Quantum Equation of Motion (qEOM)

• Quantum Equation of Motion (qEOM) is a hybrid method that prepares a variational ground state via VQE and uses a classical generalized eigenvalue problem to extract excitation energies.
• It employs a restricted operator manifold—often including singles, doubles, and occasionally triples—to project and measure fermionic excitations in molecular and lattice models.
• The approach addresses challenges such as metric conditioning and noise amplification through various parameterizations and regularization techniques to improve numerical stability.

Quantum equation of motion (qEOM) is a hybrid quantum–classical excited-state method that builds on the classical Equation-of-Motion formalism and the Variational Quantum Eigensolver (VQE). Its central idea is to prepare an approximate correlated ground state on a quantum device, span a restricted excitation manifold by acting on that state with fermionic excitation and de-excitation operators, measure the resulting projected Hamiltonian and metric matrices as expectation values of Pauli strings, and solve a generalized eigenvalue problem on a classical processor to obtain excitation energies. In current literature, qEOM appears both as a direct commutator-based formalism and as a closely related subspace method connected to quantum linear response (qLR), and it has been used for molecular excited states, response properties, thermal averages, spin-flip spectroscopy, and lattice-model excitations (Ollitrault et al., 2019, Kjellgren et al., 2024, Morrone et al., 2024).

1. Formal definition and secular equations

The starting point of qEOM is a variational approximation to the ground state, commonly written as $|\Psi_0\rangle = U(\theta)|00\ldots 0\rangle$ or, in a unitary coupled-cluster setting,

$$|\Psi_{\rm UCC}\rangle = e^{\hat T-\hat T^\dagger}|0\rangle,$$

with $\hat T$ truncated at singles and doubles in UCCSD. Excited states are then approximated by applying an operator $\hat O_I^\dagger$ or $R_I^\dagger$, expanded in a finite manifold of particle-hole excitations and de-excitations, to the prepared ground state (Ollitrault et al., 2019, Pavošević et al., 2023).

In its canonical commutator form, the exact equation-of-motion condition is written as

$$[H,\hat O_I^\dagger]|\Psi_0\rangle = \omega_I \hat O_I^\dagger |\Psi_0\rangle,$$

with $\omega_I = E_I-E_0$. After truncating the operator manifold and projecting onto the same manifold, one obtains the generalized eigenvalue problem

$A c = \omega S c,$

with

$$A_{IJ}=\langle\Psi_0|[R_I^\dagger,[H,R_J]]|\Psi_0\rangle,\qquad S_{IJ}=\langle\Psi_0|[R_I^\dagger,R_J]|\Psi_0\rangle.$$

This Hermitian double-commutator form was used by Ollitrault et al. for molecular excitation energies on noisy hardware, where the real eigenvalues $\omega$ are interpreted as excitation energies (Ollitrault et al., 2019).

A broader block formulation includes both excitation and de-excitation amplitudes. In one common notation,

$$|\Psi_{\rm UCC}\rangle = e^{\hat T-\hat T^\dagger}|0\rangle,$$0

where the blocks are built from nested commutators on the ground state. This form underlies qEOM-UCCSD, qEOM-SF-UCCSD, and several qLR-related formulations (Pavošević et al., 2023, Jensen et al., 2023).

2. Operator manifolds, metrics, and equivalent formulations

A defining structural choice in qEOM is the excitation manifold. The most common truncation uses all single and double fermionic excitations and their Hermitian adjoints; in molecular applications this is the qEOM-SD or qEOM-UCCSD setting. The same ground-state manifold can also be expressed in a quantum subspace expansion (QSE) form with matrix elements

$$|\Psi_{\rm UCC}\rangle = e^{\hat T-\hat T^\dagger}|0\rangle,$$1

For the H$$|\Psi_{\rm UCC}\rangle = e^{\hat T-\hat T^\dagger}|0\rangle,$$2O/STO-3G pipeline, it was stated that the EOM commutator form and the QSE form yield the same spectrum under exact arithmetic and complete operator pool, and the implementation used the QSE form for direct measurement (Song, 26 Jun 2026).

