Simultaneous States with Subspace-Search VQE

The Molecular Energy Levels (Excited States) tutorial found excited states one at a time with deflation. Subspace-search VQE (SSVQE) instead finds the ground state and the first few excited states all at once, in a single optimization — SubspaceVQE (fixed ansatz) and SubspaceADAPTVQE (adaptively grown ansatz).

The idea

Pick \(k\) mutually orthogonal reference determinants \(\{|\varphi_j\rangle\}\), send them all through the same parameterized unitary \(U(\vec\theta)\), and minimize the weighted energy sum

\[L(\vec\theta) = \sum_{j=0}^{k-1} w_j\, \langle\varphi_j|U^\dagger(\vec\theta)\,H\,U(\vec\theta)|\varphi_j\rangle , \qquad w_0 > w_1 > \dots > w_{k-1} > 0 .\]

Because \(U\) is unitary, the images \(U|\varphi_j\rangle\) stay orthonormal; the descending weights force the largest weight onto the lowest energy, so at the optimum \(U|\varphi_0\rangle\) is the ground state, \(U|\varphi_1\rangle\) the first excited state, and so on. Each level’s reported energy is the bare expectation value \(\langle\varphi_j|U^\dagger H U|\varphi_j\rangle\).

The reference determinants are chosen automatically as the \(k\) lowest-lying determinants of the Hartree-Fock particle-number sector (all orthonormal and with the correct electron count).

Subspace-VQE

Both drivers are ASE calculators. Attach one, evaluate the energy (which builds the Hamiltonian and runs the subspace search), and read the full spectrum off subspace_result:

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

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 = SubspaceVQE(basis="FAO", h=0.20,
                         num_states=2, weights=[2.0, 1.0], verbose=False)
atoms.get_potential_energy()               # ASE energy = ground state (eV)

result = atoms.calc.subspace_result
print(result.in_units("eV"))               # [E0, E1, ...] all levels
print(result.excitation_energies)          # [0, E1 - E0, ...] (Hartree)
print(result.levels)                        # an EnergyLevels view

num_states sets how many levels to compute; weights are the (strictly decreasing, positive) SSVQE weights, defaulting to \((k, k-1, \dots, 1)\). The result exposes energies (ascending, Hartree), optimal_energy (the ground state), the orthonormal states, and in_units("eV").

Subspace-ADAPT-VQE

SubspaceADAPTVQE grows one shared adaptive ansatz for all the states: its pool-screening gradient is the weighted sum of the per-reference gradients \(\sum_j w_j\,\langle\psi_j|[H, A_i]|\psi_j\rangle\), and the inner re-optimization minimizes the weighted energy. It records how many operators were grown.

from carcara.algorithms import SubspaceADAPTVQE

atoms.calc = SubspaceADAPTVQE(basis="FAO", h=0.20, pool="fermionic",
                              num_states=2, verbose=False,
                              gradient_tolerance=1e-4, max_iterations=20)
atoms.get_potential_energy()

result = atoms.calc.subspace_result
print(result.in_units("eV"))
print(result.num_operators)                # operators in the shared ansatz

What the levels mean

The returned energies are variational upper bounds on the exact spectrum. For orthonormal trial states the Hylleraas-Undheim-MacDonald theorem guarantees

\[E_i^{\text{SSVQE}} \ge \lambda_i \quad\text{for every } i,\]

where \(\lambda_i\) is the \(i\)-th exact eigenvalue. So the levels are ordered, orthonormal, and never below the true values.

Two practical points:

Ground state. From the default start (all-zero parameters, i.e. the Hartree-Fock reference) the optimization settles near the Hartree-Fock basin and recovers the ground state essentially exactly.
Excited-state tightness depends on the ansatz. SSVQE is exact only when the shared unitary is expressive enough to map every reference to its eigenstate simultaneously. A small ansatz (e.g. UCCSD on minimal-basis H\ :sub:2) gives looser excited-state bounds; a richer ansatz — or SubspaceADAPTVQE with a larger pool — tightens them. Increasing the ground-state weight \(w_0\) pins the ground state more firmly at the cost of the excited estimates.

A complete, runnable script (both drivers, compared to exact diagonalization) is examples/09_SubspaceVQE_H2.py.