Structurally Sparse Qubit Hamiltonians from Spectral Graph Theory

Josh Loutey

Published Jul 9, 2026• 0 citations

Abstract

We show that qubit Hamiltonians derived from spectral graph theory lattices exhibit structurally favorable Pauli decompositions for quantum simulation. After Jordan–Wigner transformation, the GeoVac lattice Hamiltonian yields $O(Q^{3.15})$ Pauli string terms in the number of qubits $Q$, compared to $O(Q^{4+})$ for conventional Gaussian-basis Hamiltonians (our own log-log fit of the published Pauli counts of Trenev et al. [ref] gives $Q^{4.25}$ for LiH and $Q^{3.92}$ for H$_2$O, spanning $Q^{3.9}$–$Q^{4.3}$). The advantage originates in angular momentum selection rules—enforced by Gaunt integrals at the two-electron integral level—which cause the electron repulsion integral (ERI) tensor density to decay as $\sim 1/M$ (empirically $\approx M^{-0.85}$–$M^{-1}$) with the number of spatial orbitals $M$, while Gaussian ERI tensors remain near-dense at $O(1)$. Beyond term counts, we quantify two operationally relevant cost metrics: the number of qubitwise-commuting (QWC) measurement groups scales as $O(Q^{3.36})$, and the Pauli 1-norm $\lambda = \sum_i |c_i|$—which controls both Trotter step counts and qubitization query complexity—scales as $O(Q^{1.69})$ with $R^2 = 0.997$. The sub-quadratic 1-norm scaling is the key result for fault-tolerant simulation: it implies that Trotter circuit depth grows far more slowly with system size than term counts alone would suggest. All Gaussian baselines are validated against published integral values (Szabo and Ostlund for H$_2$; analytical STO for He). For composed multi-center systems, the pseudopotential approximation of the composed natural geometry framework [ref] yields block-diagonal electron repulsion integrals with a universal Pauli scaling of $O(Q^{2.5})$, independent of molecular topology (exponent spread $0.02$ across three molecules). At equal qubit counts, this represents two or more orders of magnitude fewer Pauli terms than Gaussian baselines (published counts of Trenev et al. [ref], $Q^{3.9}$–$Q^{4.3}$ span fitted by us)—an advantage that grows with system size due to the $\sim 1.5$ exponent gap. These comparisons are at matched qubit counts, not matched accuracy; the composed basis carries classical equilibrium-geometry errors of 5–26% (LiH/BeH$_2$/H$_2$O; Sec. (ref)), making the framework optimal for fixed-geometry quantum simulation rather than geometry optimisation. Across the composed molecular library (35 molecules spanning $Z = 1$–56, H through Ba, including the first transition series ScH–ZnH), the composed Pauli count is exactly linear in the qubit count: $N_{\mathrm{Pauli}} = 11.10 × Q$ (exact, non-identity terms; $d$-orbital blocks give 9.23). These counts are realized under a pair-diagonal angular approximation ($m_a=m_c$, $m_b=m_d$) adopted deliberately for sparsity; the exact global-$M_L$ Coulomb rule (used in the framework's precision-physics paths, and considered but left on the table here) gives $\approx 2.5×$ more terms with the same scaling, under which the $d$-block coefficient is in fact higher, not lower (Sec. (ref)). GeoVac Paper 14. Part of the GeoVac research corpus (quantum computing group). Code, tests, and the full paper series: https://github.com/jloutey-hash/geovac. Corpus archive DOI: 10.5281/zenodo.20482394. Corpus version at deposit: v4.75.0.

DOI: 10.5281/zenodo.21286778

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Josh Loutey

Published Jul 9, 2026• 0 citations

Publisher

Abstract

O(Q^{3.15})

Pauli string terms in the number of qubits

Q

, compared to

O(Q^{4+})

for conventional Gaussian-basis Hamiltonians (our own log-log fit of the published Pauli counts of Trenev et al. [ref] gives

Q^{4.25}

for LiH and

Q^{3.92}

for H

_2

O, spanning

Q^{3.9}

–

Q^{4.3}

). The advantage originates in angular momentum selection rules—enforced by Gaunt integrals at the two-electron integral level—which cause the electron repulsion integral (ERI) tensor density to decay as

\sim 1/M

(empirically

\approx M^{-0.85}

–

M^{-1}

) with the number of spatial orbitals

M

, while Gaussian ERI tensors remain near-dense at

O(1)

. Beyond term counts, we quantify two operationally relevant cost metrics: the number of qubitwise-commuting (QWC) measurement groups scales as

O(Q^{3.36})

, and the Pauli 1-norm

\lambda = \sum_i |c_i|

—which controls both Trotter step counts and qubitization query complexity—scales as

O(Q^{1.69})

with

R^2 = 0.997

. The sub-quadratic 1-norm scaling is the key result for fault-tolerant simulation: it implies that Trotter circuit depth grows far more slowly with system size than term counts alone would suggest. All Gaussian baselines are validated against published integral values (Szabo and Ostlund for H

_2

; analytical STO for He). For composed multi-center systems, the pseudopotential approximation of the composed natural geometry framework [ref] yields block-diagonal electron repulsion integrals with a universal Pauli scaling of

O(Q^{2.5})

, independent of molecular topology (exponent spread

0.02

across three molecules). At equal qubit counts, this represents two or more orders of magnitude fewer Pauli terms than Gaussian baselines (published counts of Trenev et al. [ref],

Q^{3.9}

–

Q^{4.3}

span fitted by us)—an advantage that grows with system size due to the

\sim 1.5

exponent gap. These comparisons are at matched qubit counts, not matched accuracy; the composed basis carries classical equilibrium-geometry errors of 5–26% (LiH/BeH

_2

_2

O; Sec. (ref)), making the framework optimal for fixed-geometry quantum simulation rather than geometry optimisation. Across the composed molecular library (35 molecules spanning

Z = 1

–56, H through Ba, including the first transition series ScH–ZnH), the composed Pauli count is exactly linear in the qubit count:

N_{\mathrm{Pauli}} = 11.10 × Q

(exact, non-identity terms;

d

-orbital blocks give 9.23). These counts are realized under a pair-diagonal angular approximation (

m_a=m_c

m_b=m_d

) adopted deliberately for sparsity; the exact global-

M_L

Coulomb rule (used in the framework's precision-physics paths, and considered but left on the table here) gives

\approx 2.5×

more terms with the same scaling, under which the

d

-block coefficient is in fact higher, not lower (Sec. (ref)). GeoVac Paper 14. Part of the GeoVac research corpus (quantum computing group). Code, tests, and the full paper series: https://github.com/jloutey-hash/geovac. Corpus archive DOI: 10.5281/zenodo.20482394. Corpus version at deposit: v4.75.0.

