Molecular Energy Levels (Excited States)

Ground-state VQE finds the lowest eigenvalue of the molecular Hamiltonian. To get the energy levels — the ground state and excited states — both VQE and ADAPTVQE expose an energy_levels method built on variational quantum deflation (VQD).

After the \(m\) lowest states \(\{|\psi_j\rangle\}_{j<m}\) are found, the next one is obtained by minimizing the deflated cost

\[L_m(\vec\theta) = \langle\psi(\vec\theta)|H|\psi(\vec\theta)\rangle + \beta\sum_{j<m} |\langle\psi_j|\psi(\vec\theta)\rangle|^2 ,\]

whose global minimum is the \(m\)-th eigenstate: the penalty \(\beta\) pushes the search orthogonal to every lower state. The reported energy of each level is the bare expectation value (the penalty vanishes at the eigenstate), so it is unbiased.

VQE energy levels

Attach a VQE calculator, evaluate the ground-state energy once (this builds the Hamiltonian and configures the solver), then ask for the levels:

import numpy as np
from ase import Atoms
from carcara.algorithms import VQE

atoms = Atoms("H2", positions=[[4.0, 4.0, 3.63], [4.0, 4.0, 4.37]],
              cell=[[8.0, 0, 0], [0, 8.0, 0], [0, 0, 8.0]], pbc=True)
atoms.calc = VQE(basis="FAO", h=0.20, verbose=False)
atoms.get_potential_energy()                 # configures the solver

levels = atoms.calc.energy_levels(num_states=2, restarts=4)
print(levels.in_units("eV"))                 # ground + first excited (eV)
print(levels.excitation_energies_in_units("eV"))   # [0, E1 - E0, ...]

energy_levels returns an EnergyLevels with ascending energies (Hartree), the optimal state vectors, and convenience views: ground_state_energy, excitation_energies, gaps, and in_units("eV").

beta defaults to a robust value derived from the Hamiltonian’s coefficient 1-norm; pass it explicitly to tune. restarts runs the optimizer several times per level (seeded random starts) and keeps the best, which helps the excited-state searches escape local minima.

ADAPT-VQE energy levels

The same call works on ADAPTVQE, which grows a fresh deflated ansatz for each level — both the pool-screening gradient and the inner re-optimization carry the penalty term, so the adaptive ansatz builds itself toward the next excited state:

from carcara.algorithms import ADAPTVQE

atoms.calc = ADAPTVQE(pool="fermionic", basis="FAO", h=0.20, verbose=False)
atoms.get_potential_energy()

levels = atoms.calc.energy_levels(num_states=2)
print(levels.in_units("eV"))
print(levels.num_operators)                  # operators grown per level

What the levels are

Every energy returned is a true eigenvalue of the qubit Hamiltonian, but the levels that can be reached are those the ansatz spans from its Hartree-Fock reference. This is a physical feature of variational excited states, not an approximation error:

UCCSD-VQE optimizes all excitation amplitudes jointly, so it can reach some singly-excited levels.
ADAPT-VQE grows greedily; from a closed-shell HF reference, single excitations have zero gradient (Brillouin’s theorem), so on H\ :sub:2 it reaches the seniority-zero excited partner (the doubly-excited \(^1\Sigma_g^+\) level).

Requesting more states than the ansatz sector supports can therefore return a level more than once — that signals the reachable spectrum is exhausted.

A complete, runnable script is examples/08_energy_levels_H2.py.