ABSTRACT The analysis of natural frequency is an important problem of structural engineering, where complex structures often lead to expensive eigenvalue problems. Variational quantum eigensolvers provide a potentially efficient solution by using quantum computing, but their application is limited by the costly need to decompose system matrices into a linear combination of unitaries. In this work, we apply a recently proposed decomposition‐free method from linear systems to eigenvalue problems in structural engineering. Specifically, we develop two algorithms based on this method, the decomposition‐free variational quantum eigensolver (DF‐VQE) and generalized eigensolver (DF‐VQGE) for the standard eigenvalue problems and the generalized eigenvalue problems, respectively. In these algorithms, the Rayleigh Quotient and generalized Rayleigh Quotient cost evaluations are reformulated so that the need for matrix decomposition is bypassed. The central technical step in this application is to use vectorization to encode the system matrices directly into quantum state vectors, allowing their properties to be computed via a standard swap test quantum circuit. The proposed methods are validated on a quantum simulator with three representative structural engineering problems, including a 1D bar, a 2D beam, and a 3D truss bridge, obtaining results that align well with reference solutions from classical finite element analysis. We also discuss the quantum state preparation challenge that must be addressed before this formulation can be connected to meaningful quantum advantage.
ABSTRACT The analysis of natural frequency is an important problem of structural engineering, where complex structures often lead to expensive eigenvalue problems. Variational quantum eigensolvers provide a potentially efficient solution by using quantum computing, but their application is limited by the costly need to decompose system matrices into a linear combination of unitaries. In this work, we apply a recently proposed decomposition‐free method from linear systems to eigenvalue problems in structural engineering. Specifically, we develop two algorithms based on this method, the decomposition‐free variational quantum eigensolver (DF‐VQE) and generalized eigensolver (DF‐VQGE) for the standard eigenvalue problems and the generalized eigenvalue problems, respectively. In these algorithms, the Rayleigh Quotient and generalized Rayleigh Quotient cost evaluations are reformulated so that the need for matrix decomposition is bypassed. The central technical step in this application is to use vectorization to encode the system matrices directly into quantum state vectors, allowing their properties to be computed via a standard swap test quantum circuit. The proposed methods are validated on a quantum simulator with three representative structural engineering problems, including a 1D bar, a 2D beam, and a 3D truss bridge, obtaining results that align well with reference solutions from classical finite element analysis. We also discuss the quantum state preparation challenge that must be addressed before this formulation can be connected to meaningful quantum advantage.
Abstract The variational quantum eigensolver (VQE) is a method that uses a hybrid quantum-classical computational approach to find eigenvalues of a Hamiltonian. VQE has been proposed as an alternative to fully quantum algorithms such as quantum phase estimation (QPE) because fully quantum algorithms require quantum hardware that will not be accessible in the near future. VQE has been successfully applied to solve the electronic Schrödinger equation for a variety of small molecules. However, the scalability of this method is limited by two factors: the complexity of the quantum circuits and the complexity of the classical optimization problem. Both of these factors are affected by the choice of the variational ansatz used to represent the trial wave function. Hence, the construction of an efficient ansatz is an active area of research. Put another way, modern quantum computers are not capable of executing deep quantum circuits produced by using currently available ansatzes for problems that map onto more than several qubits. In this review, we present recent developments in the field of designing efficient ansatzes that fall into two categories—chemistry–inspired and hardware–efficient—that produce quantum circuits that are easier to run on modern hardware. We discuss the shortfalls of ansatzes originally formulated for VQE simulations, how they are addressed in more sophisticated methods, and the potential ways for further improvements.
In solid mechanics, finite element discretization of large-scale static problems produces large sparse linear systems whose solution requires substantial computation time and memory. The variational quantum linear solver (VQLS) offers a hybrid quantum-classical route, but its use in quantum finite element analysis is limited by the decomposition of nonunitary matrices, barren plateaus, and the measurement cost of expectation values. We propose a voxel-based block variational quantum linear solver (Voxel-BVQLS) that combines structured matrix decomposition, the principle of minimum potential energy, and batched quantum tests. First, we construct an LCU decomposition of the stiffness matrix from the recursive block-banded structure of voxel-grid finite element matrices, with the number of unitary terms bounded independently of the problem size. Second, we optimize the ansatz parameters using a minimum-potential-energy objective in place of a conventional VQLS loss function, thereby mitigating barren plateaus in the studied problems. Third, we introduce a block-Hadamard test whose circuit directly estimates weighted sums of multiple inner products, reducing the number of circuit configurations required per iteration. We assessed the proposed method in noiseless classical simulations using three examples. These examples show that the method reduces both the number of unitary terms in the LCU decomposition and the number of iterations required to converge, while still yielding solutions of finite accuracy. Voxel-BVQLS thus provides a structured hybrid quantum-classical framework for quantum finite element analysis on regular grids.