A second structural issue is the metric. In generalized qEOM and qLR formulations, the overlap matrix is not always the identity. This is tied to the choice of operator parameterization and, in some formulations, to whether the “killer condition” or vacuum-annihilation condition is enforced, namely that de-excitation from the exact ground state vanishes. The literature distinguishes four parameterizations (Kjellgren et al., 2024, Kumar et al., 2023):

Parameterization	Operator definition	Metric property
Naive	$$|\Psi_{\rm UCC}\rangle = e^{\hat T-\hat T^\dagger}|0\rangle,$$3	$$|\Psi_{\rm UCC}\rangle = e^{\hat T-\hat T^\dagger}|0\rangle,$$4 is not guaranteed to be the identity or invertible
Projected	$$|\Psi_{\rm UCC}\rangle = e^{\hat T-\hat T^\dagger}|0\rangle,$$5	$$|\Psi_{\rm UCC}\rangle = e^{\hat T-\hat T^\dagger}|0\rangle,$$6 is not guaranteed to be the identity or invertible
Self-consistent	$$|\Psi_{\rm UCC}\rangle = e^{\hat T-\hat T^\dagger}|0\rangle,$$7	$$|\Psi_{\rm UCC}\rangle = e^{\hat T-\hat T^\dagger}|0\rangle,$$8
State-transfer	$$|\Psi_{\rm UCC}\rangle = e^{\hat T-\hat T^\dagger}|0\rangle,$$9	$\hat T$0

This distinction is not merely formal. In the naive and projected parameterizations, the metric depends on redundant orbital rotations, and this dependence can make the metric singular. In the self-consistent and state-transfer parameterizations, $\hat T$1, so the generalized eigenvalue problem reduces to an ordinary eigenvalue problem (Kjellgren et al., 2024).

The same issue appears in the more applied analysis of subspace-based excited-state methods. For qEOM and QSE, errors in eigenvalues increase drastically with an increase in the condition number of the overlap matrix when the generalized eigenvalue equation is solved in the presence of statistical sampling errors. At very high condition numbers, the working equation may not be solvable without thresholding or related regularization, and thresholding can produce missing excited states (Kwao et al., 12 Mar 2025).

3. Quantum-computing workflow

In practical near-term implementations, qEOM is a two-stage quantum computation followed by classical diagonalization. For qEOM-SF-UCCSD/VQE, the workflow was described in three stages: state preparation and ground-state VQE, measurement of qEOM matrix elements, and classical diagonalization (Pavošević et al., 2023).

In the ground-state stage, one constructs the electronic Hamiltonian and excitation operators in fermionic second-quantized form, maps them to qubit Pauli strings via Jordan–Wigner, assembles the parametrized unitary $\hat T$2 as a product of Trotterized exponentials, and measures $\hat T$3 while updating amplitudes classically, for example with BFGS, until convergence. In the more general VQE-based description, any ansatz $\hat T$4 can be used as long as it prepares a sufficiently accurate $\hat T$5 (Pavošević et al., 2023, Ollitrault et al., 2019).

In the qEOM stage, the optimized state is repeatedly prepared on the device. For each ordered pair of excitation labels, the required nested commutators or QSE matrix elements are expanded into Pauli strings, grouped into commuting sets, and measured by shot-based protocols. The resulting matrices are then assembled and diagonalized classically. In LiH, the implementation measured each Pauli expectation at three stretch factors $\hat T$6 for zero-noise extrapolation and also corrected readout errors by inversion of the assignment-error matrix (Ollitrault et al., 2019).

The resource bottleneck is usually measurement and conditioning rather than the classical eigensolve. For qEOM-SF-UCCSD, the measurement cost scales as the square of the excitation manifold size, while each matrix element is itself a sum of Pauli terms that can be batched by commuting-group techniques. For the 2019 LiH implementation, each $\hat T$7 expands to $\hat T$8 Pauli strings and total measurements scale as $\hat T$9 (Pavošević et al., 2023, Ollitrault et al., 2019).