DOI: 10.5281/zenodo.21286778

Related papers

Structurally Sparse Molecular Hamiltonians for Near-Term Quantum Simulation

Josh Loutey

We present a library of qubit Hamiltonians for molecular quantum simulation that are structurally sparser than conventional Gaussian-basis alternatives by one to three orders of magnitude. The Hamiltonians are constructed from a spectral graph theory framework (GeoVac) that exploits natural coordinate separations in molecular electronic structure, producing block-diagonal electron repulsion integrals and Gaunt-enforced selection rules that persist through Jordan-Wigner encoding. For the composed architecture (PK-partitioned), the composed LiH Hamiltonian is near parity with minimal-basis STO-3G on Pauli count (334 vs. STO-3G's 276 reduced / 907 raw) while using more qubits; the decisive resource advantages are the $O(Q^{2.5})$ Pauli scaling and a 190 $×$ reduction versus cc-pVDZ. The balanced coupled architecture (PK-free) binds LiH with an FCI minimum at a computed $R_{\mathrm{eq}} = 3.227$ bohr (7.0% above the experimental 3.015 bohr) at $n_{\max}=2$ (878 Pauli terms, 30 qubits); a single-point evaluation at the experimental geometry reaches 0.20% energy error at $n_{\max}=3$ (84 qubits) when solved exactly. Chemistry-accuracy scope at $n_{\max}=2$ is row-conditional: first-row hydrides (LiH, BeH $_2$ , H $_2$ O, $\ldots$ ) bind with documented potential-energy-surface (PES) topology, while second-row hydrides (NaH and below) exhibit monotone overattraction under every tested configuration — frozen-core balanced, multi-zeta valence, screened cross-center, explicit-core Hartree–Fock — with no interior PES minimum at the qubit-Hamiltonian level. We document this scope boundary empirically in \S(ref); the framework's primary value proposition is structural sparsity with quantitative truncation-convergence rates (Paper 38 state-space GH convergence), not energy-accuracy parity with CCSD(T) on second-row systems. The hamiltonian() registry covers 37 systems (35 composed molecules spanning $Z=1$ –56, H through Ba, plus the He and H $_2$ reference atoms), including multi-center systems (CO, N $_2$ , F $_2$ ), mixed frozen-core molecules (NaCl), and the full first transition series as hydrides (ScH through ZnH), with Pauli scaling $O(Q^{2.5})$ confirmed universal across all three rows, compared to $O(Q^{3.9–4.3})$ for Gaussian baselines. The composed Pauli count is exactly linear in qubit count (coefficient $11.10$ , exact, for main-group molecules; $9.23$ for $d$ -orbital blocks under the pair-diagonal ERI approximation (Sec. (ref))—the exact global- $M_L$ rule makes the $d$ -block coefficient higher, not lower), so adding atoms to a molecule increases quantum cost linearly. Second- and third-row molecules use analytical frozen cores ([Ne], [Ar], [Ar]3 $d^{10}$ ); the frozen-core energy enters only through the identity operator and does not affect quantum measurement cost. Isostructural molecules with frozen cores produce identical Pauli counts across periodic table rows. The Hamiltonians are available as a Python library with Qiskit, OpenFermion, and PennyLane export. GeoVac Paper 20. Part of the GeoVac research corpus (quantum computing group). Code, tests, and the full paper series: https://github.com/jloutey-hash/geovac. Corpus archive DOI: 10.5281/zenodo.20482394. Corpus version at deposit: v4.75.0.We present a library of qubit Hamiltonians for molecular quantum simulation that are structurally sparser than conventional Gaussian-basis alternatives by one to three orders of magnitude. The Hamiltonians are constructed from a spectral graph theory framework (GeoVac) that exploits natural coordinate separations in molecular electronic structure, producing block-diagonal electron repulsion integrals and Gaunt-enforced selection rules that persist through Jordan-Wigner encoding. For the composed architecture (PK-partitioned), the composed LiH Hamiltonian is near parity with minimal-basis STO-3G on Pauli count (334 vs. STO-3G's 276 reduced / 907 raw) while using more qubits; the decisive resource advantages are the $O(Q^{2.5})$ Pauli scaling and a 190$×$ reduction versus cc-pVDZ. The balanced coupled architecture (PK-free) binds LiH with an FCI minimum at a computed $R_{\mathrm{eq}} = 3.227$ bohr (7.0% above the experimental 3.015 bohr) at $n_{\max}=2$ (878 Pauli terms, 30 qubits); a single-point evaluation at the experimental geometry reaches 0.20% energy error at $n_{\max}=3$ (84 qubits) when solved exactly. Chemistry-accuracy scope at $n_{\max}=2$ is row-conditional: first-row hydrides (LiH, BeH$_2$, H$_2$O, $\ldots$) bind with documented potential-energy-surface (PES) topology, while second-row hydrides (NaH and below) exhibit monotone overattraction under every tested configuration — frozen-core balanced, multi-zeta valence, screened cross-center, explicit-core Hartree–Fock — with no interior PES minimum at the qubit-Hamiltonian level. We document this scope boundary empirically in \S(ref); the framework's primary value proposition is structural sparsity with quantitative truncation-convergence rates (Paper 38 state-space GH convergence), not energy-accuracy parity with CCSD(T) on second-row systems. 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Jul 9, 20260 citations

