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Variational solutions to fermion-to-qubit mappings in two spatial dimensions
Jannes Nys, Giuseppe Carleo
TL;DR
Higher-dimensional fermion-to-qubit mappings introduce auxiliary degrees of freedom whose constraints complicate variational simulations. The paper develops a variational Monte Carlo framework with exact constraint-solving procedures and applies it to the 2D t-V model, recovering ground-state and low-energy spectral information with neural-network quantum states.
Problem
Higher-dimensional local fermion-to-qubit mappings require auxiliary degrees of freedom and stringent constraints that are difficult to satisfy in practical variational calculations.
Source: §1 · pp. 1–2Method
The paper uses variational Monte Carlo with exact sampling and wave-function constructions to satisfy parity and Gauss-law constraints without projection operators.
Source: §3 · p. 4, §3.1 · p. 4, §3.2 · p. 5Results
Takeaways & Limitations
Exact constraint satisfaction and explicit fermionic symmetries make variational spin-state methods applicable to the bosonized 2D fermionic system.
Source: §6 · pp. 8–9Although the constraints are local, satisfying them can introduce long-range correlations through the wave function’s sign structure.
Source: §3.4 · p. 6
Abstract
from arXiv · showhide
Through the introduction of auxiliary fermions, or an enlarged spin space, one can map local fermion Hamiltonians onto local spin Hamiltonians, at the expense of introducing a set of additional constraints. We present a variational Monte-Carlo framework to study fermionic systems through higher-dimensional (>1D) Jordan-Wigner transformations. We provide exact solutions to the parity and Gauss-law constraints that are encountered in bosonization procedures. We study the $t$-$V$ model in 2D and demonstrate how both the ground state and the low-energy excitation spectra can be retrieved in combination with neural network quantum state ansatze.
1 Introduction
Higher-dimensional fermion-to-spin mappings can preserve locality by enlarging the Hilbert space, but the resulting auxiliary degrees of freedom must satisfy stringent constraints. This work develops a variational approach for solving those constraints and studying fermionic systems with neural-network quantum states.
Background
- Higher-dimensional Jordan-Wigner strings become increasingly non-local, making numerical studies of spatially local fermionic systems more difficult.In one dimension, strings cancel for local even-parity operators; this cancellation generally fails in higher dimensions.Source: §1 · p. 1
Related approaches
- Local fermion-to-qubit mappings enlarge the Hilbert space and require additional gauge or parity constraints to remove unphysical sectors.Auxiliary modes store parity near interaction terms, while local constraints restrict states to the physical sector.Source: §1 · p. 2
Problem
- The practical application of higher-dimensional bosonization remains difficult because variational calculations must satisfy the auxiliary constraints efficiently.The paper focuses on classical variational simulation using spin-based many-body states.Source: §1 · p. 2
Approach
- The proposed framework factorizes the wave function into an exact constraint solution and a physical component optimized variationally.Neural-network quantum states parameterize the physical wave function, while locality preserves the transformed Hamiltonian’s symmetries.Source: §1 · p. 2
Motivation
- Locality-preserving mappings retain fermionic symmetries and provide access to low-energy excitation spectra in addition to ground-state energies.The mapping is designed for variational simulation of fermionic systems on classical computers.Source: §1 · p. 2
2 Bosonization
The paper bosonizes the 2D t-V model using physical and auxiliary qubits, then imposes local Gauss-law, boundary, parity, and particle-number constraints. These constraints reduce the enlarged spin Hilbert space to the physical fermionic sector while retaining the model’s symmetries.
