Anti-Hermitian Contracted Schrödinger Equation

• ACSE is an RDM-based formalism defined by the anti-Hermitian commutator residual, capturing correlated electron dynamics through two-body operators.
• It leverages cumulant reconstruction to approximate the 3-RDM, reducing computational complexity while maintaining accuracy in ground and excited state calculations.
• ACSE supports diverse applications including molecule-optimized basis construction, spin gap evaluations, and integration with hybrid quantum–classical methods.

Searching arXiv for recent and foundational papers on the Anti-Hermitian Contracted Schrödinger Equation (ACSE). The Anti-Hermitian Contracted Schrödinger Equation (ACSE) is the anti-Hermitian sector of the contracted Schrödinger equation obtained by projecting the many-electron eigenvalue problem onto the space of two-body operators. In second quantization it imposes vanishing expectation values of commutators between the Hamiltonian and all two-body excitation–de-excitation operators, thereby expressing stationarity of an exact eigenstate with respect to infinitesimal two-body unitary transformations. In practical electronic-structure theory, ACSE is an RDM-based formalism that propagates or optimizes the 2-electron reduced density matrix (2-RDM), closes the resulting hierarchy through approximate 3-RDM reconstruction, and serves not only as a correlated method for ground and excited states but also as a generator of natural orbitals, molecule-optimized Hamiltonians, and contracted quantum eigensolvers (Gidofalvi et al., 2014, Gibney et al., 2 Apr 2026).

1. Formal definition and operator structure

The starting point is the electronic Schrödinger equation,

$\hat{H}|\Psi_n\rangle = E_n|\Psi_n\rangle,$

with Hamiltonian

$$\hat{H} = \sum_{p,s} {}^{1}K^{p}_{s}\,\hat{a}^{\dagger}_{p}\hat{a}_{s} + \sum_{p,q,s,t} {}^{2}V^{p,q}_{s,t}\,\hat{a}^{\dagger}_{p}\hat{a}^{\dagger}_{q}\hat{a}_{t}\hat{a}_{s}.$$

Projecting this equation with two-body operators produces the contracted Schrödinger equation,

$$\langle \Psi_n | \hat{a}^{\dagger}_{i}\hat{a}^{\dagger}_{j}\hat{a}_{l}\hat{a}_{k}\hat{H}|\Psi_n\rangle = E_n\,{}^{2}D^{i,j}_{k,l},$$

where

$${}^{2}D^{i,j}_{k,l} = \langle \Psi_n | \hat{a}^{\dagger}_{i}\hat{a}^{\dagger}_{j}\hat{a}_{l}\hat{a}_{k}|\Psi_n\rangle$$

is the 2-RDM. The ACSE is obtained by retaining only the anti-Hermitian part of this contracted equation. A standard form used in the literature is

$$\frac{1}{2}\left\langle \Psi_n \Big| \big[\hat{a}^{\dagger}_{i}\hat{a}^{\dagger}_{j}\hat{a}_{l}\hat{a}_{k},\hat{H}\big]\Big|\Psi_n\right\rangle = 0,$$

or, equivalently up to normalization conventions,

$$\langle \Psi|[\hat{a}^{\dagger}_{i}\hat{a}^{\dagger}_{j}\hat{a}_{l}\hat{a}_{k},\hat{H}]|\Psi\rangle = 0 \quad \forall i,j,k,l.$$

This commutator form is central: it is the ACSE residual, it vanishes for an exact eigenstate, and it is the quantity used to generate iterative updates in both classical and quantum implementations (Gidofalvi et al., 2014, Smart et al., 2022).

The formal interpretation is twofold. First, the commutator condition expresses that the exact state is stationary under all infinitesimal two-body unitary transformations generated by anti-Hermitian two-body operators. Second, in the Heisenberg-picture interpretation emphasized in molecule-optimized basis work, the operator $$\hat{a}^{\dagger}_{i}\hat{a}^{\dagger}_{j}\hat{a}_{l}\hat{a}_{k}$$ is stationary under time evolution generated by $\hat{H}$ when evaluated in an eigenstate (Gidofalvi et al., 2014).

