How an Equi-ensemble Description Systematically Outperforms the Weighted-ensemble Variational Quantum Eigensolver

Technical Summary: Equi-ensemble vs. Weighted-ensemble VQE for Excited States

Problem Statement
Calculating excited-state properties in chemistry is essential for understanding photoinduced molecular behavior, yet balancing computational cost with accuracy remains a significant challenge. While the Variational Quantum Eigensolver (VQE) offers a promising quantum approach, generalizing the variational principle to many-body excited states via ensemble methods introduces complexities regarding the choice of state weights. Specifically, the Weighted-Ensemble (WE) VQE approach relies on assigning descending weights ($w_j \geq w_{j+1}$) to an ensemble of $K$ states to target the $K$ lowest eigenstates. However, this approach requires a priori knowledge of the energy ordering of the target states to assign weights correctly. If the initial guess (and its associated weights) does not match the final energy ordering, the optimization landscape becomes problematic, often leading to convergence failures or unreliable results. The paper investigates whether the Equi-Ensemble (EE) approach, where all weights are equal ($w_j = w_{j+1}$), offers a more robust alternative despite not directly yielding individual eigenenergies without post-processing.

Methodology
The authors compare two ensemble VQE strategies using the generalized variational principle for an ensemble energy $E^w_C(\theta)$:

1. Weighted-Ensemble (WE): Uses descending weights ($w_j > w_{j+1}$) to enforce a specific ordering of states during optimization.
2. Equi-Ensemble (EE): Uses equal weights ($w_j = 1/K$). In this limit, the cost function minimizes the trace of the Hamiltonian over the subspace spanned by the $K$ states. The resulting states are mutually rotated versions of the true eigenstates (quasi-diabatic) but span the correct eigen-subspace. Individual eigenenergies can be recovered via classical post-processing (e.g., diagonalization of the Hamiltonian matrix in the basis of the optimized states).

The study evaluates these methods on two distinct problems:

• Many-Body System: The formaldehyde molecule (formaldimine) exhibiting strong vibronic coupling near a conical intersection. The system is modeled at the SA-CASSCF(4,3) level. The authors test 1-GUCCSD and 2-GUCCSD ansätze with both correct and incorrect initial weight orderings relative to the true energy order.
• One-Body System: An eight-state problem derived from Quantum Density Functional Theory (Q-DFT) for a linear chain of 16 hydrogen atoms ($H_{16}$). This maps 16 spatial orbitals to 4 qubits. The study uses a hardware-efficient RyCNOT ansatz and compares optimal weights against equi-weights.

Key Results

• Sensitivity to Initial Guess in WE-VQE: In the formaldehyde case, the WE-VQE performance is highly dependent on the initial ordering of weights. When the weight order matches the true energy order, convergence is achieved. However, when the order is incorrect (a likely scenario in practical applications where final energies are unknown), the WE-VQE frequently fails to converge to the correct individual eigenstates, resulting in large errors ($10^{-2}$ Hartree) in the cost function. While the trace (subspace energy) may remain accurate, the individual states are often incorrect or stuck in local minima.
• Robustness of EE-VQE: The EE-VQE consistently converges to the correct eigen-subspace regardless of the initial state ordering. For the formaldehyde system, EE-VQE achieves high accuracy ($10^{-9}$ Hartree) with fewer iterations (approx. 300–500) compared to the WE-VQE, which requires significantly more iterations (up to 1400) and more complex ansätze (2-GUCCSD) to attempt to correct the ordering.
• Q-DFT Performance: In the $H_{16}$ chain, the WE-VQE exhibits non-democratic errors, where the highest-energy state (associated with the lowest weight) dominates the error ($>10^{-1}$ Hartree). In contrast, the EE-VQE combined with classical post-processing diagonalization reduces errors by two orders of magnitude for short bond distances and maintains democratic error distribution across all states.
• Convergence Dynamics: The WE-VQE often gets trapped in local minima corresponding to the correct subspace but with incorrect state ordering. Escaping these minima to find the correct individual ordering requires the optimizer to temporarily deteriorate the subspace quality, leading to slow and unstable convergence. The EE-VQE avoids this by optimizing the subspace directly.

Significance and Claims
The paper concludes that the Equi-Ensemble approach is superior to the Weighted-Ensemble approach for extracting excited-state properties on quantum computers. The authors argue that the additional complexity and risk of convergence failure introduced by enforcing ordered weights in WE-VQE are detrimental.

Key claims include:

1. Reliability: EE-VQE systematically provides the correct eigen-subspace without requiring prior knowledge of the energy ordering, whereas WE-VQE is prone to failure if the initial guess is imperfect.
2. Efficiency: EE-VQE converges faster and with less computational overhead (fewer iterations and simpler ansätze) than WE-VQE.
3. Post-Processing Viability: While EE-VQE does not yield individual eigenenergies directly, the required post-processing (classical diagonalization or rotation) has negligible computational overhead compared to the quantum circuit execution. This overhead is justified by the robustness and accuracy gained.
4. Applicability: The EE approach is particularly advantageous for systems with avoided crossings, conical intersections, or large ensembles (like in Q-DFT) where state ordering is difficult to predict or where only the subspace energy is required.

The authors assert that researchers should prefer the equi-ensemble description with post-processing over the weighted-ensemble approach to ensure reliable and efficient convergence to the targeted excited-state subspace.