Structural Sparsity for Quantum Simulation: A Synthesis of the GeoVac Quantum-Computing Arc

Josh Loutey

The Geometric Vacuum (GeoVac) framework discretizes electronic structure on natural geometries, producing sparse graph Hamiltonians whose angular content is fixed by Gaunt selection rules. This synthesis assembles the framework's quantum-computing arc—the four Group 4 papers—into a single resource-oriented narrative. The spine is: the qubit encoding and its Pauli/ $1$ -norm scaling [ref]; the benchmarked resource library across three periodic-table rows [ref]; the $S_N$ representation-theoretic foundation that classifies the atomic building blocks [ref]; and the nuclear shell-model extension that delineates exactly which part of the framework transfers [ref]. The unifying value proposition is structural, not accuracy-based: after the Jordan–Wigner transformation the GeoVac lattice Hamiltonian yields $O(Q^{3.15})$ Pauli terms for atoms and $O(Q^{2.5})$ for composed multi-center molecules (versus $O(Q^{3.9–4.3})$ for Gaussian baselines), with a sub-quadratic Pauli $1$ -norm $\lambda = O(Q^{1.69})$ ( $R^2=0.997$ ) that controls Trotter depth and qubitization query cost. Two disciplines run through the arc. First, we separate the robust scaling exponents—unchanged under the choice of electron-repulsion selection rule—from the absolute multipliers, which depend on a stated pair-diagonal angular approximation and are disclosed as such. Second, every comparison is made at matched qubit count, not matched accuracy: the composed basis carries classical equilibrium-geometry errors of $5$ – $26%$ , so the framework is positioned for fixed-geometry NISQ/VQE simulation (Pauli count, QWC measurement groups), not for fault-tolerant geometry optimisation. The nuclear extension is honest in the same way: its binding energies are encoding-validation benchmarks far from experiment, and the Fock projection rigidity theorem states precisely that the conformal $S^3$ structure does not transfer to the harmonic-oscillator basis—only the angular machinery and the qubit encoding do. GeoVac synthesis document. Part of the GeoVac research corpus (synthesis / reading guide group). Code, tests, and the full paper series: https://github.com/jloutey-hash/geovac. Corpus archive DOI: 10.5281/zenodo.20482394. Corpus version at deposit: v4.75.0.The Geometric Vacuum (GeoVac) framework discretizes electronic structure on natural geometries, producing sparse graph Hamiltonians whose angular content is fixed by Gaunt selection rules. This synthesis assembles the framework's quantum-computing arc—the four Group 4 papers—into a single resource-oriented narrative. The spine is: the qubit encoding and its Pauli/$1$-norm scaling [ref]; the benchmarked resource library across three periodic-table rows [ref]; the $S_N$ representation-theoretic foundation that classifies the atomic building blocks [ref]; and the nuclear shell-model extension that delineates exactly which part of the framework transfers [ref]. The unifying value proposition is structural, not accuracy-based: after the Jordan–Wigner transformation the GeoVac lattice Hamiltonian yields $O(Q^{3.15})$ Pauli terms for atoms and $O(Q^{2.5})$ for composed multi-center molecules (versus $O(Q^{3.9–4.3})$ for Gaussian baselines), with a sub-quadratic Pauli $1$-norm $\lambda = O(Q^{1.69})$ ($R^2=0.997$) that controls Trotter depth and qubitization query cost. Two disciplines run through the arc. First, we separate the robust scaling exponents—unchanged under the choice of electron-repulsion selection rule—from the absolute multipliers, which depend on a stated pair-diagonal angular approximation and are disclosed as such. Second, every comparison is made at matched qubit count, not matched accuracy: the composed basis carries classical equilibrium-geometry errors of $5$–$26%$, so the framework is positioned for fixed-geometry NISQ/VQE simulation (Pauli count, QWC measurement groups), not for fault-tolerant geometry optimisation. The nuclear extension is honest in the same way: its binding energies are encoding-validation benchmarks far from experiment, and the Fock projection rigidity theorem states precisely that the conformal $S^3$ structure does not transfer to the harmonic-oscillator basis—only the angular machinery and the qubit encoding do. GeoVac synthesis document. Part of the GeoVac research corpus (synthesis / reading guide group). Code, tests, and the full paper series: https://github.com/jloutey-hash/geovac. Corpus archive DOI: 10.5281/zenodo.20482394. Corpus version at deposit: v4.75.0.