2 Bosonization
- The 2D square lattice hosts physical and auxiliary qubits at each site, with the t-V model defined using fermionic occupation operators.The construction uses periodic lattice coordinates and restricts the Hamiltonian to even-parity fermionic operators.Source: §2 · p. 3
2.1 Local constraints
- The auxiliary system obeys a local Gauss-law constraint that links its plaquette operators to the physical configuration.For each physical configuration, the auxiliary system corresponds to an eigenstate of a dynamical Wen plaquette model.Source: §2.1 · p. 3
- The local constraint is kinematic, depending on lattice topology rather than the Hamiltonian being studied.Its structure resembles a Chern-Simons Gauss law or flux attachment and can produce topologically degenerate auxiliary sectors.Source: §2.1 · p. 3
2.2 Parity constraints
- The construction restricts the system to even-by-even square tori and imposes Wilson-loop identities associated with fermionic boundary conditions.These non-contractable-loop constraints supplement the local Gauss law after bosonization.Source: §2.2 · pp. 3–4
- Fixing the fermion number adds a particle-number constraint, while the chemical potential can be set to zero because it contributes only a constant energy.The parity and boundary constraints are expressed as spin-operator identities in the bosonized system.Source: §2.2 · p. 4
3 Variational Monte Carlo approach
The paper develops variational Monte Carlo methods for bosonized fermionic systems while enforcing parity and Gauss-law constraints. It introduces sampling and wave-function constructions that retain physicality and support neural-network variational states.
3 Variational Monte Carlo approach
- The framework targets ground and low-lying excited states by minimizing a parametrized wave function's variational energy with Markov-chain sampling.Samples must remain within the constrained Hilbert space.Source: §3 · p. 4
3.1 Solving constraints in VMC
- Constraint-preserving Monte Carlo updates satisfy diagonal parity and number restrictions, while initial samples are constructed to obey the required global constraints.The updates are based on the free-fermion Hamiltonian, and periodicity constraints determine remaining auxiliary qubits.Source: §3.1 · p. 4
- Gauss-law constraints require wave-function enforcement because sampling alone cannot satisfy their non-diagonal operators, and soft energy penalties do not reliably recover the physical spectrum.The Gauss-law projection is block diagonal but exponentially large and induces non-local auxiliary effects.Source: §3.1 · pp. 4–5
3.2 Canonical sample reduction
- Canonical sample reduction repeatedly applies constraint operators to map each auxiliary configuration to a reduced sample, with O(N) processing per configuration.A fixed site ordering determines the sequence of reductions and associated sign factors.Source: §3.2 · p. 5
3.3 Doubly canonical sample reduction / 3.4 Vacuum reduction
- Doubly canonical reduction incorporates symmetry operations before reduction, whereas vacuum reduction infers signs from the vacuum and enables determinant-free neural-network amplitudes.The resulting ansatz exactly fulfills the Gauss law, allowing an RBM to represent the physical variational part without explicit antisymmetrization.Source: §3.3 · p. 6, §3.4 · p. 6
4 Quantum state symmetries
The symmetry representation combines bare fermionic transformations with local auxiliary-mode transformations, enabling variational states restricted to chosen irreducible representations. The construction replaces non-local parity factors with local operators and can be evaluated in O(N) time.
4 Quantum state symmetries
- Symmetry transformations are decomposed as U_T = V_T T^b, combining a bare transformation with a tensor product of local auxiliary-mode operators.The auxiliary operators replace the non-local parity factors that arise in Jordan-Wigner transformations.Source: §4 · p. 6
- Character-weighted symmetry operations construct quantum-state ansatze within a chosen irreducible representation of the symmetry group.The framework determines the auxiliary transformations for translations, rotations, and reflections in the C4v group.Source: §4 · p. 6
- Figure 2 compares canonical, doubly canonical, and vacuum reductions against Exact Diagonalization for 2×L and 4×L lattices.The plotted cases use V/t = 0.01 for 2×L and V/t = 0.1 for 4×L.Source: Figure 2 · p. 7
- The local parity representation avoids sorting configurations, because auxiliary modes encode the parity of each symmetry transformation.This makes the symmetry methods O(N) rather than requiring the more costly parity evaluations used by other determinant-free approaches.Source: §4 · p. 7
5 Results
The results show that constraint-solving strategies differ substantially in variational tractability, while vacuum reduction closely reproduces exact ground-state energies. Symmetry-restricted ansatze further improve optimization and recover low-energy spectra in the 4×4 system.
