Marco Metodológico

Since the development of the variational quantum eigensolver (VQE) algorithm by Peruzzo et al.52_{, numerous studies and demonstrations of VQE focused on approximating ground}

states of physical systems. In principle, the VQE algorithm was designed as a modular framework, treating each component (i.e. state preparation, energy estimation, classical op- timization) as a black box that could be easily improved and/or extended. Recent studies have leveraged this flexibility, specifically applying different formulations of objective func- tions to compute excited states, which are fundamental to understanding photochemical properties and reactivities of molecules. In the following subsections, we highlight several methods extending the original VQE algorithm for approximating excited states for molec- ular systems.

Folded Spectrum and Lagrangian-Based Approaches. The first and perhaps the simplest extension consists of the application of the folded spectrum method, which utilizes a variational method to converge to the eigenvector closest to a shift parameter λ.

This is achieved by variationally minimizing the operator Hλ = (H− λI)2 according to

Peruzzo et al.52_{. This methodology, though relatively straightforward to implement, requires}

a quadratic increase in the number of terms of the effective Hamiltonian. This translates to a significant increase in the number of measurements needed, especially in the case of quantum chemistry Hamiltonians316_.

the VQE calculation to construct the Lagrangian216_:

L = H +X

i

λi(Si− siI)2 (79)

where λi are energy multipliers, Si are sets of operators associated with the desired sym-

metries, and si are the desired expectation values for these sets of operators. The set of

operators Si, e.g. spin numbers, accounts for symmetries whereby the energies are mini-

mized by the appropriate excited states (with respect to the original Hamiltonian). We note that S2

i and Si must be efficiently measurable on the quantum computer to ensure

the efficiency of the method. By solving VQE for the Lagrangian instead of the original Hamiltonian, it is possible to converge to an approximation of the excited state.

More recently, two new Lagrangian-based approaches were introduced for calculating excited state energies without the measurement overhead360_{. The first doubles the circuit}

depth to measure overlaps between the ground state and the prepared state to put a penalty on the overlap with the ground state. Choosing this penalty to be large enough ensures that the first excited state becomes the minimizer of the new cost function. The second method works by a similar principle, except the size of the quantum register is doubled instead of the circuit depth. A SWAP test, a circuit construction to compare two quantum states (or quantum registers) by computing the overlap, is then applied to incorporate a penalty for the prepared state having overlap with the ground state.

Linear Response: Quantum Subspace Expansion (QSE). More recently, a method- ology based on linear response has been developed320 _{and demonstrated on existing hard-}

ware58_{. In summary, this framework, called the Quantum Subspace Expansion (QSE), ex-}

tends the VQE algorithm and requires additional measurements to estimate the excited state energies. That is, after obtaining the the ground state |ψi of a molecule using VQE, an ap- proximate subspace of low-energy excited states is found by taking the linear combinations

of states of the form Oi|ψi, where the Oi are physically motivated quantum operators. For

example, in fermionic systems, these operators could correspond to fermionic excitation op- erators. In the algorithm, the matrix elements_{hψ| O}iHOj|ψi are computed on the quantum

device. The classical computation then diagonalizes the matrix to find the excited state en- ergies. While the QSE method benefits from the low coherence time requirements of VQE, the quality of the excited states obtained is subject not only to the quality of the ansatz employed in VQE but also to the errors induced by the linear-response expansion.

Witness-Assisted Variational Eigenspectra Solver (WAVES). An alternative proto- col that also utilizes VQE as a subroutine to compute the ground state is the witness-assisted variational eigenspectra solver (WAVES)316_{. The objective function in WAVES is augmented}

to include the energy (E) as well as an approximation for the entropy (a purity term Tr[ρ2 C]):

Fobj(P, E) = E − T · Tr[ρ2C]. (80)

In this setup, a control ancilla qubit is considered along with the trial state. Here, the control qubit behaves as an “eigenstate witness” where its entropy measurement nears zero if the optimized trial state is arbitrarily close to an eigenstate of the Hamiltonian.

A tunable parameter T (that can be pre-optimized) is used to bias towards excited states. In the first iteration of WAVES, T is set to 0 (i.e. implementing regular VQE) to compute the ground state. Then, T is tuned such that when the objective function is optimized, the resulting trial states correspond to approximate excited states. These states are fed into the iterative phase estimation algorithm (IPEA) to extract the corresponding excited state energies. For near-term devices, the IPEA procedure could be replaced by a Hamiltonian averaging approach320_.