Recent research has demonstrated the exponential potential of hybrid quantum–classical algorithms (HQAs) in solving electromagnetic (EM) problems. However, the optimization objective of HQAs typically focuses on the magnitude of electric fields in EM problems, neglecting phase information. Consequently, the resulting quantum solution is inherently normalized to unity and subject to a global phase offset. This limitation prevents the calculation of the correct magnitude and phase of S-parameters. To address this challenge, we propose a block-encoding-based HQA (BEHQA) integrating a specialized quantum method for S-parameter extraction with magnitude and phase. The proposed method employs the specialized quantum method to reconstruct both the magnitude scaling and the phase offset, ensuring correct S-parameter extraction. Besides, HQAs fundamentally operate on unitary matrices, necessitating the explicit decomposition of nonunitary finite element method (FEM) matrices into many unitary operators. To mitigate the overhead associated with matrix decomposition, we introduce the block-encoding to embed the FEM matrix directly as a sub-block within a unitary operator. This strategy eliminates the need for decomposing FEM matrices to unitary operators, thereby reducing circuit complexity. For efficient optimization, we also derive a novel cost function specifically designed for the proposed BEHQA. The algorithm employs a strongly entangling ansatz (SEA) and gradient-based optimization to minimize the loss function. The effectiveness of BEHQA is demonstrated through its application to two microwave components, showcasing its feasibility for quantum-driven EM simulations.
Recent research has demonstrated the exponential potential of hybrid quantum–classical algorithms (HQAs) in solving electromagnetic (EM) problems. However, the optimization objective of HQAs typically focuses on the magnitude of electric fields in EM problems, neglecting phase information. Consequently, the resulting quantum solution is inherently normalized to unity and subject to a global phase offset. This limitation prevents the calculation of the correct magnitude and phase of S-parameters. To address this challenge, we propose a block-encoding-based HQA (BEHQA) integrating a specialized quantum method for S-parameter extraction with magnitude and phase. The proposed method employs the specialized quantum method to reconstruct both the magnitude scaling and the phase offset, ensuring correct S-parameter extraction. Besides, HQAs fundamentally operate on unitary matrices, necessitating the explicit decomposition of nonunitary finite element method (FEM) matrices into many unitary operators. To mitigate the overhead associated with matrix decomposition, we introduce the block-encoding to embed the FEM matrix directly as a sub-block within a unitary operator. This strategy eliminates the need for decomposing FEM matrices to unitary operators, thereby reducing circuit complexity. For efficient optimization, we also derive a novel cost function specifically designed for the proposed BEHQA. The algorithm employs a strongly entangling ansatz (SEA) and gradient-based optimization to minimize the loss function. The effectiveness of BEHQA is demonstrated through its application to two microwave components, showcasing its feasibility for quantum-driven EM simulations.
Quantum computing in the electromagnetic (EM) field represents a transformative frontier with significant potential for advancement. Recent studies have demonstrated the potential of quantum computing for solving waveguide mode problems, solving finite element method (FEM) equations for EM problems, and scaling the FEM solution. Among these applications, the Harrow–Hassidim–Lloyd (HHL) algorithm is an algorithm with large quantum advantages and has attracted attention. However, existing quantum methods remain constrained by the problem that the quantum-based FEM solution only obtains electric field magnitude and lacks phase information. The lack of phase information results in the inability to solve an important characteristic of EM problems: scattering parameters (S-parameters). To address the above issue, this work proposes a novel quantum method of calculating multiport S-parameter magnitudes based on FEM and HHL algorithm for microwave components. The proposed method introduces novel mathematical formulations optimized for quantum architectures. To streamline the quantum computation process and avoid the need for multiple quantum circuit simulations for each S-parameter, we design an integrated quantum circuit capable of computing magnitudes of all multiport S-parameters within a single execution. Furthermore, a specific matrix-scaling method is also developed to simplify the proposed circuits. The specific matrix-scaling method is used to improve the condition of the FEM matrix and reduce the complexity of the designed quantum circuits. The proposed method is validated using two microwave structures, demonstrating how our proposed quantum method computes S-parameter magnitudes in practical EM applications.