5 Results
- Canonical and doubly canonical reductions are exact constraint solutions but are difficult to optimize because their auxiliary factors impose non-trivial sign structures.The doubly canonical method does not remove sign-learning challenges for configurations outside the symmetry-related samples or when the ground state lies in a nontrivial irrep.Source: §5 · p. 7
- Figure 3 compares ED energy levels with VMC optimization inside translation irreps and without symmetry restrictions for Nf ∈ {4, 6, 8} on a 4×4 lattice.The comparison spans V/t ∈ {10^-2, 10^-1, 1, 10}.Source: Figure 3 · p. 8
- Vacuum reduction closely matches Exact Diagonalization and the snaked Jordan-Wigner approach, outperforming canonical reduction increasingly on larger 4×L lattices.Canonical and vacuum methods are similar on 2×L lattices, but vacuum reduction yields lower energies for 4×L systems at large L.Source: §5 · pp. 7–8
- Symmetrizing the vacuum-reduction ansatz removes an optimization local minimum and improves ground-state estimates, especially when the energy gap is small.The unsymmetrized ansatz is accurate except for Nf = 6, whose ground state has a challenging sign structure and small relevant-sector gap.Source: §5 · p. 8
6 Conclusions
The paper introduces exact, projection-free solutions of parity and Gauss-law constraints for local fermion-to-qubit mappings and applies them to the 2D t-V model. The resulting symmetry-aware neural-network ansatze reproduce ground states and access low-energy spectra, with extensions suggested by the topology-dependent constraints.
6 Conclusions
- The study develops multiple exact approaches for satisfying parity and Gauss-law constraints without applying projection operators.These constraints are required to remove unphysical sectors introduced by local fermion-to-qubit mappings.Source: §6 · pp. 8–9
- Applying the method to the 2D square-lattice t-V model shows that constraint handling directly affects reliable ground-state representation.Imposing a sign structure on the vacuum simplifies the variational challenge, although the Gauss constraint retains non-local effects.Source: §6 · p. 9
- Keeping fermionic symmetries manifest enables restriction of the variational ansatz to chosen irreducible representations and recovery of low-energy spectra with neural-network quantum states.The conclusion links symmetry restriction to the study of excited states rather than only ground-state energies.Source: §6 · p. 9
- Because the Gauss constraint depends only on lattice topology, the approach is positioned for application to other square-lattice Hamiltonians and future extensions to other lattices and dimensions.The paper also identifies equivalent bosonic systems as a future direction.Source: §6 · p. 9
B Details of the bosonization procedure
The bosonization procedure introduces Majorana operators to express the t-V model in a form suitable for a local fermion-to-qubit mapping. The supplied details identify the operator representation and the Hamiltonian terms used in the construction.
B Details of the bosonization procedure
- The procedure introduces two Majorana fermion operators, γ1 and γ2, and states their commutation relations for the t-V model.The derivation is restricted to the even fermion sector.Source: B Details of the bosonization procedure · p. 10
- The Hamiltonian terms relevant to the t-V model are rewritten in terms of the Majorana operators.This provides the operator-level starting point for the subsequent bosonization mapping.Source: B Details of the bosonization procedure · p. 10
B.1 Constructing bosonic operators
The construction introduces auxiliary Majorana modes that separate internal and spatial degrees of freedom, then combines them into bosonic operators whose site-local pairing avoids non-local Jordan-Wigner strings.
