Excited Pfaffians in Quantum Systems

• Excited Pfaffians are advanced quantum states that generalize Pfaffian wave functions to systematically represent orthogonal excited states in many-body systems.
• They integrate variational Monte Carlo methods with neural-network architectures to efficiently scale and accurately reproduce complex excitation spectra.
• Their applications span quantum chemistry, gauge theories, and topological phases, providing insights into phenomena like nonabelian statistics and fractional quantum Hall effects.

Excited Pfaffians are a class of quantum states and computational architectures that systematically generalize the concept of Pfaffian wave functions to excited states across a range of many-body contexts, including strongly correlated electrons, gauge theories, and neural quantum states. Their applications range from the simulation of atomic and molecular spectra using neural networks, to the study of excited sectors in lattice gauge theory, and the characterization of nonabelian excitations in topological phases such as the $\nu=5/2$ fractional quantum Hall effect. Excited Pfaffians enable efficient and scalable representation, sampling, and discrimination of orthogonal excited states, and provide unique insight into the structure and dynamics of quantum systems that exhibit pairing correlations or nontrivial topology.

1. Mathematical Foundation of the Pfaffian Ansatz

The Pfaffian is a generalization of the determinant for antisymmetric matrices, making it foundational in the construction of antisymmetric many-fermion wave functions. For $N_e$ electrons with coordinates $x = (r_1, ..., r_{N_e})$, the standard fermionic Pfaffian wave function takes the form: $\psi_{\text{Pf}}(x) = \mathrm{Pf}[A(x)],$ where $A(x) \in \mathbb{R}^{N_e \times N_e}$ is a learned skew-symmetric pairing matrix, often parameterized via neural networks or, in special cases, in terms of pairwise orbitals or projection operators. This structure ensures antisymmetry under exchange and allows the recovery of both mean-field Hartree–Fock (via determinants) and correlated-pairing states (as in BCS theory) as special cases (Gao et al., 15 Mar 2026).

Excited Pfaffian constructions extend this ansatz by introducing state-specific modifications—either by varying the antisymmetrizer block, by explicit operator insertion, or by basis expansion—to systematically build sets of orthogonal excited states.

2. Excited Pfaffians in Variational Monte Carlo and Neural Networks

The Excited Pfaffian architecture, as introduced in variational quantum Monte Carlo (VMC) contexts, enables simultaneous and efficient modeling of multiple excited states with shared neural-network structure. The core innovation is to express $S$ orthogonalized excited states as: $$\psi_s(x) = \sum_{k=1}^M \mathrm{Pf}[\Phi_k(x) A_{s,k} \Phi_k(x)^T],$$ where the orbital matrices $\Phi_k(x)$ and corresponding embedding network are globally shared, and the only state-specific parameters are the low-rank antisymmetrizer blocks $A_{s,k}$. State orthogonality is enforced by maintaining $\det(A_s^T A_t) \approx \delta_{st}$. In the Hartree–Fock limit, this construction reduces to use of orbital-selection matrices, yielding determinants as a special case (Gao et al., 15 Mar 2026).

Parameter sharing across states yields asymptotically constant evaluation cost in $N_e$0, contrasting with the linear or worse scaling in conventional approaches—empirically, scaling as $N_e$1 up to $N_e$2. This structure is particularly amenable to large-scale VMC optimization via shared-sample multi-state importance sampling.

3. Multi-State Importance Sampling and Variational Optimization

Robust training of multiple excited states with strong orthogonality is critical in quantum state simulation. Conventional approaches require independent sampling from each state's $N_e$3 measure, causing variance in overlap estimates to scale as $N_e$4. The Multi-State Importance Sampling (MSIS) protocol pools samples from all $N_e$5 states via the mixture

$N_e$6

enabling unbiased and variance-reduced estimation of pairwise overlaps: $N_e$7 The variance reduction is exponential in $N_e$8 and independent of state count, yielding efficient scaling and high accuracy in orthogonality-enforced VMC (Gao et al., 15 Mar 2026).

The excitation manifolds are optimized by a composite loss comprising statewise energy minimization, explicit overlap penalties, and (as needed) spin-purity enforcement via "spin snapping" terms.

