network backflow results, while 4 × 4 is benchmarked against exact diagonalization.Source: §III · p. 5
A. Benchmarks in the square lattice
The parametrized-constraint ansatz performs strongly in square-lattice energy benchmarks, with accuracy improving as neural-network width or hidden-fermion number increases. Its error also compares favorably with the cited state-of-the-art benchmark and AFQMC results.
A. Benchmarks in the square lattice
- At n = 1/2, increasing α systematically lowers energy error, which does not significantly rise with U at the largest α and is orders of magnitude below Slater-RBM error at U = 10.The benchmark uses a 4 × 4 lattice with ˜N = 8.Source: A. Benchmarks in the square lattice · p. 6, Figure 3 · p. 6
- Energy-variance extrapolation supplies additional ground-state energy estimates because direct extrapolation of error to α →∞ is challenging.Source: A. Benchmarks in the square lattice · p. 6
- Increasing the number of hidden fermions improves accuracy more than increasing neural-network depth, which yields only marginal energy-error gains.Source: A. Benchmarks in the square lattice · p. 6
- Physically motivated rigid constraints yield only marginal accuracy improvement over Slater-Jastrow, while parametrizing the constraint is advantageous.Source: A. Benchmarks in the square lattice · p. 7
- At n = 5/8, the parametrized-constraint determinant has half the relative error reported for Ref.42 without symmetry projection, and projection further improves it.The comparison is for the 4 × 4 lattice; the projected state uses relevant symmetry subspaces.Source: Figure 3 · p. 6, A. Benchmarks in the square lattice · p. 7
- At half filling on square geometries, the hidden-fermion determinant energies agree better with AFQMC than the neural-network Jastrow results from Ref.9.The cited comparison covers increasingly larger system sizes.Source: A. Benchmarks in the square lattice · p. 7
B. Increasing system size and stripe order at 1/8 hole-doped rectangular geometries
The hidden fermion determinant state is tested on 1/8-doped 4×L rectangles, using a soft orbital constraint to guide stripe order. With periodic boundaries on both sides, it beats the Slater-Jastrow and neural network backflow ansätze; with open long-side boundaries, it does not outperform DMRG.
Setup and stripe guidance
- For 1/8-doped, zero-magnetization 4×L rectangles, the study targets λ = 8 stripes and guides the orbitals with a soft mean-field constraint.The constraint is optimized first, then merged into the variational orbitals for continued energy optimization.Source: B. Increasing system size and stripe… · p. 7
PBC-OBC results
- For PBC-OBC, hidden fermion determinant energies agree well with DMRG, but the best L = 16 energy lies between the DMRG stripe energies and does not outperform DMRG.At L = 8, tuning S stabilizes both half-filled and filled stripe configurations.Source: B. Increasing system size and stripe… · p. 7, Figure 4 · p. 8
PBC-PBC results
- On 4×4, relative error is reduced by over one order of magnitude versus neural network backflow; at 4×8 and 4×16, energy per site is lower than both comparators.The PBC-PBC comparison also includes the Slater-Jastrow ansatz and the 4×4 ED energy reference.Source: B. Increasing system size and stripe… · pp. 7–8, Figure 4 · p. 8
- In the 4×16 PBC-PBC state, hole-density maxima repeat every eight sites and coincide with antiferromagnetic domain walls.The staggered magnetization is modulated along the rectangle; these features agree with observations from other numerical methods.Source: B. Increasing system size and stripe… · p. 8, Figure 4 · p. 8
IV. CONCLUSIONS
The conclusions summarize the exactly constrained, neural-network-optimized hidden-fermion Slater determinant, its universality and variational accuracy for the square-lattice Hubbard model, and its links to other correlated wave functions. They also discuss comparisons between hidden- and visible-space states and examine how hidden-fermion count, network depth, and energy-variance extrapolation affect accuracy.
IV. CONCLUSIONS
- The neural-network-optimized ansatz achieves state-of-the-art variational accuracy for the square-lattice Hubbard ground state without relying on imposed symmetries, unlike standard approaches [9,42].The authors identify systems with few exploitable symmetries, molecular Hamiltonians, and disordered interacting-fermion models as potential applications.Source: §IV · p. 8
- The authors envision applying the approach to molecular Hamiltonians and disordered interacting-fermion models, which lack exploitable lattice or translational symmetries.They specifically discuss molecular ground-state calculations and fully connected lattices with random exchange interactions; the physical SU(2) case requires computational approaches [53].Source: §IV · p. 8
Appendix A: Universality of the hidden fermion determinant state in the lattice
- The hidden-fermion determinant is universal in a fixed-particle Fock subspace: an explicit construction matches arbitrary target amplitudes, while a universal function approximator can avoid combinatorially many hidden modes.The constraint function relates visible configurations to hidden configurations and is optimized together with the hidden-sector orbitals.Source: §IV · p. 8, Appendix A · p. 11
Appendix B: Configuration interaction wavefunctions from the hidden fermion determinant state
- With ˜N hidden fermions, the determinant can compactly represent configuration interaction with all single through ˜N-tuple excitations to the ˜N lowest virtual orbitals.For ˜N greater than 1, the construction constrains which CI coefficients can vary independently; position-dependent bosonic prefactors can add Jastrow-like factors.Source: Appendix B · p. 11, §3 · p. 13, §4 · pp. 13–14
1. Explicit construction of the Gutzwiller wave function in the Slater determinant hidden fermion state
