Variational Quantum Deflation

The variational quantum eigensolver (VQE) finds the ground state energy of a Hamiltonian by minimizing the expectation value E(theta) = <psi(theta)|H|psi(theta)> over parameterized circuit parameters theta. By the variational principle, this minimization converges to the ground state energy from above. VQE is well suited to near-term hardware because it requires only short circuits with classical optimization in the loop, but it inherently targets only the lowest eigenvalue. Many physically interesting quantities, including absorption spectra, photochemical reaction barriers, and conical intersections, require knowledge of multiple excited states above the ground state.

Variational quantum deflation extends VQE to excited states by augmenting the cost function with penalty terms that orthogonalize each new state against all previously found states. After finding the k-th state |phi_k(theta_k)>, the cost function for the (k+1)-th state becomes E_{k+1}(theta) = <psi(theta)|H|psi(theta)> + sum_j beta_j |<phi_j|psi(theta)>|^2, where beta_j are positive penalty weights chosen large enough that any state overlapping with |phi_j> is energetically penalized above the next true eigenvalue. The overlap terms |<phi_j|psi(theta)>|^2 are measured via a SWAP test or destructive SWAP on the quantum hardware. Minimizing this modified cost function drives the variational ansatz to the next lowest eigenstate not yet in the deflated set.

Excited state ordering and convergence depend on choosing beta values carefully. If beta is too small, the penalty is insufficient and the optimizer may collapse back to a lower state. If beta is too large, the optimization landscape becomes distorted and convergence slows. In practice, beta is set to a value exceeding the gap between eigenvalues by a safety margin estimated from chemical knowledge or a pilot calculation. Qiskit Nature provides VQD implementations (under the excited states solvers framework) that automate the overlap measurements and handle the sequential optimization loop. A competing approach, the quantum equation of motion (qEOM) method, computes excited states in a single step by solving a generalized eigenvalue problem built from VQE-derived operators, and can be more shot-efficient for small excitation manifolds.

The primary application domain for VQD is photochemistry, where conical intersections between ground and excited potential energy surfaces govern non-radiative decay, photoisomerization (as in retinal in vision), and singlet fission in organic photovoltaics. Accurately locating these intersections requires simultaneous knowledge of at least two state energies as a function of nuclear geometry, exactly the information VQD supplies. VQD is also used for computing reaction pathway energy profiles involving electronically excited intermediates. Compared to classical multireference methods (CASSCF, MRCI) that become intractable as active space size grows, VQD offers a potential route to treating larger active spaces on future fault-tolerant hardware, though current NISQ implementations are limited by circuit depth and overlap measurement noise.