Abstract Extracting eigenvalues and eigenvectors of exponentially large matrices will be an important application of near-term quantum computers. The variational quantum eigensolver (VQE) treats the case when the matrix is a Hamiltonian. Here, we address the case when the matrix is a density matrix ρ . We introduce the variational quantum state eigensolver (VQSE), which is analogous to VQE in that it variationally learns the largest eigenvalues of ρ as well as a gate sequence V that prepares the corresponding eigenvectors. VQSE exploits the connection between diagonalization and majorization to define a cost function C=Tr(ρ~H) C = Tr ( ρ ̃ H ) where H is a non-degenerate Hamiltonian. Due to Schur-concavity, C is minimized when ρ~=VρV† ρ ̃ = V ρ V † is diagonal in the eigenbasis of H . VQSE only requires a single copy of ρ (only n qubits) per iteration of the VQSE algorithm, making it amenable for near-term implementation. We heuristically demonstrate two applications of VQSE: (1) Principal component analysis, and (2) Error mitigation.
Accurately predicting the properties of molecules and materials from first principles requires solving the electronic structure problem. However, the computational cost of solving the electronic structure problem scales prohibitively in the number of atoms, and the usual approximate methods designed to reduce the overhead are often inaccurate for strongly correlated systems. For this reason, quantum algorithms have been developed as a promising alternative. In this work, we develop algebraic tools to improve the efficiency of three quantum algorithms for the electronic structure problem, addressing key bottlenecks for both near-term and fault-tolerant quantum computers. The first is the variational quantum eigensolver (VQE), which suffers from a large measurement overhead needed to estimate the Hamiltonian expectation value. To this end, we introduce a general class of measurable fragments of the electronic Hamiltonian, which enable partitioning schemes that significantly reduce the sampling cost for energy estimation. The second are quantum subspace methods (QSMs). We introduce a new QSM, the quantum seniority-based subspace expansion algorithm, which is a scalable alternative to VQE that alleviates VQE's state preparation, measurement, and optimization challenges while reducing qubit requirements. The third is quantum phase estimation (QPE), which requires an encoding of the electronic Hamiltonian directly on the quantum computer, the cost of which scales indirectly with the Hamiltonian spectral range. To address this, we develop linear programming approaches to implement the Block Invariant Symmetry Shift protocol, a method which reduces the spectral range of the Hamiltonian, improving its block-encoding cost for QPE. Collectively, these methods demonstrate that one can exploit the Hamiltonian's algebraic structure to mitigate computational costs, accelerating the path toward practical quantum advantage for electronic structure.
Many quantum algorithms are developed to evaluate eigenvalues for Hermitian matrices. However, few practical approach exists for the eigenanalysis of non-Hermintian ones, such as arising from modern power systems. The main difficulty lies in the fact that, as the eigenvector matrix of a general matrix can be non-unitary, solving a general eigenvalue problem is inherently incompatible with existing unitary-gate-based quantum methods. To fill this gap, this paper introduces a Variational Quantum Universal Eigensolver (VQUE), which is deployable on noisy intermediate scale quantum computers. Our new contributions include: (1) The first universal variational quantum algorithm capable of evaluating the eigenvalues of non-Hermitian matrices-Inspired by Schur's triangularization theory, VQUE unitarizes the eigenvalue problem to a procedure of searching unitary transformation matrices via quantum devices; (2) A Quantum Process Snapshot technique is devised to make VQUE maintain the potential quantum advantage inherited from the original variational quantum eigensolver-With additional [Formula: see text] quantum gates, this method efficiently identifies whether a unitary operator is triangular with respect to a given basis; (3) Successful deployment and validation of VQUE on a real noisy quantum computer, which demonstrates the algorithm's feasibility. We also undertake a comprehensive parametric study to validate VQUE's scalability, generality, and performance in realistic applications.
Abstract Variational quantum‐classical hybrid algorithms are emerging as important tools for simulating quantum chemistry with quantum devices. These algorithms can be applied to evaluate various molecular properties, including potential energy surfaces. Here in, recent progresses on the development of the so‐called variational quantum eigensolver (VQE) are surveyed. The eigensolver aims at reducing the consumption of quantum resources as much as possible. The key feature of VQE is that variation quantum states are optimized by a feedback process, where the measurement of the Hamiltonian is implemented term by term. This approach avoids the need of encoding all of the information about the molecular Hamiltonian in a quantum circuit. The VQE method is also compatible with classical methods in quantum chemistry, such as unitary coupled‐cluster ansatz. Furthermore, basic elements of VQE are covered, such as qubit encoding, mapping rules of the fermionic operators, ansatz preparation, together with several techniques for improving the performance, including constraining, and error mitigation.