2. Reduced-density-matrix hierarchy and closure

ACSE is formulated in terms of reduced density matrices rather than an explicit many-determinant wavefunction. The 1-RDM is obtained by contraction of the 2-RDM,

$${}^{1}D^{i}_{k} = \frac{1}{N-1}\sum_j {}^{2}D^{i,j}_{k,j}.$$

This contraction is the route by which ACSE yields energies, natural orbitals, and orbital occupations (Gidofalvi et al., 2014).

A decisive practical advantage of ACSE relative to the full contracted Schrödinger equation is the reduction in RDM rank. The full CSE depends on the 2-, 3-, and 4-RDMs, whereas the ACSE depends only on the 2- and 3-RDMs. The residual can therefore be written as a functional of the 2-RDM, the 3-RDM, and the reduced Hamiltonian integrals, after which the remaining 3-RDM dependence must be approximated to close the equations (Gidofalvi et al., 2014, Boyn et al., 2021).

The standard closure is cumulant reconstruction of the 3-RDM from lower-order RDMs. One form used in ACSE implementations is the Valdemoro reconstruction,

$${}^3D^{ijk}_{qst} \approx {}^1D^{i}_{q}\wedge {}^1D^{j}_{s}\wedge {}^1D^{k}_{t} + 3\,{}^2\Delta^{ij}_{qs}\wedge {}^1D^{k}_{t},$$

with

$$\hat{H} = \sum_{p,s} {}^{1}K^{p}_{s}\,\hat{a}^{\dagger}_{p}\hat{a}_{s} + \sum_{p,q,s,t} {}^{2}V^{p,q}_{s,t}\,\hat{a}^{\dagger}_{p}\hat{a}^{\dagger}_{q}\hat{a}_{t}\hat{a}_{s}.$$0

In the implementation described for hybrid quantum–classical all-electron correlation, the three-body cumulant $$\hat{H} = \sum_{p,s} {}^{1}K^{p}_{s}\,\hat{a}^{\dagger}_{p}\hat{a}_{s} + \sum_{p,q,s,t} {}^{2}V^{p,q}_{s,t}\,\hat{a}^{\dagger}_{p}\hat{a}^{\dagger}_{q}\hat{a}_{t}\hat{a}_{s}.$$1 is set to zero. In the later open-source implementation, both Valdemoro and Nakatsuji–Yasuda reconstructions are exposed, with markedly different robustness properties across correlation regimes (Boyn et al., 2021, Gibney et al., 2 Apr 2026).

Because the ACSE operates directly on reduced density matrices, $$\hat{H} = \sum_{p,s} {}^{1}K^{p}_{s}\,\hat{a}^{\dagger}_{p}\hat{a}_{s} + \sum_{p,q,s,t} {}^{2}V^{p,q}_{s,t}\,\hat{a}^{\dagger}_{p}\hat{a}^{\dagger}_{q}\hat{a}_{t}\hat{a}_{s}.$$2-representability is an intrinsic issue. The hybrid QACSE literature emphasizes that physically meaningful 2-RDMs must satisfy $$\hat{H} = \sum_{p,s} {}^{1}K^{p}_{s}\,\hat{a}^{\dagger}_{p}\hat{a}_{s} + \sum_{p,q,s,t} {}^{2}V^{p,q}_{s,t}\,\hat{a}^{\dagger}_{p}\hat{a}^{\dagger}_{q}\hat{a}_{t}\hat{a}_{s}.$$3-representability conditions and uses them explicitly as an error-mitigation mechanism, while classical ACSE workflows rely on physical initial 2-RDMs and controlled propagation from those initial conditions (Boyn et al., 2021, Smart et al., 2021).

3. Propagation, residual-driven optimization, and orbital transformations

Rather than solving the commutator equations as a monolithic nonlinear system, ACSE is commonly solved as a differential flow in a fictitious parameter $$\hat{H} = \sum_{p,s} {}^{1}K^{p}_{s}\,\hat{a}^{\dagger}_{p}\hat{a}_{s} + \sum_{p,q,s,t} {}^{2}V^{p,q}_{s,t}\,\hat{a}^{\dagger}_{p}\hat{a}^{\dagger}_{q}\hat{a}_{t}\hat{a}_{s}.$$4. In one standard formulation,

$$\hat{H} = \sum_{p,s} {}^{1}K^{p}_{s}\,\hat{a}^{\dagger}_{p}\hat{a}_{s} + \sum_{p,q,s,t} {}^{2}V^{p,q}_{s,t}\,\hat{a}^{\dagger}_{p}\hat{a}^{\dagger}_{q}\hat{a}_{t}\hat{a}_{s}.$$5