Jul 9, 20260 citations

Nuclear Shell Model Hamiltonians on the Hyperspherical Lattice

Josh Loutey

We present structurally sparse qubit Hamiltonians for the nuclear shell model, constructed by applying the GeoVac angular-sparsity framework to the harmonic oscillator (HO) single-particle basis with a Mayer–Jensen spin–orbit term. The HO shell closures $2, 8, 20, 40, 70, 112$ emerge directly from graph state counting, and the physical magic numbers $2, 8, 20, 28, 50, 82, 126$ are recovered with a spin–orbit strength $v_{ls}/\hbar\omega \approx 0.17$ and a Nilsson $\ell(\ell+1)$ correction $d_{\ell\ell}/\hbar\omega \approx 0.02$ . We build the first two-species qubit encoding for the deuteron ( $1p+1n$ , $16$ qubits, $592$ non-identity Pauli terms, $1$ -norm $\approx 342$ MeV) using the Minnesota nucleon–nucleon potential and Moshinsky–Talmi brackets, and extend the architecture to helium-4 ( $2p+2n$ , $16$ qubits, $712$ Pauli terms) with intra-species antisymmetrization and proton–proton Coulomb repulsion. Despite a Hilbert-space dimension $12.25×$ larger than the deuteron, He-4 requires only $1.20×$ more Pauli terms — a direct consequence of the angular selection rules that carry through to multi-particle systems. A composed nuclear–electronic deuterium encoding ( $26$ qubits) validates the multi-block architecture and quantifies the $\sim 10^{13}$ coefficient ratio between nuclear, electronic, and hyperfine scales. We state and apply the Fock projection rigidity theorem: the $S^3$ conformal projection that makes electronic Hamiltonians equivalent to a graph Laplacian is unique to the Coulomb potential and does not extend to HO or Woods–Saxon systems. For nuclear Hamiltonians, the GeoVac contribution is therefore the angular machinery and qubit encoding, not the conformal projection. All nuclear physics inputs (Minnesota parameters, spin–orbit strength, $\hbar\omega$ ) follow the standard shell-model literature; the quantitative binding energies at $N_{\mathrm{shells}}=2$ are far from experiment and should be read as encoding-validation benchmarks, not as a nuclear-structure calculation. GeoVac Paper 23. Part of the GeoVac research corpus (quantum computing group). Code, tests, and the full paper series: https://github.com/jloutey-hash/geovac. Corpus archive DOI: 10.5281/zenodo.20482394. Corpus version at deposit: v4.75.0.We present structurally sparse qubit Hamiltonians for the nuclear shell model, constructed by applying the GeoVac angular-sparsity framework to the harmonic oscillator (HO) single-particle basis with a Mayer–Jensen spin–orbit term. The HO shell closures $2, 8, 20, 40, 70, 112$ emerge directly from graph state counting, and the physical magic numbers $2, 8, 20, 28, 50, 82, 126$ are recovered with a spin–orbit strength $v_{ls}/\hbar\omega \approx 0.17$ and a Nilsson $\ell(\ell+1)$ correction $d_{\ell\ell}/\hbar\omega \approx 0.02$. We build the first two-species qubit encoding for the deuteron ($1p+1n$, $16$ qubits, $592$ non-identity Pauli terms, $1$-norm $\approx 342$ MeV) using the Minnesota nucleon–nucleon potential and Moshinsky–Talmi brackets, and extend the architecture to helium-4 ($2p+2n$, $16$ qubits, $712$ Pauli terms) with intra-species antisymmetrization and proton–proton Coulomb repulsion. Despite a Hilbert-space dimension $12.25×$ larger than the deuteron, He-4 requires only $1.20×$ more Pauli terms — a direct consequence of the angular selection rules that carry through to multi-particle systems. A composed nuclear–electronic deuterium encoding ($26$ qubits) validates the multi-block architecture and quantifies the $\sim 10^{13}$ coefficient ratio between nuclear, electronic, and hyperfine scales. We state and apply the Fock projection rigidity theorem: the $S^3$ conformal projection that makes electronic Hamiltonians equivalent to a graph Laplacian is unique to the Coulomb potential and does not extend to HO or Woods–Saxon systems. For nuclear Hamiltonians, the GeoVac contribution is therefore the angular machinery and qubit encoding, not the conformal projection. All nuclear physics inputs (Minnesota parameters, spin–orbit strength, $\hbar\omega$) follow the standard shell-model literature; the quantitative binding energies at $N_{\mathrm{shells}}=2$ are far from experiment and should be read as encoding-validation benchmarks, not as a nuclear-structure calculation. GeoVac Paper 23. Part of the GeoVac research corpus (quantum computing group). Code, tests, and the full paper series: https://github.com/jloutey-hash/geovac. Corpus archive DOI: 10.5281/zenodo.20482394. Corpus version at deposit: v4.75.0.

Jul 9, 20260 citations

Entanglement Structure of the Angular Momentum Eigenbasis: Energy-Entanglement Decoupling and Basis-Intrinsic Sparsity

Josh Loutey

We characterize the entanglement structure of multi-electron atoms in the angular momentum eigenbasis—the basis whose orbitals are products of hydrogenic radial functions and spherical harmonics, labeled by quantum numbers $(n, l, m)$ . Four computational results are presented. First, energy-entanglement decoupling: the off-diagonal one-body Hamiltonian (graph Laplacian topology, coupling constant $\kappa = -1/16$ ) contributes $10$ – $39%$ of the correlation energy but less than $0.2%$ of the entanglement entropy; the off-diagonal electron repulsion contributes $100%$ of the entanglement. Entanglement entropy scales as $S \sim Z^{-2.56}$ across He-like ions. Second, basis-intrinsic sparsity: among orbital bases for systems with spherical symmetry, the angular momentum eigenbasis is the unique basis producing sparse electron repulsion integrals (ERIs). Any unitary rotation of orbital indices, no matter how small, fills the ERI tensor from $42%$ to $99%$ density in a step-function transition at the identity. The nonzero ERI count is $Z$ -independent, confirming that sparsity is purely geometric. Third, entanglement network topology: orbital mutual information maps reveal a hub migration pattern— $1s \to 2s \to 2p$ —that tracks the partially filled shell across the first row of the periodic table. Fourth, core-valence decoupling: the core-valence mutual information falls to $\approx 2×10^{-3}$ by $Z = 4$ and below $10^{-3}$ for $Z \geq 5$ , quantitatively justifying the composed fiber bundle factorization used in the GeoVac molecular framework. In the composed molecular architecture, per-block entanglement is $R$ -independent, with core/bond entropy ratios of $50×$ at matched qubit count. These results establish that the angular momentum eigenbasis decouples one-body energy transport from two-body entanglement, and that the $O(Q^{2.5})$ Pauli scaling of the GeoVac qubit Hamiltonians is a direct consequence of this basis choice. This is Paper 26 in the GeoVac series. GeoVac Paper 26. Part of the GeoVac research corpus (precision observations group). Code, tests, and the full paper series: https://github.com/jloutey-hash/geovac. Corpus archive DOI: 10.5281/zenodo.20482394. Corpus version at deposit: v4.75.0.We characterize the entanglement structure of multi-electron atoms in the angular momentum eigenbasis—the basis whose orbitals are products of hydrogenic radial functions and spherical harmonics, labeled by quantum numbers $(n, l, m)$. Four computational results are presented. First, energy-entanglement decoupling: the off-diagonal one-body Hamiltonian (graph Laplacian topology, coupling constant $\kappa = -1/16$) contributes $10$–$39%$ of the correlation energy but less than $0.2%$ of the entanglement entropy; the off-diagonal electron repulsion contributes $100%$ of the entanglement. Entanglement entropy scales as $S \sim Z^{-2.56}$ across He-like ions. Second, basis-intrinsic sparsity: among orbital bases for systems with spherical symmetry, the angular momentum eigenbasis is the unique basis producing sparse electron repulsion integrals (ERIs). Any unitary rotation of orbital indices, no matter how small, fills the ERI tensor from $42%$ to $99%$ density in a step-function transition at the identity. The nonzero ERI count is $Z$-independent, confirming that sparsity is purely geometric. Third, entanglement network topology: orbital mutual information maps reveal a hub migration pattern—$1s \to 2s \to 2p$—that tracks the partially filled shell across the first row of the periodic table. Fourth, core-valence decoupling: the core-valence mutual information falls to $\approx 2×10^{-3}$ by $Z = 4$ and below $10^{-3}$ for $Z \geq 5$, quantitatively justifying the composed fiber bundle factorization used in the GeoVac molecular framework. In the composed molecular architecture, per-block entanglement is $R$-independent, with core/bond entropy ratios of $50×$ at matched qubit count. These results establish that the angular momentum eigenbasis decouples one-body energy transport from two-body entanglement, and that the $O(Q^{2.5})$ Pauli scaling of the GeoVac qubit Hamiltonians is a direct consequence of this basis choice. This is Paper 26 in the GeoVac series. GeoVac Paper 26. Part of the GeoVac research corpus (precision observations group). Code, tests, and the full paper series: https://github.com/jloutey-hash/geovac. Corpus archive DOI: 10.5281/zenodo.20482394. Corpus version at deposit: v4.75.0.