4. Spin-flip, orbital optimization, and higher excitations

One major extension of qEOM is the spin-flip construction introduced by Pavošević et al. In qEOM-SF-UCCSD, the reference is a high-spin triplet $\hat O_I^\dagger$0, and the excitation manifold is restricted to operators that flip one $\hat O_I^\dagger$1 spin. The generalized eigenvalue machinery is unchanged, but the resulting $\hat O_I^\dagger$2 are singlet energy gaps measured from the triplet reference. This formulation was designed to access singlet-diradical target states and was reported to outperform conventional UCCSD/VQE for the cis-trans isomerization of ethylene and the automerization of cyclobutadiene; the predicted qEOM-SF-UCCSD/VQE barrier heights for these two problems were stated to be in good agreement with the experimentally determined values (Pavošević et al., 2023).

A second extension enlarges the excitation manifold beyond singles and doubles. In qEOM-SDT, the operator $\hat O_I^\dagger$3 is truncated at singles, doubles, and triples, and the triples space is reduced by point group symmetry and perturbation-theory screening. In the formulation reported in 2024, the scaling was reduced from $\hat O_I^\dagger$4 to $\hat O_I^\dagger$5, and a perturbation correction was introduced for neglected triple excitations. On challenging cases such as the $\hat O_I^\dagger$6 state of $\hat O_I^\dagger$7, the $\hat O_I^\dagger$8 state of $\hat O_I^\dagger$9, and full dissociation of HF, the reported energy errors were less than $R_I^\dagger$0 eV (Zheng et al., 2024).

A third extension combines qEOM with orbital optimization. In oo-VQE-qEOM, orbital rotations $R_I^\dagger$1 are optimized together with circuit parameters, and the excitation manifold includes both active-space spin-adapted singles and doubles $R_I^\dagger$2 and singlet one-electron excitations $R_I^\dagger$3 between inactive, active, and virtual spaces. Noise-free simulations on BeH$R_I^\dagger$4, H$R_I^\dagger$5, and H$R_I^\dagger$6O were reported to reproduce conventional classical CASSCF calculations, with oo-UCCSD-qEOMSD absorption and circular-dichroism spectra closely tracking CASSCF-LR benchmarks (Jensen et al., 2023).

5. Benchmarks and scientific applications

The first hardware-oriented molecular benchmark in the present corpus is LiH on IBM Q Poughkeepsie. In a 4-qubit active-space implementation, a minimal “UCC-inspired” circuit with 6 CNOTs, 8 fixed single-qubit rotations, and a single $R_I^\dagger$7 was used as the VQE reference. At the equilibrium Li–H distance of $R_I^\dagger$8 Å, the first five excitation energies were measured; unmitigated errors were $R_I^\dagger$9 Ha, while mitigated errors were $$[H,\hat O_I^\dagger]|\Psi_0\rangle = \omega_I \hat O_I^\dagger |\Psi_0\rangle,$$0–$$[H,\hat O_I^\dagger]|\Psi_0\rangle = \omega_I \hat O_I^\dagger |\Psi_0\rangle,$$1 Ha, all within or near chemical accuracy $$[H,\hat O_I^\dagger]|\Psi_0\rangle = \omega_I \hat O_I^\dagger |\Psi_0\rangle,$$2 Ha). As the bond is stretched, excited states remained within $$[H,\hat O_I^\dagger]|\Psi_0\rangle = \omega_I \hat O_I^\dagger |\Psi_0\rangle,$$3 Ha after mitigation (Ollitrault et al., 2019).