with

$$\hat{H} = \sum_{p,s} {}^{1}K^{p}_{s}\,\hat{a}^{\dagger}_{p}\hat{a}_{s} + \sum_{p,q,s,t} {}^{2}V^{p,q}_{s,t}\,\hat{a}^{\dagger}_{p}\hat{a}^{\dagger}_{q}\hat{a}_{t}\hat{a}_{s}.$$6

and

$$\hat{H} = \sum_{p,s} {}^{1}K^{p}_{s}\,\hat{a}^{\dagger}_{p}\hat{a}_{s} + \sum_{p,q,s,t} {}^{2}V^{p,q}_{s,t}\,\hat{a}^{\dagger}_{p}\hat{a}^{\dagger}_{q}\hat{a}_{t}\hat{a}_{s}.$$7

Thus the generator is built directly from the instantaneous ACSE residual, and the flow drives that residual toward zero (Gidofalvi et al., 2014).

This propagation viewpoint supports several distinct uses of ACSE. In correlated electronic-structure calculations it is the solver itself. In molecule-optimized basis construction, a short low-cost ACSE evolution is used only to generate an approximate correlated 1-RDM, whose eigenvectors define approximate natural orbitals. In that setting, the workflow is: perform MCSCF in a large basis, propagate the ACSE briefly from $$\hat{H} = \sum_{p,s} {}^{1}K^{p}_{s}\,\hat{a}^{\dagger}_{p}\hat{a}_{s} + \sum_{p,q,s,t} {}^{2}V^{p,q}_{s,t}\,\hat{a}^{\dagger}_{p}\hat{a}^{\dagger}_{q}\hat{a}_{t}\hat{a}_{s}.$$8 to $$\hat{H} = \sum_{p,s} {}^{1}K^{p}_{s}\,\hat{a}^{\dagger}_{p}\hat{a}_{s} + \sum_{p,q,s,t} {}^{2}V^{p,q}_{s,t}\,\hat{a}^{\dagger}_{p}\hat{a}^{\dagger}_{q}\hat{a}_{t}\hat{a}_{s}.$$9, contract the resulting 2-RDM to the 1-RDM, diagonalize the virtual–virtual block, order orbitals by occupation number, and truncate by fixed rank to match the size of a smaller target basis (Gidofalvi et al., 2014).

The same paper recasts this orbital optimization as a Hamiltonian transformation problem. A one-body anti-Hermitian operator,

$$\langle \Psi_n | \hat{a}^{\dagger}_{i}\hat{a}^{\dagger}_{j}\hat{a}_{l}\hat{a}_{k}\hat{H}|\Psi_n\rangle = E_n\,{}^{2}D^{i,j}_{k,l},$$0

induces

$$\langle \Psi_n | \hat{a}^{\dagger}_{i}\hat{a}^{\dagger}_{j}\hat{a}_{l}\hat{a}_{k}\hat{H}|\Psi_n\rangle = E_n\,{}^{2}D^{i,j}_{k,l},$$1

so natural-orbital truncation can be viewed as a one-electron unitary transformation followed by rank truncation. The same framework suggests a generalization to two-body unitary transformations,

$$\langle \Psi_n | \hat{a}^{\dagger}_{i}\hat{a}^{\dagger}_{j}\hat{a}_{l}\hat{a}_{k}\hat{H}|\Psi_n\rangle = E_n\,{}^{2}D^{i,j}_{k,l},$$2

with $$\langle \Psi_n | \hat{a}^{\dagger}_{i}\hat{a}^{\dagger}_{j}\hat{a}_{l}\hat{a}_{k}\hat{H}|\Psi_n\rangle = E_n\,{}^{2}D^{i,j}_{k,l},$$3 of the same algebraic form as the ACSE generator. The paper does not implement this two-body Hamiltonian optimization, but it identifies ACSE as the natural framework for doing so (Gidofalvi et al., 2014).

