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arXiv:2306.11844

Refining the weighted subspace-search variational quantum eigensolver: compression of ansätze into a single pure state and optimization of weights

Cheng-Lin Hong, Luis Colmenárez, Lexin Ding, Carlos L. Benavides-Riveros, Christian Schilling

Published Jun 20, 2023• 7 citations
arXivHTMLPDF

Abstract

The weighted subspace-search variational quantum eigensolver (SSVQE) is a prominent algorithm for calculating excited-state properties of molecular quantum systems. In this work, we elaborate on some of its fundamental features with the aim of improving its practical realization. First, we demonstrate that the initial ansätze for various excited states could be prepared into a single pure state through a minimal number of ancilla qubits, followed by the optimization of a subsequent global unitary rotation in the targeted subspace. Since the ancillas' sole purpose is to purify an underlying ensemble $ρ_{\boldsymbol{w}}$ state with spectral weights $\boldsymbol{w}$, their measurement would just collapse $ρ_{\boldsymbol{w}}$ with probabilities $w_j$ to one of its eigenstates $|Ψ_j \rangle$. We thus observe that our realization of SSVQE is equivalent to the original SSVQE improved by importance sampling. Then, we elaborate by numerical means on the potential influence of the auxiliary weights $\boldsymbol{w}$ on the accuracy of the sought-after eigenstates and eigenenergies. Clear trends are discovered which are contrasted with some recent mathematical results concerning the ensemble variational principle that underlies SSVQE.