Jul 9, 20260 citations

Quantum Cut Sparsifiers

Arpon Basu, Joshua Brakensiek, Pravesh K. Kothari +1 more

In this paper, we continue a line of research initiated by Basu, Brakensiek, and Putterman [2026] studying the sparsifiability of Hamiltonians. We focus particularly on the sparsifiability of the widely-studied Quantum Cut (QC) Hamiltonians. Our main result is that in an $n$ -qubit system, any $n$ -qubit QC Hamiltonian can be sparsified to $\widetilde{O}(n /\varepsilon^2)$ many terms while preserving the energy of every state up to a factor of $1 \pm \varepsilon$ . Our result can be interpreted as giving an importance sampling scheme for the edges of an arbitrary graph $G$ such that the \emph{Kikuchi} graph at level $\ell$ of the sampled graph is a spectral approximation to the Kikuchi graph of $G$ . Importantly, the \emph{same} sampling scheme works simultaneously for all $\ell$ . The natural approach of leverage score sampling, analyzed via matrix concentration inequalities, yields a polynomially worse bound in our setting because the underlying matrices have dimension $\sim 2^n$ . Instead, our approach relies on decomposing the action of these matrices into invariant subspaces. Then, by using an operator-valued inequality of Alon and Kozma [Ann. Henri Poincaré, 2020], itself building on an \emph{octopus inequality} of Caputo, Liggett, and Richthammer [J. AMS, 2010], we extend our sparsification technique to all expander graphs. We then invoke expander decomposition to extend our sparsifier to all graphs.In this paper, we continue a line of research initiated by Basu, Brakensiek, and Putterman [2026] studying the sparsifiability of Hamiltonians. We focus particularly on the sparsifiability of the widely-studied Quantum Cut (QC) Hamiltonians. Our main result is that in an $n$-qubit system, any $n$-qubit QC Hamiltonian can be sparsified to $\widetilde{O}(n /\varepsilon^2)$ many terms while preserving the energy of every state up to a factor of $1 \pm \varepsilon$. Our result can be interpreted as giving an importance sampling scheme for the edges of an arbitrary graph $G$ such that the \emph{Kikuchi} graph at level $\ell$ of the sampled graph is a spectral approximation to the Kikuchi graph of $G$. Importantly, the \emph{same} sampling scheme works simultaneously for all $\ell$. The natural approach of leverage score sampling, analyzed via matrix concentration inequalities, yields a polynomially worse bound in our setting because the underlying matrices have dimension $\sim 2^n$. Instead, our approach relies on decomposing the action of these matrices into invariant subspaces. Then, by using an operator-valued inequality of Alon and Kozma [Ann. Henri Poincaré, 2020], itself building on an \emph{octopus inequality} of Caputo, Liggett, and Richthammer [J. AMS, 2010], we extend our sparsification technique to all expander graphs. We then invoke expander decomposition to extend our sparsifier to all graphs.

quant-pharXiv:2606.09728Jun 9, 20260 citations

Entropy as a Projection Artifact: One-Body Operators are Entanglement-Inert on Sparse Lattices