4. Quantum and multistate generalizations

ACSE has also become the organizing principle behind contracted quantum eigensolvers. In the CQE framework, the ACSE residual

$$\langle \Psi_n | \hat{a}^{\dagger}_{i}\hat{a}^{\dagger}_{j}\hat{a}_{l}\hat{a}_{k}\hat{H}|\Psi_n\rangle = E_n\,{}^{2}D^{i,j}_{k,l},$$4

is treated as the gradient of the energy with respect to infinitesimal two-body unitary transformations. The wavefunction is then updated iteratively by exponentials of anti-Hermitian two-body operators, so that the state after each step is related by

$$\langle \Psi_n | \hat{a}^{\dagger}_{i}\hat{a}^{\dagger}_{j}\hat{a}_{l}\hat{a}_{k}\hat{H}|\Psi_n\rangle = E_n\,{}^{2}D^{i,j}_{k,l},$$5

This transforms the ACSE from a purely RDM equation into a residual-driven wavefunction optimization on quantum hardware, where the relevant commutators are measured rather than reconstructed from approximate higher RDMs (Smart et al., 2022, Smart et al., 2021).

The same formalism has been extended to excited states. ES-CQE introduces a projected or deflated setting in which previously obtained states are removed by a projector $$\langle \Psi_n | \hat{a}^{\dagger}_{i}\hat{a}^{\dagger}_{j}\hat{a}_{l}\hat{a}_{k}\hat{H}|\Psi_n\rangle = E_n\,{}^{2}D^{i,j}_{k,l},$$6, leading to an anti-Hermitian contracted projected Schrödinger equation (ACPSE). The ACPSE residual contains the usual ACSE commutator term plus projection terms involving overlaps and 2-electron transition density matrices. Vanishing ACPSE residuals define stationarity in the projected subspace, and the resulting excited states are obtained without constructing a generalized eigenvalue problem (Smart et al., 2023).

A parallel generalization replaces a single pure state by a weighted ensemble. The ensemble ACSE,

$$\langle \Psi_n | \hat{a}^{\dagger}_{i}\hat{a}^{\dagger}_{j}\hat{a}_{l}\hat{a}_{k}\hat{H}|\Psi_n\rangle = E_n\,{}^{2}D^{i,j}_{k,l},$$7

underlies purification-based and weighted-random parallel CQE algorithms for simultaneous optimization of several eigenstates. In this setting the residual of the ensemble ACSE is the gradient of the ensemble Rayleigh–Ritz objective, and the same two-body anti-Hermitian update is applied across the entire state set (Benavides-Riveros et al., 2023).

Convergence acceleration has been studied by reinterpreting CQE as optimization in a local parameter space. Quasi-Newton, limited-memory BFGS, and nonlinear conjugate-gradient variants use residual differences to approximate curvature and yield superlinear convergence near the solution. This point matters especially for NISQ implementations because fewer macro-iterations reduce noise accumulation (Smart et al., 2022).

5. Representative applications and quantitative behavior

ACSE has been deployed across basis-set acceleration, strongly correlated ground states, spin gaps, all-electron post-processing of quantum data, and open-source benchmarks for weakly and strongly correlated molecules and transition-metal systems.

