Excited-State Methods

Plain VQE finds the ground state — the lowest eigenvalue of $H$ — because the variational principle bounds the energy from below by $E_0$ . But chemistry and materials science also need excited states: optical spectra, reaction barriers, and band gaps are all energy differences between eigenstates. This lesson surveys how the variational toolbox is extended upward.

The obstacle: the variational principle only sees the bottom

For any normalized trial state, $\langle\psi|H|\psi\rangle \ge E_0$ , with equality only at the ground state. Minimizing the energy therefore always slides toward $E_0$ . To reach $E_1, E_2, \dots$ we must change the optimization so that the ground state is no longer the answer it converges to.

Method 1: Orthogonality-constrained VQE (VQD)

Variational Quantum Deflation finds excited states one at a time. After obtaining the ground state $|\psi_0\rangle$ , minimize a modified cost for the next state that penalizes overlap with the states already found:

C_k(\boldsymbol{\theta}) = \langle\psi_k|H|\psi_k\rangle + \sum_{j<k}\beta_j\,\bigl|\langle\psi_j|\psi_k\rangle\bigr|^2.

The penalty weights $\beta_j$ must exceed the relevant energy gaps so that any overlap with a lower state costs more energy than it saves. Driven down, the optimizer is forced into the orthogonal complement, where the lowest available eigenstate is now $E_k$ . The overlap terms $|\langle\psi_j|\psi_k\rangle|^2$ are themselves measured on hardware (e.g. via a swap test or a compute-uncompute circuit).

Method 2: Subspace-search VQE (SSVQE)

Instead of one state at a time, SSVQE optimizes a single ansatz on several mutually orthogonal inputs simultaneously. Feed in orthogonal reference states $|r_0\rangle, |r_1\rangle, \dots, |r_{k}\rangle$ (for instance distinct computational-basis strings) and minimize a weighted sum of their energies,

C(\boldsymbol{\theta}) = \sum_{j} w_j\,\langle r_j|U^\dagger(\boldsymbol{\theta})\,H\,U(\boldsymbol{\theta})|r_j\rangle, \qquad w_0 > w_1 > \dots > 0.

Because a unitary preserves orthogonality, the outputs $U(\boldsymbol{\theta})|r_j\rangle$ stay orthogonal automatically — no explicit overlap penalty is needed. With strictly decreasing weights the minimum aligns each output with a successive eigenstate, recovering $E_0$ through $E_k$ in one optimization.

Method 3: Quantum subspace expansion

A third route keeps the ground-state circuit fixed and expands a small basis around it. Choose a set of cheap excitation operators $\{O_i\}$ (e.g. single Pauli excitations) and form the projected Hamiltonian and overlap matrices

\mathcal{H}_{ij} = \langle\psi_0|O_i^\dagger H O_j|\psi_0\rangle, \qquad \mathcal{S}_{ij} = \langle\psi_0|O_i^\dagger O_j|\psi_0\rangle.

Both are measured on the quantum device, then a small generalized eigenvalue problem $\mathcal{H}\,\mathbf{c} = E\,\mathcal{S}\,\mathbf{c}$ is solved on a classical computer. Its eigenvalues approximate the low-lying spectrum, including excited states. The quantum cost is only the matrix elements; the diagonalization is classical and tiny.

Choosing a method

VQD is simple and accurate for the lowest few states, but sequential and overlap-measurement heavy.
SSVQE gets several states from one optimization and avoids overlap penalties, but the shared ansatz must be expressive enough for all targeted states at once.
Subspace expansion is cheap and noise-robust (it post-processes a fixed ground-state circuit), but its accuracy is capped by the chosen excitation basis.

Key takeaways

The variational principle only locates $E_0$ ; reaching excited states requires orthogonality.
VQD adds explicit overlap penalties; SSVQE exploits unitary orthogonality with weighted inputs; subspace expansion solves a small generalized eigenproblem from measured matrices.
All three keep the heavy lifting on hardware (expectation values, overlaps) and the diagonalization classical.