A more complete systems study appears in the H$$[H,\hat O_I^\dagger]|\Psi_0\rangle = \omega_I \hat O_I^\dagger |\Psi_0\rangle,$$4O/STO-3G reproducible pipeline. There, the bare qubit Hamiltonian interleaved cation $$[H,\hat O_I^\dagger]|\Psi_0\rangle = \omega_I \hat O_I^\dagger |\Psi_0\rangle,$$5 states below the neutral manifold, variational deflation inherited the contamination and inverted the spectrum, whereas the qEOM subspace method restored the neutral excited-state ladder to sub-milli-Hartree accuracy. With exact or UCCSD ground states, qEOM recovered all seven neutral gaps with errors $$[H,\hat O_I^\dagger]|\Psi_0\rangle = \omega_I \hat O_I^\dagger |\Psi_0\rangle,$$6 Ha $$[H,\hat O_I^\dagger]|\Psi_0\rangle = \omega_I \hat O_I^\dagger |\Psi_0\rangle,$$7 eV, and a realistic measurement model compressed $$[H,\hat O_I^\dagger]|\Psi_0\rangle = \omega_I \hat O_I^\dagger |\Psi_0\rangle,$$8 distinct Pauli strings to $$[H,\hat O_I^\dagger]|\Psi_0\rangle = \omega_I \hat O_I^\dagger |\Psi_0\rangle,$$9 qubit-wise commuting measurement bases. Under full nonlinear generalized-eigenvalue trials, the matrix-aware hybrid shot allocation was reported to have a $\omega_I = E_I-E_0$0 chance that all seven excitation gaps lie within chemical accuracy $\omega_I = E_I-E_0$1 eV) at $\omega_I = E_I-E_0$2 total shots, while the remaining bottleneck was identified as per-circuit two-qubit gate fidelity rather than measurement cost (Song, 26 Jun 2026).

qEOM has also been extended beyond excitation energies. In the qLR/qEOM response-property framework, dynamic isotropic polarizabilities and specific rotations were computed from qEOM-derived states, and the reported errors versus FCI were substantially smaller than CCSD-LR in strong-correlation regimes; for H$\omega_I = E_I-E_0$3O at $\omega_I = E_I-E_0$4 Å, the listed errors were $\omega_I = E_I-E_0$5 for qLR(sc), $\omega_I = E_I-E_0$6 for qLR(proj), and $\omega_I = E_I-E_0$7 for CCSD-LR, while for chiral $\omega_I = E_I-E_0$8 at $\omega_I = E_I-E_0$9 the reported specific rotations were $A c = \omega S c,$0, $A c = \omega S c,$1, and $A c = \omega S c,$2, respectively (Kumar et al., 2023). In a distinct application, qEOM combined with informationally complete POVMs was used to reconstruct thermal states of ethylene and butadiene; for butadiene, the generalized eigenproblem became unstable at $A c = \omega S c,$3–$A c = \omega S c,$4 shots but stabilized at $A c = \omega S c,$5 shots, with $A c = \omega S c,$6 Ha for all $A c = \omega S c,$7 once sampling noise was controlled (Morrone et al., 2024).

The method is not restricted to molecular chemistry. It has been used for the Lipkin model on IBM quantum machines, where the use of symmetries and the Gray code reduced quantum resources and improved results, and for the half-filled one-dimensional $A c = \omega S c,$8 Fermi–Hubbard model on a chain with $A c = \omega S c,$9 sites and periodic boundary conditions, where qEOM was used to compute charge and spin gaps, the spectral function, and spin-spin dynamic correlations (Hlatshwayo et al., 2022, Bespalova et al., 2024).