Josh Loutey

We characterize the operator-theoretic origin of entanglement entropy in the GeoVac Coulomb lattice, a finite spectral graph realization of the hydrogenic angular momentum eigenbasis on $S^{3}$ . Five results are presented. First, the von Neumann entropy of any single-particle reduced density matrix derived from a non-degenerate ground state of a purely one-body fermionic Hamiltonian on a finite lattice is identically zero; numerically, we confirm $S_{\mathrm{kin}}/S_{\mathrm{full}} \sim 10^{-14}$ for He at $n_{\max}=2,3$ in the angular momentum eigenbasis. All ground-state entanglement is generated by the two-body interaction $V_{ee}$ , with the non-degeneracy qualifier essential: open-shell and dissociated-bond states escape the floor. Second, the holographic area-law scaling $S_n = k\log A_n$ with $A_n\propto n^{4}$ , asserted on one-particle grounds in earlier framework papers, is rigorously recovered as a pair-counting statement: $A_n=g_n^{2}$ , where $g_n=2n^{2}$ is the shell degeneracy. The factor of four is the two-body signature. Third, in the Coulomb lattice the cusp concentrates as a single ``hot node'' on the $(1s,1s)$ pair-state of the $V_{ee}$ graph; entanglement is co-located with cusp mass. Fourth, on the Bargmann–Segal harmonic oscillator lattice the ground-state entropy of any two-fermion closed-shell state interacting via any central two-body potential is identically zero, a consequence of Moshinsky–Talmi total-quanta conservation; this is a rigidity statement structurally parallel to the Fock and Bargmann–Segal theorems of Paper 24. Fifth, Coulomb-lattice block entropy obeys a universal scaling $S_B = A (\tilde{w}_B/\delta_B)^{\gamma}$ , where $\tilde{w}_B$ is the dimensionless $V_{ee}$ mass off-diagonal in the one-body eigenbasis and $\delta_B$ the dimensionless first $H_1$ gap; the local exponent extrapolates to $\gamma_{\infty} \approx 1.96$ in the joint $Z, n_{\max}\to\infty$ limit (Richardson on $n_{\max}=2,3,4,5$ ), slightly below the non-degenerate second-order Rayleigh–Schrödinger value of $2$ , with the residual gap attributed to multi-shell aggregation in the basis-extension limit. Entropy is slotted into the exchange-constant taxonomy as a projection-class observable: zero on the combinatorial graph, finite only after the metric embedding of $1/r_{12}$ . GeoVac Paper 27. Part of the GeoVac research corpus (precision observations group). Code, tests, and the full paper series: https://github.com/jloutey-hash/geovac. Corpus archive DOI: 10.5281/zenodo.20482394. Corpus version at deposit: v4.75.0.We characterize the operator-theoretic origin of entanglement entropy in the GeoVac Coulomb lattice, a finite spectral graph realization of the hydrogenic angular momentum eigenbasis on $S^{3}$. Five results are presented. First, the von Neumann entropy of any single-particle reduced density matrix derived from a non-degenerate ground state of a purely one-body fermionic Hamiltonian on a finite lattice is identically zero; numerically, we confirm $S_{\mathrm{kin}}/S_{\mathrm{full}} \sim 10^{-14}$ for He at $n_{\max}=2,3$ in the angular momentum eigenbasis. All ground-state entanglement is generated by the two-body interaction $V_{ee}$, with the non-degeneracy qualifier essential: open-shell and dissociated-bond states escape the floor. Second, the holographic area-law scaling $S_n = k\log A_n$ with $A_n\propto n^{4}$, asserted on one-particle grounds in earlier framework papers, is rigorously recovered as a pair-counting statement: $A_n=g_n^{2}$, where $g_n=2n^{2}$ is the shell degeneracy. The factor of four is the two-body signature. Third, in the Coulomb lattice the cusp concentrates as a single ``hot node'' on the $(1s,1s)$ pair-state of the $V_{ee}$ graph; entanglement is co-located with cusp mass. Fourth, on the Bargmann–Segal harmonic oscillator lattice the ground-state entropy of any two-fermion closed-shell state interacting via any central two-body potential is identically zero, a consequence of Moshinsky–Talmi total-quanta conservation; this is a rigidity statement structurally parallel to the Fock and Bargmann–Segal theorems of Paper 24. Fifth, Coulomb-lattice block entropy obeys a universal scaling $S_B = A (\tilde{w}_B/\delta_B)^{\gamma}$, where $\tilde{w}_B$ is the dimensionless $V_{ee}$ mass off-diagonal in the one-body eigenbasis and $\delta_B$ the dimensionless first $H_1$ gap; the local exponent extrapolates to $\gamma_{\infty} \approx 1.96$ in the joint $Z, n_{\max}\to\infty$ limit (Richardson on $n_{\max}=2,3,4,5$), slightly below the non-degenerate second-order Rayleigh–Schrödinger value of $2$, with the residual gap attributed to multi-shell aggregation in the basis-extension limit. Entropy is slotted into the exchange-constant taxonomy as a projection-class observable: zero on the combinatorial graph, finite only after the metric embedding of $1/r_{12}$. GeoVac Paper 27. Part of the GeoVac research corpus (precision observations group). Code, tests, and the full paper series: https://github.com/jloutey-hash/geovac. Corpus archive DOI: 10.5281/zenodo.20482394. Corpus version at deposit: v4.75.0.

Jul 9, 20260 citations

O(Q^{3.15})

Pauli string terms in the number of qubits

Q

, compared to

O(Q^{4+})

for conventional Gaussian-basis Hamiltonians (our own log-log fit of the published Pauli counts of Trenev et al. [ref] gives

Q^{4.25}

for LiH and

Q^{3.92}

for H

_2

O, spanning

Q^{3.9}

–

Q^{4.3}

\sim 1/M

(empirically

\approx M^{-0.85}

–

M^{-1}

) with the number of spatial orbitals

M

, while Gaussian ERI tensors remain near-dense at

O(1)

. Beyond term counts, we quantify two operationally relevant cost metrics: the number of qubitwise-commuting (QWC) measurement groups scales as

O(Q^{3.36})

, and the Pauli 1-norm

\lambda = \sum_i |c_i|

—which controls both Trotter step counts and qubitization query complexity—scales as

O(Q^{1.69})

with

R^2 = 0.997

_2

O(Q^{2.5})

, independent of molecular topology (exponent spread

0.02

across three molecules). At equal qubit counts, this represents two or more orders of magnitude fewer Pauli terms than Gaussian baselines (published counts of Trenev et al. [ref],

Q^{3.9}

–

Q^{4.3}

span fitted by us)—an advantage that grows with system size due to the

\sim 1.5

exponent gap. These comparisons are at matched qubit counts, not matched accuracy; the composed basis carries classical equilibrium-geometry errors of 5–26% (LiH/BeH

_2

_2

O; Sec. (ref)), making the framework optimal for fixed-geometry quantum simulation rather than geometry optimisation. Across the composed molecular library (35 molecules spanning

Z = 1

–56, H through Ba, including the first transition series ScH–ZnH), the composed Pauli count is exactly linear in the qubit count:

N_{\mathrm{Pauli}} = 11.10 × Q

(exact, non-identity terms;

d

-orbital blocks give 9.23). These counts are realized under a pair-diagonal angular approximation (

m_a=m_c

m_b=m_d

) adopted deliberately for sparsity; the exact global-

M_L

Coulomb rule (used in the framework's precision-physics paths, and considered but left on the table here) gives