Domain	Representative result	Citation
Molecule-optimized basis construction	For hydrogen fluoride, the ACSE curve in a molecule-optimized basis of polarized-double-zeta rank has nonparallelity error $$\langle \Psi_n | \hat{a}^{\dagger}_{i}\hat{a}^{\dagger}_{j}\hat{a}_{l}\hat{a}_{k}\hat{H}|\Psi_n\rangle = E_n\,{}^{2}D^{i,j}_{k,l},$$8 a.u., compared with $$\langle \Psi_n | \hat{a}^{\dagger}_{i}\hat{a}^{\dagger}_{j}\hat{a}_{l}\hat{a}_{k}\hat{H}|\Psi_n\rangle = E_n\,{}^{2}D^{i,j}_{k,l},$$9 a.u. for the standard polarized double-zeta basis; for $${}^{2}D^{i,j}_{k,l} = \langle \Psi_n | \hat{a}^{\dagger}_{i}\hat{a}^{\dagger}_{j}\hat{a}_{l}\hat{a}_{k}|\Psi_n\rangle$$0, ACSE in TZ/DZ[0.01] reduces NPE from $${}^{2}D^{i,j}_{k,l} = \langle \Psi_n | \hat{a}^{\dagger}_{i}\hat{a}^{\dagger}_{j}\hat{a}_{l}\hat{a}_{k}|\Psi_n\rangle$$1 a.u. to $${}^{2}D^{i,j}_{k,l} = \langle \Psi_n | \hat{a}^{\dagger}_{i}\hat{a}^{\dagger}_{j}\hat{a}_{l}\hat{a}_{k}|\Psi_n\rangle$$2 a.u. relative to the TZ reference	(Gidofalvi et al., 2014)
Spin-sensitive biradical chemistry	Spin-averaged CASSCF/ACSE gives mean signed error $${}^{2}D^{i,j}_{k,l} = \langle \Psi_n | \hat{a}^{\dagger}_{i}\hat{a}^{\dagger}_{j}\hat{a}_{l}\hat{a}_{k}|\Psi_n\rangle$$3 kcal/mol and mean absolute error $${}^{2}D^{i,j}_{k,l} = \langle \Psi_n | \hat{a}^{\dagger}_{i}\hat{a}^{\dagger}_{j}\hat{a}_{l}\hat{a}_{k}|\Psi_n\rangle$$4 kcal/mol for singlet–triplet gaps of a main-group biradicaloid set	(Boyn et al., 2021)
Hybrid quantum–classical all-electron correlation	For the benzyne isomers, QACSE/ACSE with [4,4] active spaces yields meta–ortho gaps of $${}^{2}D^{i,j}_{k,l} = \langle \Psi_n | \hat{a}^{\dagger}_{i}\hat{a}^{\dagger}_{j}\hat{a}_{l}\hat{a}_{k}|\Psi_n\rangle$$5 kcal/mol for both 3- and 4-qubit seeds and para–ortho gaps of $${}^{2}D^{i,j}_{k,l} = \langle \Psi_n | \hat{a}^{\dagger}_{i}\hat{a}^{\dagger}_{j}\hat{a}_{l}\hat{a}_{k}|\Psi_n\rangle$$6 and $${}^{2}D^{i,j}_{k,l} = \langle \Psi_n | \hat{a}^{\dagger}_{i}\hat{a}^{\dagger}_{j}\hat{a}_{l}\hat{a}_{k}|\Psi_n\rangle$$7 kcal/mol, reaching experimental confidence intervals or within $${}^{2}D^{i,j}_{k,l} = \langle \Psi_n | \hat{a}^{\dagger}_{i}\hat{a}^{\dagger}_{j}\hat{a}_{l}\hat{a}_{k}|\Psi_n\rangle$$8 kcal/mol of them	(Boyn et al., 2021)
Open-source benchmark suite	In ethylene $${}^{2}D^{i,j}_{k,l} = \langle \Psi_n | \hat{a}^{\dagger}_{i}\hat{a}^{\dagger}_{j}\hat{a}_{l}\hat{a}_{k}|\Psi_n\rangle$$9 excitation energies with a [2,2] active space, ACSE with Valdemoro reconstruction gives mean unsigned errors $$\frac{1}{2}\left\langle \Psi_n \Big| \big[\hat{a}^{\dagger}_{i}\hat{a}^{\dagger}_{j}\hat{a}_{l}\hat{a}_{k},\hat{H}\big]\Big|\Psi_n\right\rangle = 0,$$0–$$\frac{1}{2}\left\langle \Psi_n \Big| \big[\hat{a}^{\dagger}_{i}\hat{a}^{\dagger}_{j}\hat{a}_{l}\hat{a}_{k},\hat{H}\big]\Big|\Psi_n\right\rangle = 0,$$1 eV versus DMRG; for $$\frac{1}{2}\left\langle \Psi_n \Big| \big[\hat{a}^{\dagger}_{i}\hat{a}^{\dagger}_{j}\hat{a}_{l}\hat{a}_{k},\hat{H}\big]\Big|\Psi_n\right\rangle = 0,$$2 spin splittings in def2-TZVP, ACSE gives mean unsigned error $$\frac{1}{2}\left\langle \Psi_n \Big| \big[\hat{a}^{\dagger}_{i}\hat{a}^{\dagger}_{j}\hat{a}_{l}\hat{a}_{k},\hat{H}\big]\Big|\Psi_n\right\rangle = 0,$$3 kcal/mol	(Gibney et al., 2 Apr 2026)

These results define a recurring pattern. First, ACSE is often most valuable for relative energies, barrier profiles, and nonparallelity rather than merely pointwise absolute energies. Second, when seeded with multireference 2-RDMs, ACSE adds dynamic correlation while preserving the underlying multiconfigurational character. Third, the method is sufficiently flexible to support both classical and hybrid quantum–classical workflows, including all-electron post-processing of noisy active-space quantum simulations (Boyn et al., 2021, Gibney et al., 2 Apr 2026).