6. Instabilities, noise amplification, and methodological outlook

The main methodological difficulty in qEOM is the generalized eigenvalue problem itself. In qEOM and QSE, finite-shot errors perturb both the projected Hamiltonian and the metric, and the sensitivity of the computed excitation energies grows with the condition number of the overlap matrix. A quantitative analysis on H$$A_{IJ}=\langle\Psi_0|[R_I^\dagger,[H,R_J]]|\Psi_0\rangle,\qquad S_{IJ}=\langle\Psi_0|[R_I^\dagger,R_J]|\Psi_0\rangle.$$0 classified $$A_{IJ}=\langle\Psi_0|[R_I^\dagger,[H,R_J]]|\Psi_0\rangle,\qquad S_{IJ}=\langle\Psi_0|[R_I^\dagger,R_J]|\Psi_0\rangle.$$1–$$A_{IJ}=\langle\Psi_0|[R_I^\dagger,[H,R_J]]|\Psi_0\rangle,\qquad S_{IJ}=\langle\Psi_0|[R_I^\dagger,R_J]|\Psi_0\rangle.$$2 as “low,” $$A_{IJ}=\langle\Psi_0|[R_I^\dagger,[H,R_J]]|\Psi_0\rangle,\qquad S_{IJ}=\langle\Psi_0|[R_I^\dagger,R_J]|\Psi_0\rangle.$$3–$$A_{IJ}=\langle\Psi_0|[R_I^\dagger,[H,R_J]]|\Psi_0\rangle,\qquad S_{IJ}=\langle\Psi_0|[R_I^\dagger,R_J]|\Psi_0\rangle.$$4 as “medium,” and $$A_{IJ}=\langle\Psi_0|[R_I^\dagger,[H,R_J]]|\Psi_0\rangle,\qquad S_{IJ}=\langle\Psi_0|[R_I^\dagger,R_J]|\Psi_0\rangle.$$5 as “high”; in the medium regime, QSE/qEOM errors increase sharply, and in the high regime the generalized eigenproblem may fail without additional regularization (Kwao et al., 12 Mar 2025).

A closely related formal instability arises from the metric structure of naive and projected parameterizations. Because the metric depends on redundant orbital rotations in these variants, singular points can appear even in an idealized noise-less setting. The analysis of divergences in qEOM and qLR showed that such singularities produce divergences in excitation energies and that finite quantum-sampling noise amplifies the problem dramatically near the singular region, whereas self-consistent and state-transfer parameterizations avoid the pathology because $$A_{IJ}=\langle\Psi_0|[R_I^\dagger,[H,R_J]]|\Psi_0\rangle,\qquad S_{IJ}=\langle\Psi_0|[R_I^\dagger,R_J]|\Psi_0\rangle.$$6 (Kjellgren et al., 2024).

Regularization can restore solvability but not always completeness. Thresholding, or truncated SVD of the overlap matrix, can remove singular directions, but in the analyzed H$$A_{IJ}=\langle\Psi_0|[R_I^\dagger,[H,R_J]]|\Psi_0\rangle,\qquad S_{IJ}=\langle\Psi_0|[R_I^\dagger,R_J]|\Psi_0\rangle.$$7 examples it produced missing excited states. Tikhonov regularization is gentler, but shifts all excitation energies slightly. This has motivated the use of formulations whose working equation is an ordinary eigenvalue problem, such as q-sc-EOM, or operator constructions with analytically orthonormal metrics (Kwao et al., 12 Mar 2025).

Hardware noise remains a second major limitation. In the Lipkin-model studies, higher configuration complexity improved the noiseless results but amplified device errors, and zero-noise extrapolation was introduced to reduce systematic biases. In the H$$A_{IJ}=\langle\Psi_0|[R_I^\dagger,[H,R_J]]|\Psi_0\rangle,\qquad S_{IJ}=\langle\Psi_0|[R_I^\dagger,R_J]|\Psi_0\rangle.$$8O/STO-3G pipeline, even a compact 29-operator ADAPT-VQE ansatz left single-circuit fidelities well below $$A_{IJ}=\langle\Psi_0|[R_I^\dagger,[H,R_J]]|\Psi_0\rangle,\qquad S_{IJ}=\langle\Psi_0|[R_I^\dagger,R_J]|\Psi_0\rangle.$$9 on current hardware modalities, making accurate state preparation rather than measurement cost the binding constraint (Hlatshwayo et al., 2023, Song, 26 Jun 2026).

These findings suggest a bifurcated outlook. On one side, qEOM continues to expand through spin-flip constructions, orbital optimization, triples corrections, symmetry-respecting subspace methods, and finite-temperature variants. On the other, the most urgent technical questions concern metric conditioning, shot allocation, regularization, and error mitigation, since the practical success of the method depends not only on the expressive quality of the operator manifold but also on the numerical stability of the generalized eigenvalue problem under realistic sampling and gate noise (Pavošević et al., 2023, Zheng et al., 2024, Kwao et al., 12 Mar 2025).