\approx 2.5×

more terms with the same scaling, under which the

d

O(Q^{2.5})

Pauli scaling and a 190

×

reduction versus cc-pVDZ. The balanced coupled architecture (PK-free) binds LiH with an FCI minimum at a computed

R_{\mathrm{eq}} = 3.227

bohr (7.0% above the experimental 3.015 bohr) at

n_{\max}=2

(878 Pauli terms, 30 qubits); a single-point evaluation at the experimental geometry reaches 0.20% energy error at

n_{\max}=3

(84 qubits) when solved exactly. Chemistry-accuracy scope at

n_{\max}=2

is row-conditional: first-row hydrides (LiH, BeH

_2

, H

_2

\ldots

) bind with documented potential-energy-surface (PES) topology, while second-row hydrides (NaH and below) exhibit monotone overattraction under every tested configuration — frozen-core balanced, multi-zeta valence, screened cross-center, explicit-core Hartree–Fock — with no interior PES minimum at the qubit-Hamiltonian level. We document this scope boundary empirically in \S(ref); the framework's primary value proposition is structural sparsity with quantitative truncation-convergence rates (Paper 38 state-space GH convergence), not energy-accuracy parity with CCSD(T) on second-row systems. The hamiltonian() registry covers 37 systems (35 composed molecules spanning

Z=1

–56, H through Ba, plus the He and H

_2

reference atoms), including multi-center systems (CO, N

_2

, F

_2

), mixed frozen-core molecules (NaCl), and the full first transition series as hydrides (ScH through ZnH), with Pauli scaling

O(Q^{2.5})

confirmed universal across all three rows, compared to

O(Q^{3.9–4.3})

for Gaussian baselines. The composed Pauli count is exactly linear in qubit count (coefficient

11.10

, exact, for main-group molecules;

9.23

for

d

-orbital blocks under the pair-diagonal ERI approximation (Sec. (ref))—the exact global-

M_L

rule makes the

d

-block coefficient higher, not lower), so adding atoms to a molecule increases quantum cost linearly. Second- and third-row molecules use analytical frozen cores ([Ne], [Ar], [Ar]3

d^{10}

); the frozen-core energy enters only through the identity operator and does not affect quantum measurement cost. Isostructural molecules with frozen cores produce identical Pauli counts across periodic table rows. The Hamiltonians are available as a Python library with Qiskit, OpenFermion, and PennyLane export. GeoVac Paper 20. Part of the GeoVac research corpus (quantum computing group). Code, tests, and the full paper series: https://github.com/jloutey-hash/geovac. Corpus archive DOI: 10.5281/zenodo.20482394. Corpus version at deposit: v4.75.0.

1

-norm scaling [ref]; the benchmarked resource library across three periodic-table rows [ref]; the

S_N

representation-theoretic foundation that classifies the atomic building blocks [ref]; and the nuclear shell-model extension that delineates exactly which part of the framework transfers [ref]. The unifying value proposition is structural, not accuracy-based: after the Jordan–Wigner transformation the GeoVac lattice Hamiltonian yields

O(Q^{3.15})

Pauli terms for atoms and

O(Q^{2.5})

for composed multi-center molecules (versus

O(Q^{3.9–4.3})

for Gaussian baselines), with a sub-quadratic Pauli

1

-norm

\lambda = O(Q^{1.69})

(

R^2=0.997

) that controls Trotter depth and qubitization query cost. Two disciplines run through the arc. First, we separate the robust scaling exponents—unchanged under the choice of electron-repulsion selection rule—from the absolute multipliers, which depend on a stated pair-diagonal angular approximation and are disclosed as such. Second, every comparison is made at matched qubit count, not matched accuracy: the composed basis carries classical equilibrium-geometry errors of

5

–

26%

, so the framework is positioned for fixed-geometry NISQ/VQE simulation (Pauli count, QWC measurement groups), not for fault-tolerant geometry optimisation. The nuclear extension is honest in the same way: its binding energies are encoding-validation benchmarks far from experiment, and the Fock projection rigidity theorem states precisely that the conformal

S^3

structure does not transfer to the harmonic-oscillator basis—only the angular machinery and the qubit encoding do. GeoVac synthesis document. Part of the GeoVac research corpus (synthesis / reading guide group). Code, tests, and the full paper series: https://github.com/jloutey-hash/geovac. Corpus archive DOI: 10.5281/zenodo.20482394. Corpus version at deposit: v4.75.0.

2, 8, 20, 40, 70, 112

emerge directly from graph state counting, and the physical magic numbers

2, 8, 20, 28, 50, 82, 126

are recovered with a spin–orbit strength

v_{ls}/\hbar\omega \approx 0.17

and a Nilsson

\ell(\ell+1)

correction

d_{\ell\ell}/\hbar\omega \approx 0.02

. We build the first two-species qubit encoding for the deuteron (

1p+1n

16

qubits,

592

non-identity Pauli terms,

1

-norm

\approx 342

MeV) using the Minnesota nucleon–nucleon potential and Moshinsky–Talmi brackets, and extend the architecture to helium-4 (

2p+2n

16

qubits,

712

Pauli terms) with intra-species antisymmetrization and proton–proton Coulomb repulsion. Despite a Hilbert-space dimension

12.25×

larger than the deuteron, He-4 requires only

1.20×

more Pauli terms — a direct consequence of the angular selection rules that carry through to multi-particle systems. A composed nuclear–electronic deuterium encoding (

26

qubits) validates the multi-block architecture and quantifies the

\sim 10^{13}

coefficient ratio between nuclear, electronic, and hyperfine scales. We state and apply the Fock projection rigidity theorem: the

S^3

conformal projection that makes electronic Hamiltonians equivalent to a graph Laplacian is unique to the Coulomb potential and does not extend to HO or Woods–Saxon systems. For nuclear Hamiltonians, the GeoVac contribution is therefore the angular machinery and qubit encoding, not the conformal projection. All nuclear physics inputs (Minnesota parameters, spin–orbit strength,

\hbar\omega

) follow the standard shell-model literature; the quantitative binding energies at

N_{\mathrm{shells}}=2

are far from experiment and should be read as encoding-validation benchmarks, not as a nuclear-structure calculation. GeoVac Paper 23. Part of the GeoVac research corpus (quantum computing group). Code, tests, and the full paper series: https://github.com/jloutey-hash/geovac. Corpus archive DOI: 10.5281/zenodo.20482394. Corpus version at deposit: v4.75.0.