B.1 Constructing bosonic operators
- Auxiliary η and χ Majorana modes separate internal degrees of freedom from spatial edge degrees of freedom on the square lattice.Two η modes describe spinless-fermion internal structure, while four χ modes correspond to the square lattice’s edge attachments.Source: §B.1 · p. 11
- The resulting Θ, Λ, and Υ operators are bosonic because they commute at different lattice sites.Source: §B.1 · p. 11
- Pairing operators at the same lattice site eliminates non-local Jordan-Wigner strings after mapping the Hamiltonian to qubits.The construction inserts auxiliary operators so the relevant expressions contain only same-site operator pairs.Source: §B.1 · p. 11
- The ordering and edge directions of the χ modes determine the signs of the constructed operator terms.The conventions are illustrated in Figure 4 and enter the signs of Eq. (25).Source: §B.1 · p. 11, Figure 4 · p. 12
B.2 Parton decomposition
The parton decomposition enlarges each site’s Hilbert space with auxiliary fermions, then uses an on-site Jordan-Wigner transformation and parity constraints to obtain a reduced qubit representation.
B.2 Parton decomposition
- The parton Hilbert space is 2^3-dimensional at each site, and because there are 3 fermion modes {a†, b†, c†}, the on-site representation requires 3 qubits.Source: §B.2 · p. 12, §B.3 · p. 12
B.3 On-site Jordan-Wigner transformation
- The on-site Jordan-Wigner transformation reproduces the required same-site anticommutation relations without introducing intersite Jordan-Wigner strings.The bosonic operators already commute across distinct sites, so only their on-site parton components require Jordan-Wigner treatment.Source: §B.3 · p. 12
B.4 Constraints
- An on-site parity constraint removes one redundant auxiliary degree of freedom because the parity operator commutes with the Hamiltonian.The third spin can therefore be absorbed into the remaining spins, yielding the Hamiltonian expression in Eq. (2).Source: §B.4 · p. 13
B.5 Constraints
The bosonized construction requires local Gauss-law and global Wilson-loop constraints, while also specifying lattice symmetries, optimization settings, boundary handling, and an odd-parity modification.
B.5 Constraints
- The auxiliary system is constrained by a Gauss-law condition derived from plaquette identities that are trivial in the fermionic representation.The same construction produces constraints associated with non-contractable Wilson loops on even-by-even tori.Source: §B.5 · p. 13
C Symmetries
- The mapping specifies transformations for lattice rotations, mirrors, and translations through corresponding operations on the auxiliary degrees of freedom.Source: C Symmetries · p. 14
D Optimization details
- Variational optimization uses MCMC with SGD for 12.5k steps, 4096 samples per step, and 16–64 parallel walkers.Gradients are obtained by automatic differentiation and decorrelated with Stochastic Reconfiguration.Source: D Optimization details · p. 14
E Open boundary conditions
- Open boundaries require fewer spins at edge sites because their coordination numbers are lower, allowing unused spins to be decoupled.The method is stated to generalize to Gauss constraints of the form used in Eq. (3).Source: E Open boundary conditions · p. 14
F Odd fermion sector
- The derivation and experiments target even fermion parity; odd-parity states require a lattice defect and reversed edge direction, producing a Jordan-Wigner-string-like effect.Source: F Odd fermion sector · p. 15
G Scaling behavior
Local fermion-to-qubit mappings make Hamiltonian-term evaluation constant-time, while probability-amplitude evaluation and constraint-aware sampling scale linearly per sample.
G Scaling behavior
- Hamiltonian terms can be evaluated in constant time with the local mapping, instead of scaling with Jordan-Wigner-string length.Source: G Scaling behavior · p. 15
- Probability-amplitude evaluation, vacuum preparation, and the bosonized operator procedure each scale as O(N) per sample.For the bosonized operator, any path between fermion operators may be used because the Gauss constraint makes the path irrelevant.Source: G Scaling behavior · p. 15
H Pseudocodes
- Initial samples satisfy fixed fermion-number constraints by randomly permuting physical occupations and inferring auxiliary qubits from periodicity constraints.Source: H Pseudocodes · p. 15
- Canonical sample reduction processes sites in a specified ordering and applies auxiliary plaquette operations when the corresponding auxiliary qubit is occupied.Source: H Pseudocodes · p. 16