Quantum approach to address electromagnetic (EM) computing presents a promising problems. Recently, preliminary conducted on the application of the Harrow/Hassidim/Lloyd (HHL) quantum method to solve finite-element-method (FEM) equations in the EM domain. Since quantum measurements are typically limited to measuring the magnitude rather than the phase, only the magnitude of the electric field can be calculated by existing quantum computing methods. However, without access to phase information of the electric field, the final S-parameters cannot be solved in a traditional computing way. This paper proposes a new quantum method to calculate magnitudes of the two-port S-parameters of lossless waveguides. Novel formulations are derived to leverage two inner products to develop the complete calculation. The correctness and accuracy of the proposed method are demonstrated using a waveguide example.
This manuscript presents the Quantum Finite Element Method (Q-FEM) developed for use in noisy intermediate-scale quantum (NISQ) computers and employs the variational quantum linear solver (VQLS) algorithm. The proposed method leverages the classical FEM procedure to perform the unitary decomposition of the stiffness matrix and employs generator functions to design explicit quantum circuits corresponding to the unitaries. Q-FEM keeps the structure of the finite element discretization intact allowing for the use of variable element lengths and material coefficients in FEM discretization. The proposed method is tested on a steady-state heat equation discretized using linear and quadratic shape functions. Numerical verification studies are performed on the IBM QISKIT simulator and it is demonstrated that Q-FEM is effective in converging to the correct solution for a variety of problems and model discretizations, including with different element lengths, variable coefficients, and different boundary conditions. The formalism developed herein is general and can be extended to problems with higher dimensions. However, numerical examples also demonstrate that the number of parameters for the variational ansatz scale exponentially with the number of qubits, and increases the odds of convergence. Moreover, the deterioration of system conditioning with problem size results in barren plateaus and convergence difficulties.
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.
In quantum computing, every operation must be unitary to ensure reversibility and preserve probability amplitudes, which is a fundamental requirement for any quantum process. However, many real-world computations in areas such as machine learning, signal processing, and dynamical modeling inherently involve non-unitary matrices. Since these cannot be directly implemented on quantum hardware, a practical workaround is required. One promising solution is block-encoding, a framework that embeds a non-unitary matrix A in C^{n x n} into a larger unitary matrix U_A in C^{2n x 2n}, enabling the simulation of non-unitary actions within quantum circuits through ancilla preparation and post-selection. This thesis presents a novel and efficient method for block-encoding based on Gram-Schmidt orthogonalization. The algorithm constructs a unitary U_A with A embedded as the top-left block and remaining components filled to ensure global unitarity. Compared to traditional Singular Value Decomposition (SVD)-based approaches, the proposed method achieves at least 20% improvement in execution time, without compromising output fidelity or numerical stability. A corresponding quantum circuit is developed to extract A|psi> (that is, the transformed version of the input quantum state |psi> after applying the linear operator A) using post-selection on ancilla qubits, with the success probability determined by
Evaluating the entanglement spectrum is essential for characterizing exotic quantum phases such as quantum criticality and topological order. However, for large quantum many-body systems, this task is hindered by the exponential measurement complexity of standard tomographic techniques. To address this challenge, we introduce a hybrid quantum-classical variational framework for partial singular value decomposition of bipartite states, built on the canonical form of matrix product states. We employ a deflation-based optimization approach to sequentially extract dominant and subdominant Schmidt components of target states. Because hardware noise and finite circuit depth can compromise the mutual orthogonality of these extracted vectors, we propose an improved deflation algorithm incorporating explicit classical orthogonality correction. This classical post-processing acts as an error-filtering mechanism, enabling shallow and suboptimal quantum circuits. As a result, numerical accuracy is decoupled from quantum circuit optimization, mitigating optimization difficulties caused by barren plateaus and hardware noise. Furthermore, shallow ansatzes enable a concurrent execution strategy. Overlap matrices are evaluated by classical tensor network contractions, while cross terms between the target state and the extracted vectors are computed using an auxiliary reference state. This concurrent hybrid design improves computational throughput and bypasses the overhead of controlled target-state preparation. Numerical benchmarks on the ground states of one- and two-dimensional Heisenberg models demonstrate improved accuracy and numerical stability. By mitigating hurdles of circuit depth, optimization hardness, and measurement complexity, our framework provides a robust pathway for large-scale entanglement spectrum estimation on advanced near-term quantum devices.