(n, l, m)

. Four computational results are presented. First, energy-entanglement decoupling: the off-diagonal one-body Hamiltonian (graph Laplacian topology, coupling constant

\kappa = -1/16

) contributes

10

–

39%

of the correlation energy but less than

0.2%

of the entanglement entropy; the off-diagonal electron repulsion contributes

100%

of the entanglement. Entanglement entropy scales as

S \sim Z^{-2.56}

across He-like ions. Second, basis-intrinsic sparsity: among orbital bases for systems with spherical symmetry, the angular momentum eigenbasis is the unique basis producing sparse electron repulsion integrals (ERIs). Any unitary rotation of orbital indices, no matter how small, fills the ERI tensor from

42%

99%

density in a step-function transition at the identity. The nonzero ERI count is

Z

-independent, confirming that sparsity is purely geometric. Third, entanglement network topology: orbital mutual information maps reveal a hub migration pattern—

1s \to 2s \to 2p

—that tracks the partially filled shell across the first row of the periodic table. Fourth, core-valence decoupling: the core-valence mutual information falls to

\approx 2×10^{-3}

Z = 4

and below

10^{-3}

for

Z \geq 5

, quantitatively justifying the composed fiber bundle factorization used in the GeoVac molecular framework. In the composed molecular architecture, per-block entanglement is

R

-independent, with core/bond entropy ratios of

50×

at matched qubit count. These results establish that the angular momentum eigenbasis decouples one-body energy transport from two-body entanglement, and that the

O(Q^{2.5})

Pauli scaling of the GeoVac qubit Hamiltonians is a direct consequence of this basis choice. This is Paper 26 in the GeoVac series. GeoVac Paper 26. Part of the GeoVac research corpus (precision observations group). Code, tests, and the full paper series: https://github.com/jloutey-hash/geovac. Corpus archive DOI: 10.5281/zenodo.20482394. Corpus version at deposit: v4.75.0.

n

-qubit system, any

n

-qubit QC Hamiltonian can be sparsified to

\widetilde{O}(n /\varepsilon^2)

many terms while preserving the energy of every state up to a factor of

1 \pm \varepsilon

. Our result can be interpreted as giving an importance sampling scheme for the edges of an arbitrary graph

G

such that the \emph{Kikuchi} graph at level

\ell

of the sampled graph is a spectral approximation to the Kikuchi graph of

G

. Importantly, the \emph{same} sampling scheme works simultaneously for all

\ell

. The natural approach of leverage score sampling, analyzed via matrix concentration inequalities, yields a polynomially worse bound in our setting because the underlying matrices have dimension

\sim 2^n

. Instead, our approach relies on decomposing the action of these matrices into invariant subspaces. Then, by using an operator-valued inequality of Alon and Kozma [Ann. Henri Poincaré, 2020], itself building on an \emph{octopus inequality} of Caputo, Liggett, and Richthammer [J. AMS, 2010], we extend our sparsification technique to all expander graphs. We then invoke expander decomposition to extend our sparsifier to all graphs.

We characterize the operator-theoretic origin of entanglement entropy in the GeoVac Coulomb lattice, a finite spectral graph realization of the hydrogenic angular momentum eigenbasis on

S^{3}

. Five results are presented. First, the von Neumann entropy of any single-particle reduced density matrix derived from a non-degenerate ground state of a purely one-body fermionic Hamiltonian on a finite lattice is identically zero; numerically, we confirm

S_{\mathrm{kin}}/S_{\mathrm{full}} \sim 10^{-14}

for He at

n_{\max}=2,3

in the angular momentum eigenbasis. All ground-state entanglement is generated by the two-body interaction

V_{ee}

, with the non-degeneracy qualifier essential: open-shell and dissociated-bond states escape the floor. Second, the holographic area-law scaling

S_n = k\log A_n

with

A_n\propto n^{4}

, asserted on one-particle grounds in earlier framework papers, is rigorously recovered as a pair-counting statement:

A_n=g_n^{2}

, where

g_n=2n^{2}

is the shell degeneracy. The factor of four is the two-body signature. Third, in the Coulomb lattice the cusp concentrates as a single ``hot node'' on the

(1s,1s)

pair-state of the

V_{ee}

graph; entanglement is co-located with cusp mass. Fourth, on the Bargmann–Segal harmonic oscillator lattice the ground-state entropy of any two-fermion closed-shell state interacting via any central two-body potential is identically zero, a consequence of Moshinsky–Talmi total-quanta conservation; this is a rigidity statement structurally parallel to the Fock and Bargmann–Segal theorems of Paper 24. Fifth, Coulomb-lattice block entropy obeys a universal scaling

S_B = A (\tilde{w}_B/\delta_B)^{\gamma}

, where

\tilde{w}_B

is the dimensionless

V_{ee}

mass off-diagonal in the one-body eigenbasis and

\delta_B

the dimensionless first

H_1

gap; the local exponent extrapolates to

\gamma_{\infty} \approx 1.96

in the joint

Z, n_{\max}\to\infty

limit (Richardson on

n_{\max}=2,3,4,5

), slightly below the non-degenerate second-order Rayleigh–Schrödinger value of

2

, with the residual gap attributed to multi-shell aggregation in the basis-extension limit. Entropy is slotted into the exchange-constant taxonomy as a projection-class observable: zero on the combinatorial graph, finite only after the metric embedding of

1/r_{12}

. GeoVac Paper 27. Part of the GeoVac research corpus (precision observations group). Code, tests, and the full paper series: https://github.com/jloutey-hash/geovac. Corpus archive DOI: 10.5281/zenodo.20482394. Corpus version at deposit: v4.75.0.