- A particular choice of disjointly supported hidden and visible orbitals and constraint function reproduces the Gutzwiller projection of an arbitrary Slater determinant.The appendix also describes a Gutzwiller-inspired constraint based on the number of doubly occupied sites.Source: §1 · p. 14, a. Gutzwiller-inspired constraint… · p. 15
d. Benchmarks
- In the 4 × 4 Hubbard model at quarter occupation, Gutzwiller-inspired and unrestricted hidden-spin constraints have errors comparable to unrestricted Slater determinants with Gutzwiller and Jastrow factors, with marginal improvements at some U.The Jastrow-inspired constraint has error comparable to a single unrestricted Slater determinant; these results show that constraint choice matters for capturing correlated phenomena.Source: Figure 5 · p. 15, d. Benchmarks · p. 16
b. Unrestricted hidden spin ansatz
- The unrestricted hidden-spin ansatz uses indistinguishable hidden fermions and offers flexibility beyond Gutzwiller single-site correlations.It can reproduce the Gutzwiller ansatz for particular orbitals and is related to the hidden-spin formalism, which captures the Mott transition at mean-field level.Source: b. Unrestricted hidden spin ansatz · p. 15
d. Benchmarks
- At smaller U, correlated enlarged-space ansätze have lower energy error than the visible-space counterpart, while at larger U the correlated unrestricted hidden-spin ansatz notably outperforms it.The enlarged-space states use a complex E-RBM factor over visible and hidden occupations; the comparison also includes an unrestricted visible-space Slater determinant with a complex RBM factor.Source: d. Benchmarks · p. 16
- The number of constraint functions is doubly exponential, making brute-force search intractable despite the hidden-fermion formalism’s performance with carefully chosen constraints.Source: d. Benchmarks · p. 16
fermion determinant state
- Accuracy improves with network depth up to six hidden layers, but deeper optimization can become challenging and reach a mean-field local minimum; increasing hidden-fermion count lowers error at fixed depth.Source: fermion determinant state · p. 17
- A depth-2 network with eight hidden fermions gives the lowest error across the tested coupling strengths and is the minimal architecture satisfying the universal approximation theorem.This configuration was selected as the starting point for studying network width, which is its sole expressivity control parameter.Source: fermion determinant state · p. 17
Appendix E: Energy-variance extrapolation in Fig. 3 (a)
- Energy-variance extrapolation offers a better ground-state estimate from converged energies and variances, although extrapolating directly from neural-network complexity is not well motivated.The variational energy and variance show a linear trend; linear regression across network complexities yields an estimate from the zero-variance intercept.Source: Appendix E · p. 17
Appendix F: Energy benchmarks in the L × L Hubbard model at half filling.
At half filling, the hidden fermion determinant state keeps relative energy errors at or below O(10^-3) as square-lattice size grows, outperforming a reported neural-network Jastrow accuracy. The appendix also describes energy extrapolation, NetKet implementation, and converged energy benchmarks across several lattice settings.
Appendix F: Energy benchmarks in the L × L Hubbard model at half filling.
- Variational energies plotted against variance show an approximately linear trend; their zero-variance intercepts estimate the ground-state energy.Figure 7 presents separate coupling-constant panels and hidden-to-input unit ratios α.Source: Figure 7 · p. 18
- Relative energy errors stay at O(10^-3) or lower as L increases, and are at least twofold more accurate than the neural-network wave function reported in Ref.9.Figure 8 compares errors with AFQMC energies for 4 × 4, 6 × 6, and 8 × 8 lattices; errors increase with system size.Source: Appendix F · p. 17, Figure 8 · p. 18
Appendix G: First quantized fermions on the open-sourced library NetKet: a VMC library for distinguishable particles
- The NetKet implementation maps fermion configurations to local spin-model states, avoiding unphysical states in the Markov chain through valid initialization and exchange transitions.The local Hilbert space has dimension N+1; its states encode vertex locations or absence.Source: Appendix G · p. 18
Appendix H: Converged values of variational energies for different trial wave functions
- The appendix collects converged ground-state energies for the hidden fermion determinant state and other trial states, with exact-diagonalization benchmarks for smaller systems.It gives numerical values for the Hubbard model across the listed settings.Source: Appendix H · p. 19
1. 4 × 4 Hubbard model with n = 1/2
- For the 4 × 4 half-filled lattice, Table I reports parametrized-constraint HFDS and ED energies, while Table II compares other trial states and physically motivated constraints.Table II also includes HFDS variants with an added RBM factor in the augmented space.Source: §1 · p. 19, Table I · p. 19, Table II · p. 19
2. 4 × 4 Hubbard model with n = 5/8
- For the 4 × 4 lattice at n = 5/8, Table III reports fully parametrized HFDS energies, ED references, and energies after symmetry-subspace projections.The projections include C4 rotation, momentum K = 0 translation, and their intersection.Source: §2 · p. 20, Table III · p. 20
3. 4 × L Hubbard model with n = 7/8 and U = 8
- Table IV reports variational energies per site for 4 × L rectangles at n = 7/8 and U = 8, with periodic or open boundary conditions along the long side.Periodic boundary conditions are used along the short side.Source: §3 · p. 20, Table IV · p. 20
4. L × L Hubbard model with n = 1
- Table V reports HFDS energies per site for half-filled L × L lattices with periodic boundary conditions along one side and anti-periodic conditions along the other.The hidden-fermion count and hidden-unit densities are specified in the supplementary information.Source: §4 · p. 20, Table V · p. 21