A related line of work concerns orbital dependence. Applying ACSE to $$\frac{1}{2}\left\langle \Psi_n \Big| \big[\hat{a}^{\dagger}_{i}\hat{a}^{\dagger}_{j}\hat{a}_{l}\hat{a}_{k},\hat{H}\big]\Big|\Psi_n\right\rangle = 0,$$4 dissociation, biradical singlet–triplet gaps, and bicyclobutane transition states, one study found that CASCI alone is strongly MO-dependent, whereas CASCI/ACSE is markedly less sensitive. In that survey, DFT orbitals were identified as a cost-effective alternative to CASSCF orbitals for post-CI dynamic-correlation calculations, especially with functionals such as M06-L, M06-2X, and $$\frac{1}{2}\left\langle \Psi_n \Big| \big[\hat{a}^{\dagger}_{i}\hat{a}^{\dagger}_{j}\hat{a}_{l}\hat{a}_{k},\hat{H}\big]\Big|\Psi_n\right\rangle = 0,$$5B97X-D (Boyn et al., 2022).

6. Limitations, misconceptions, and current directions

A common misconception is that solving the anti-Hermitian part is identical to solving the full Schrödinger equation in every implementation. The data do not support that statement. The full CSE is formally equivalent to the full Schrödinger equation, whereas ACSE is a reduced condition whose practical accuracy depends on cumulant reconstruction, update strategy, and representability control. In approximate classical implementations the energies are not guaranteed to be upper bounds, because the 3-RDM is reconstructed rather than exact (Gidofalvi et al., 2014, Boyn et al., 2021).

The dominant approximation is the treatment of the 3-RDM. Valdemoro reconstruction neglects the three-body cumulant; Nakatsuji–Yasuda reconstruction approximates that cumulant as a bilinear functional of the 2-cumulant around a Hartree–Fock reference. The open-source benchmarks show that Valdemoro is generally more robust, especially with active–active propagation disabled, whereas Nakatsuji–Yasuda can be less stable in strongly multireference and excited-state settings (Gibney et al., 2 Apr 2026).

Another misconception is that two-body anti-Hermitian stationarity is always sufficient for excited states. ES-CQE shows explicit counterexamples: in highly symmetric restricted subspaces, ACPSE can be spuriously satisfied by non-eigenstates, and states dominated by higher-than-double excitation character can be difficult to reach with a two-body unitary ansatz. The rectangular $$\frac{1}{2}\left\langle \Psi_n \Big| \big[\hat{a}^{\dagger}_{i}\hat{a}^{\dagger}_{j}\hat{a}_{l}\hat{a}_{k},\hat{H}\big]\Big|\Psi_n\right\rangle = 0,$$6 analysis therefore identifies symmetry-induced failures and quadruple-excitation-dominated states as intrinsic limitations of a purely two-body ACSE/ACPSE description, suggesting that higher-body contracted equations would be needed in such cases (Smart et al., 2023).

Methodologically, current development points in several directions already identified in the literature. Open-source classical ACSE now exists for ground and excited states, but its present implementation uses simple Euler integration and does not yet exploit symmetry or parallelization. The same benchmark paper explicitly identifies improved multi-reference 3-RDM reconstruction, DIIS or quasi-Newton accelerators, and more advanced propagation strategies as natural next steps (Gibney et al., 2 Apr 2026). On the quantum side, quasi-second-order CQE updates, projected excited-state variants, and ensemble formulations indicate that ACSE has evolved from a classical 2-RDM theory into a broader framework for residual-driven many-body optimization on both classical and quantum architectures (Smart et al., 2022, Benavides-Riveros et al., 2023).

In contemporary electronic-structure theory, ACSE therefore occupies a specific position: it is neither a conventional wavefunction expansion method nor a purely variational 2-RDM optimization, but a contracted equation-of-motion formalism centered on the anti-Hermitian commutator residual. Its practical identity is defined by that residual, by approximate closure of the RDM hierarchy, and by its ability to bridge strongly correlated classical chemistry, orbital and Hamiltonian optimization, and quantum algorithms for ground and excited states.