Subspace-search variational quantum eigensolver for excited states (1810.09434v2)

Abstract: The variational quantum eigensolver (VQE), a variational algorithm to obtain an approximated ground state of a given Hamiltonian, is an appealing application of near-term quantum computers. The original work [A. Peruzzo et al.; \textit{Nat. Commun.}; \textbf{5}, 4213 (2014)] focused only on finding a ground state, whereas the excited states can also induce interesting phenomena in molecules and materials. Calculating excited states is, in general, a more difficult task than finding ground states for classical computers. To extend the framework to excited states, we here propose an algorithm, the subspace-search variational quantum eigensolver (SSVQE). This algorithm searches a low energy subspace by supplying orthogonal input states to the variational ansatz and relies on the unitarity of transformations to ensure the orthogonality of output states. The $k$-th excited state is obtained as the highest energy state in the low energy subspace. The proposed algorithm consists only of two parameter optimization procedures and does not employ any ancilla qubits. The disuse of the ancilla qubits is a great improvement from the existing proposals for excited states, which have utilized the swap test, making our proposal a truly near-term quantum algorithm. We further generalize the SSVQE to obtain all excited states up to the $k$-th by only a single optimization procedure. From numerical simulations, we verify the proposed algorithms. This work greatly extends the applicable domain of the VQE to excited states and their related properties like a transition amplitude without sacrificing any feasibility of it.

• The paper introduces the Subspace-search Variational Quantum Eigensolver (SSVQE), a novel method extending VQE to compute excited states of a Hamiltonian on near-term quantum computers.
• SSVQE works by constructing and optimizing orthogonal input states to find a low-energy subspace, offering variations for finding multiple states or targeting specific ones without extra qubits.
• Numerical simulations validated SSVQE's accuracy on a transverse Ising model and helium hydride molecule, highlighting its potential for efficient applications in quantum chemistry on NISQ devices.

Overview of the Subspace-search Variational Quantum Eigensolver Paper

The paper "Subspace-search variational quantum eigensolver for excited states" by Ken M Nakanishi, Kosuke Mitarai, and Keisuke Fujii introduces an innovative method for calculating excited states of a Hamiltonian using near-term quantum computers. This method, titled the Subspace-search Variational Quantum Eigensolver (SSVQE), extends the capabilities of the traditional Variational Quantum Eigensolver (VQE), a hybrid quantum-classical algorithm predominantly used to find ground states.

The authors address a core challenge in quantum computation: obtaining accurate excited states in quantum systems, which has significant implications in understanding chemical reactions and physical properties. While classical computational approaches to this task are often computationally intensive and provide limited accuracy, SSVQE leverages the emerging technology of Noisy Intermediate-Scale Quantum (NISQ) devices.

Methodology

The SSVQE algorithm constructs orthogonal input states for its parameters, searching for a low-energy subspace of the Hamiltonian. By ensuring that the transformations maintain orthogonality, this approach bypasses the need for additional qubits or complex swap tests used in prior methods. The essence of the proposal involves two main steps: optimizing the variational parameters to find a low-energy subspace and then further refining these results to obtain the precise excited states.

The algorithm's versatility is enhanced with two variations: a generalized form of SSVQE capable of finding multiple excited states simultaneously and a weighted version that efficiently targets a specific excited state. These innovations allow SSVQE to function with improved computational efficiency and applicability to larger systems, compared to earlier quantum algorithms.

Experimental Validations

To validate the proposed algorithms, the authors conducted numerical simulations using a transverse Ising model with 4 qubits, showcasing that SSVQE can accurately capture excited states. The simulations demonstrated the convergence and fidelity of the method, and its capability was further proven on a molecular Hamiltonian, specifically for the helium hydride molecule.

Implications and Future Directions

The implications of this research are profound for the future of quantum computing in chemistry and materials science. The enhanced efficiency and reduced resource requirements make SSVQE a strong candidate for exploitation in near-term quantum devices. It extends the domain of feasible problems for quantum algorithms, making them viable for studying more complex molecular systems and their excited states.

Theoretically, this work enriches the VQE framework, reinforcing its flexibility and adaptability. Practically, the ability to compute transition amplitudes using this method has direct applications in studying properties like permittivity and emission rates, laying groundwork for future experimental validations.

Future developments may focus on refining the ansatz circuits further, exploring additional algorithms within the SSVQE framework, and conducting experimental implementations on larger quantum systems. This work aligns with the growing interest in harnessing the computational power of quantum machines and addressing challenges that are currently computationally prohibitive in classical domains.

In summary, the SSVQE algorithm presents a significant advancement toward practical quantum computing applications, offering an efficient route to explore excited states within a feasible computational framework on NISQ devices.