4. Excited Pfaffians in Fractional Quantum Hall Systems

In the context of the fractional quantum Hall effect (FQHE) at filling fraction $N_e$9, the Moore–Read (Pfaffian) state provides a paradigmatic example. The ground-state wave function is

$x = (r_1, ..., r_{N_e})$0

encoding p-wave pairing of composite fermions (Wojs et al., 2010).

Excited Pfaffian states in this context include:

• Neutral fermion (magnetoroton) modes, constructed by breaking a pair and generating quasiparticle–quasihole excitations;
• Charged quasihole and quasiparticle states, formed by modifying pairwise correlations and inserting analytic factors tied to additional charge coordinates.

Exact diagonalization in the presence of Landau level mixing demonstrates that such Pfaffian-based excitations accurately describe experimental spectra and support the emergence of nonabelian statistics via Majorana zero modes (Wojs et al., 2010).

5. Excited Pfaffian Sectors in SO(2N) Gauge Theories

In lattice gauge theory, generalized Pfaffian operators offer access to novel excited-state sectors orthogonal to those built from trace operators. In SO(2N) gauge groups, a closed path $x = (r_1, ..., r_{N_e})$1 defines a parallel transport operator $x = (r_1, ..., r_{N_e})$2; the Pfaffian is constructed from $x = (r_1, ..., r_{N_e})$3 such matrices: $x = (r_1, ..., r_{N_e})$4 producing gauge-invariant states which, for small $x = (r_1, ..., r_{N_e})$5, match spectra not accessible via traces—for example, negative $x = (r_1, ..., r_{N_e})$6-parity glueballs in SO(6) corresponding to SU(4) negative-$x = (r_1, ..., r_{N_e})$7 states (Teper, 2018).

The excited Pfaffian spectrum features its own hierarchy, e.g., degenerate but orthogonal multiplets in SO(4) (mirroring SU(2)$x = (r_1, ..., r_{N_e})$8SU(2)), and a sequence of increasing masses and string tensions as $x = (r_1, ..., r_{N_e})$9 grows, with the Pfaffian sector decoupling in the large-$\psi_{\text{Pf}}(x) = \mathrm{Pf}[A(x)],$0 limit.

6. Numerical Results and Applications

Excited Pfaffian frameworks have demonstrated leading performance in several quantum benchmarks:

• On the carbon dimer ($\psi_{\text{Pf}}(x) = \mathrm{Pf}[A(x)],$1) potential energy curve, a jointly trained Excited Pfaffian model reproduces 12 singlet/triplet states at up to 200$\psi_{\text{Pf}}(x) = \mathrm{Pf}[A(x)],$2 reduced computational cost over natural excited-state (NES) baselines, using 40$\psi_{\text{Pf}}(x) = \mathrm{Pf}[A(x)],$3 fewer samples (Gao et al., 15 Mar 2026).
• For the beryllium atom, all 33 excited states up to ionization are reproduced with sub-millihartree error within 231 A100 GPU-hours—over two orders of magnitude faster than NES.
• Cross-molecule models trained on up to 12 distinct molecules yield excitation energies matching or surpassing prior VMC and experiment, with 15–60$\psi_{\text{Pf}}(x) = \mathrm{Pf}[A(x)],$4 sample reduction.

In the lattice context, explicit extraction of excited Pfaffian glueball and string spectra affirms the physical significance of these sectors in low-$\psi_{\text{Pf}}(x) = \mathrm{Pf}[A(x)],$5 SO(2N) gauge theory (Teper, 2018).

7. Physical Significance and Generalization

Excited Pfaffian ansätze constitute a unifying framework for constructing and analyzing multi-state quantum wave functions. Their structural economy and orthogonality properties enable both state-of-the-art quantum simulation (including generalization across structures and states) and new insight into the symmetry and topological structure of many-body spectra. They resolve longstanding puzzles in gauge theory spectra (SO(2N) versus SU(N)), provide efficient neural architectures for broad quantum chemistry applications, and clarify the robustness and physical observability of nonabelian statistics in topological phases (Wojs et al., 2010, Teper, 2018, Gao et al., 15 Mar 2026).