DOI
10.48550/arxiv.2306.11844

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The weighted subspace-search variational quantum eigensolver (SSVQE) is a prominent algorithm for calculating excited-state properties of molecular quantum systems. In this work, we elaborate on some of its fundamental features with the aim of improving its practical realization. First, we demonstrate that the initial ansätze for various excited states could be prepared into a single pure state through a minimal number of ancilla qubits, followed by the optimization of a subsequent global unitary rotation in the targeted subspace. Since the ancillas' sole purpose is to purify an underlying ensemble ρwρ_{\boldsymbol{w}}ρw​ state with spectral weights w\boldsymbol{w}w, their measurement would just collapse ρwρ_{\boldsymbol{w}}ρw​ with probabilities wjw_jwj​ to one of its eigenstates ∣Ψj⟩|Ψ_j \rangle∣Ψj​⟩. We thus observe that our realization of SSVQE is equivalent to the original SSVQE improved by importance sampling. Then, we elaborate by numerical means on the potential influence of the auxiliary weights w\boldsymbol{w}w on the accuracy of the sought-after eigenstates and eigenenergies. Clear trends are discovered which are contrasted with some recent mathematical results concerning the ensemble variational principle that underlies SSVQE.
computations of the ground and excited electronic states. While substantial progress has been made in designing physically informed quantum algorithms to address the ground state under the constraints of near-term quantum hardware, relatively fewer developments exist for reliably computing excited state energetics, presumably due to the growing complexity of the quantum circuits for excited eigenroots. In this work, we exploit the inherent point-group and spin symmetries of molecular systems to construct symmetry-adapted sets of orthogonal reference states which are evolved with a common symmetry-scalar unitary, enabling the simultaneous determination of ground and excited eigenroots. By minimizing the energy functional over a restricted symmetry-adapted correlated manifold, the weighted subspace-search variational quantum eigensolver enables us to directly target multiple eigenroots within each spatial and spin symmetry sector. This physically motivated selection of references and the unitary confer substantial flexibility to the weighted energy landscape, largely allowing the optimization to avoid numerical traps and converge directly to the targeted eigenstates. More crucially, symmetry resolution also enables access to a larger number of eigenstates without substantially increasing the complexity of the cost function. Through benchmarking on moderate to strongly correlated molecular systems, we demonstrate the accuracy and advantages of our approach relative to several established excited state algorithms on quantum computers.
Abstract The use of the variational quantum eigensolver (VQE) on near-term quantum computers for solving Kohn–Sham Hamiltonian is still underexplored. In this study, we apply the weighted subspace search variational quantum eigensolver (weighted SSVQE) to obtain the occupied KS orbitals, their corresponding energies, and the total energy, exploring the effects of varying entanglement unitary blocks, circuit depths, and optimization strategies. While the choice of optimization algorithm has minimal impact on the total energy, it significantly influences convergence behavior, stability, and computational time. Our results further reveal the existence of an optimal weight configuration that achieves both faster convergence and improved energy accuracy. Furthermore, our findings highlight the critical role of ansatz depth and entanglement unitary blocks on total energy calculations across molecular geometries. The most effective design begins with a low-expressibility, low-entangling ansatz, such as linear entanglement unitary blocks, and repeats it until a sufficient number of layers is achieved. Using homoatomic and heteroatomic molecule as test sets, the approach remains valid and effective across systems exhibiting different types of electron correlation. This approach proves highly efficient, enabling shallow circuits that avoid convergence issues and closely match standard density functional calculations. This work thus offers a design strategy for shallow, convergence-friendly VQE circuits that match standard density functional energies.
Accurate excited state energies are essential for interpreting fluorescence and photochemical processes, yet variational quantum eigensolvers (VQEs) on noisy intermediate-scale quantum (NISQ) hardware have largely been confined to ground states. Subspace-search VQE (SSVQE) offers a NISQ-friendly route to excited states without ancilla-based overlap measurements, however it has lacked an experimental demonstration since it requires the coherent preparation and manipulation of multiple orthogonal basis states under a shared variational unitary. Here we realize SSVQE on a photonic platform by encoding orthogonal qudit states in the orbital angular momentum of single photons and optimizing a shared variational circuit. We estimate excited state energies for a four-dimensional HeH+\rm{HeH^+}HeH+ molecule and an eight-dimensional Heisenberg chain, thereby demonstrating scalable excited state eigensolving in a high-dimensional Hilbert space. Our results establish photonic qudits as a resource-efficient pathway to excited state quantum simulation.
Within the evolving domain of quantum computational chemistry, the Variational Quantum Eigensolver (VQE) has been developed to explore not only the ground state but also the excited states of molecules. In this study, we compare the performance of Variational Quantum Deflation (VQD) and Subspace-Search Variational Quantum Eigensolver (SSVQE) methods in determining the low-lying excited states of LiHLiHLiH. Our investigation reveals that while VQD exhibits a slight advantage in accuracy, SSVQE stands out for its efficiency, allowing the determination of all low-lying excited states through a single parameter optimization procedure. We further evaluate the effectiveness of optimizers, including Gradient Descent (GD), Quantum Natural Gradient (QNG), and Adam optimizer, in obtaining LiHLiHLiH's first excited state, with the Adam optimizer demonstrating superior efficiency in requiring the fewest iterations. Moreover, we propose a novel approach combining Folded Spectrum VQE (FS-VQE) with either VQD or SSVQE, enabling the exploration of highly excited states. We test the new approaches for finding all three H4H_4H4​'s excited states. Folded Spectrum SSVQE (FS-SSVQE) can find all three highly excited states near −1.0-1.0−1.0 Ha with only one optimizing procedure, but the procedure converges slowly. In contrast, although Folded spectrum VQD (FS-VQD) gets highly excited states with individual optimizing procedures, the optimizing procedure converges faster.
The Variational Quantum Eigensolver (VQE), as a hybrid quantum-classical algorithm, is an important tool for effective quantum computing in the current noisy intermediate-scale quantum (NISQ) era. However, the traditional hardware-efficient ansatz without taking into account symmetries requires more computational resources to explore the unnecessary regions in the Hilbert space. The conventional Subspace-Search VQE (SSVQE) algorithm, which can calculate excited states, is also unable to effectively handle degenerate states since the loss function only contains the expectation value of the Hamiltonian. In this study, the energy eigenstates of the one-dimensional Fermi-Hubbard model with two lattice sites and the two-dimensional Hubbard model with four lattice sites are calculated. By incorporating symmetries into the quantum circuits and loss function, we find that both the ground state and excited state calculations are improved greatly compared to the case without symmetries. The enhancement in excited state calculations is particularly significant. This is because quantum circuits that conserve the particle number are used, and appropriate penalty terms are added to the loss function, enabling the optimization process to correctly identify degenerate states. The results are verified through repeated simulations.