Excited-State Methods¶

Excited-state workflows either build reduced eigenproblems around a converged reference state or solve for excited states directly with variational penalties.

(1)¶\[L_n = \langle \psi_n|H|\psi_n\rangle + \beta \sum_{k<n} |\langle \psi_k|\psi_n\rangle|^2\]

The VQD workflow uses the deflated objective in (1) to discourage collapse back to lower-energy states.

Excited states require additional structure beyond ground-state minimization.

Two main approaches are implemented.

Post-VQE linear-algebra methods¶

Construct reduced eigenproblems around a converged reference state.

Methods:

QSE
EOM-QSE
LR-VQE
EOM-VQE

These rely on:

high-quality ground-state reference
well-conditioned reduced manifolds

Generally noiseless-only due to reliance on statevector information.

QSE¶

Construct operator-generated subspace:

\[ |\phi_i\rangle = O_i|\psi\rangle \]

Solve:

\[ Hc = ESc \]

where:

\[ H_{ij} = \langle \psi|O_i^\dagger H O_j|\psi\rangle \]

\[ S_{ij} = \langle \psi|O_i^\dagger O_j|\psi\rangle \]

EOM-QSE¶

Commutator-based reduced problem:

\[ A_{ij} = \langle \psi|O_i^\dagger[H,O_j]|\psi\rangle \]

Solve:

\[ Ac = \omega Sc \]

Typically non-Hermitian.

Positive real-dominant roots selected.

LR-VQE¶

Tangent-space linear response around converged parameters:

\[ S_{ij} = \langle \partial_i\psi|\partial_j\psi\rangle \]

\[ A_{ij} = \langle \partial_i\psi|(H-E_0)|\partial_j\psi\rangle \]

Solve:

\[ Ac = \omega Sc \]

Approximate excited energies:

\[ E_k = E_0 + \omega_k \]

Corresponds to a Tamm–Dancoff style approximation.

EOM-VQE¶

Full-response tangent-space formulation.

Produces paired roots:

\[ \pm \omega \]

Positive solutions correspond to excitation energies.

More expressive but more numerically sensitive.

Variational excited states¶

Solve excited states directly.

SSVQE¶

Shared unitary applied to orthogonal inputs:

\[ |\psi_k(\theta)\rangle = U(\theta)|\phi_k\rangle \]

Minimize:

\[ \sum_k w_k \langle \psi_k|H|\psi_k\rangle \]

VQD¶

Sequential deflation:

\[ L_n = \langle \psi_n|H|\psi_n\rangle + \beta \sum_{k<n} |\langle \psi_k|\psi_n\rangle|^2 \]

Supports noisy evaluation using density matrices.

Excited-state summary¶

method	category	noise
QSE	operator subspace	noiseless
EOM-QSE	operator EOM	noiseless
LR-VQE	tangent response	noiseless
EOM-VQE	full response	noiseless
SSVQE	variational	supported
VQD	deflation	supported