Excited States

VQE Excited-State Methods

Scope

This document describes the excited-state methods implemented in the VQE stack, including:

• mathematical formulation

• differences between methods

• implementation details specific to this repository

• practical guidance

These methods operate on a shared Hamiltonian and ansatz infrastructure.

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Overview

Excited-state methods in this repository fall into two categories:

1. Post-VQE methods

Require a converged noiseless VQE reference state:

• LR-VQE

• EOM-VQE

• QSE

• EOM-QSE

They construct and solve a reduced eigenvalue problem.

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2. Variational methods

Do not require a prior VQE run:

• SSVQE

• VQD

They directly optimize excited states using modified objectives.

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Comparison Summary

Method	Type	Core object	EVP type	Noise	Key limitation
LR-VQE	Post-VQE	Tangent space	Hermitian	No	linearized approximation
EOM-VQE	Post-VQE	Full response	structured EVP	No	numerically delicate
QSE	Post-VQE	Operator subspace	Hermitian	No	operator pool dependence
EOM-QSE	Post-VQE	Commutator manifold	Non-Hermitian	No	complex spectrum / filtering
SSVQE	Variational	Multi-state unitary	optimization	Yes	harder with many states
VQD	Variational	Deflated objective	optimization	Yes	depends on prior convergence

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Post-VQE Methods

LR-VQE

Idea

Linear-response approximation using the tangent space of the ansatz at the optimum.

Let:

• $|\psi_0\rangle = |\psi(\theta^*)\rangle$

• $|\partial_i \psi\rangle = \partial_{\theta_i} |\psi(\theta)\rangle$

Core problem

$$ A c = \omega S c $$

where:

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

Excitation energies:

$$ E_k \approx E_0 + \omega_k $$

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Implementation details (this repo)

• tangent vectors computed via finite differences (fd_eps)

• matrices explicitly constructed via QNodes

• Hermitianization applied to stabilize:

  • $A \leftarrow (A + A^\dagger)/2$

  • $S \leftarrow (S + S^\dagger)/2$

• overlap filtering / rank truncation:

  • eigenvalues of $S$ below eps are removed

• generalized EVP solved on reduced subspace

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Properties

• Hermitian EVP

• numerically stable relative to EOM-VQE

• approximation limited to local linear response

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EOM-VQE

Idea

Extends LR-VQE to a full-response equation-of-motion formulation.

Core problem

Generalized eigenvalue problem with paired roots:

$$ \omega \in {+\omega_k, -\omega_k} $$

Physical energies:

$$ E_k \approx E_0 + \omega_k, \quad \omega_k > 0 $$

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Implementation details

• same tangent construction as LR-VQE

• augmented response structure (full-response matrices)

• positive-root selection using omega_eps

• overlap filtering / rank truncation

• explicit stabilization:

  • orthonormalization

  • Hermitianization steps

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Properties

• richer physics than LR-VQE

• more sensitive to:

  • conditioning of $S$

  • numerical noise in derivatives

• can produce unstable or spurious roots

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QSE

Idea

Construct a subspace spanned by operators acting on the reference state:

$$ |\phi_i\rangle = O_i |\psi\rangle $$

Core problem

$$ H c = E S c $$

where:

$$ H_{ij} = \langle \psi | O_i^\dagger H O_j | \psi \rangle $$ $$ S_{ij} = \langle \psi | O_i^\dagger O_j | \psi \rangle $$

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Implementation details

• operator pool:

  • typically Hamiltonian-derived or truncated (top-k, max_ops)

• matrices built via QNodes

• Hermitian EVP

• optional truncation via:

  • operator selection

  • overlap threshold (eps)

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Properties

• physically intuitive subspace

• strongly dependent on operator pool quality

• relatively stable numerically

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EOM-QSE

Idea

Uses a commutator-based equation of motion in operator space.

Core problem

$$ A c = \omega S c $$

where:

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

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Implementation details

• same operator pool as QSE

• non-Hermitian matrix $A$

• eigenvalues may be complex

• filtering applied:

  • discard roots with large imaginary parts (imag_tol)

  • keep positive real-dominant roots (omega_eps)

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Properties

• non-Hermitian EVP

• can produce richer spectra

• requires careful filtering

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Variational Methods

SSVQE

Idea

Optimize multiple states simultaneously using a shared unitary.

$$ |\psi_k(\theta)\rangle = U(\theta)|\phi_k\rangle $$

Objective

$$ \mathcal{L}(\theta) = \sum_k w_k \langle \psi_k(\theta)|H|\psi_k(\theta)\rangle $$

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Implementation details

• orthogonality enforced via orthogonal input states

• single shared parameter vector

• weights control energy ordering

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Properties

• no explicit overlap penalty

• scales poorly with many states

• compatible with noisy simulation

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VQD

Idea

Sequentially compute states with deflation penalties.

Objective

$$ \mathcal{L}_n(\theta)

\langle \psi(\theta)|H|\psi(\theta)\rangle + \beta \sum_{k<n} \mathcal{O}(\psi_k,\psi) $$

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Overlap models

Noiseless:

$$ |\langle \psi_k|\psi\rangle|^2 $$

Noisy:

$$ \mathrm{Tr}(\rho_k \rho) $$

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Implementation details

• overlap computed via:

  • adjoint circuit (statevector)

  • density matrix overlap (noisy)

• optional beta scheduling:

  • ramp-up strategies

• sequential optimization

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Properties

• flexible and intuitive

• sensitive to:

  • choice of $\beta$

  • convergence of previous states

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Numerical Stability Considerations

Across all post-VQE methods:

Ill-conditioning

• overlap matrix $S$ may be near-singular

• handled via:

  • eigenvalue thresholding (eps)

  • rank truncation

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Finite-difference sensitivity

• tangent methods depend on fd_eps

• too small → numerical noise

• too large → approximation error

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Root selection

For EOM methods:

• discard:

  • complex roots (large imaginary part)

  • negative or near-zero roots

• keep:

  • positive, real-dominant roots

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Practical Guidance

• use QSE for:

  • stable, simple excited-state estimates

• use LR-VQE for:

  • tangent-space physics

  • controlled approximations

• use EOM-VQE / EOM-QSE for:

  • richer spectra (with care)

• use SSVQE for:

  • small multi-state problems

  • noisy simulations

• use VQD for:

  • scalable sequential workflows

  • noisy environments

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Key Takeaway

All methods in this repository:

• share a common VQE reference and Hamiltonian

• differ primarily in:

  • subspace construction

  • eigenproblem structure

  • numerical stability

Understanding these differences is critical for selecting the appropriate method for a given problem.