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IEEE Microwave and Wireless Technology Letters

Estimation of Waveguide Eigenmodes Based on Subspace-Search Variational Quantum Algorithm

Zhuo‐Wei Miao, Fanxu Meng

Published Aug 27, 2024• 1 citation
Publisher

Abstract

The variational quantum eigensolver (VQE), a variational algorithm to approximate the ground state of the given Hamiltonian, is the leading candidate for receiving quantum advantage on noisy intermediate-scale quantum (NISQ) devices. The eigenmode analysis within waveguides, a canonical problem in electromagnetics, can be reformulated as an eigenvalue problem adopting the finite difference method. Therefore, in this work, a subspace-search-based VQE is applied for the computation of the eigenmodes. The proposed algorithm has the promise to show exponential efficiency outperforming over the classical algorithms. Compared with the existing counterparts, our work makes the additional orthogonality constrain unnecessary in the cost function, meantime, avoids the costly inner product evaluation in the cost function, where a few number of ancillary qubits and deeper quantum circuits are indispensable. Comprehensive experimental results show that the proposed framework significantly provides more accurate eigenmode estimation with fewer iterations and shows more favorable resource efficiency.

DOI
10.1109/lmwt.2024.3440058

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The variational quantum eigensolver (VQE), a variational algorithm to approximate the ground state of the given Hamiltonian, is the leading candidate for receiving quantum advantage on noisy intermediate-scale quantum (NISQ) devices. The eigenmode analysis within waveguides, a canonical problem in electromagnetics, can be reformulated as an eigenvalue problem adopting the finite difference method. Therefore, in this work, a subspace-search-based VQE is applied for the computation of the eigenmodes. The proposed algorithm has the promise to show exponential efficiency outperforming over the classical algorithms. Compared with the existing counterparts, our work makes the additional orthogonality constrain unnecessary in the cost function, meantime, avoids the costly inner product evaluation in the cost function, where a few number of ancillary qubits and deeper quantum circuits are indispensable. Comprehensive experimental results show that the proposed framework significantly provides more accurate eigenmode estimation with fewer iterations and shows more favorable resource efficiency.
Current quantum computers (QCs) belong to the noisy intermediate-scale quantum (NISQ) class, characterized by noisy qubits, limited qubit capabilities, and limited circuit depth. These limitations have led to the development of hybrid quantum-classical algorithms that split the computational cost between classical and quantum hardware. Among the hybrid algorithms, the variational quantum eigensolver (VQE) is mentioned. The VQE is a variational quantum algorithm designed to estimate the eigenvalues and eigenvectors of a system on universal-gate quantum architectures. A canonical problem in electromagnetics is the computation of eigenmodes within waveguides. Following the finite difference method, the wave equation can be recast as an eigenvalue problem. This work exploits the quantum superposition and entanglement in quantum computing to solve the square waveguide mode problem. This algorithm is expected to demonstrate exponentially efficiency over classical computational techniques as the qubit count increases. The simulations were performed on IBM’s three-qubit quantum simulator, Qasm IBM Simulator. A shot-based simulation was performed considering computationally based measurements of the quantum hardware. The results of the probabilistic read-out, reported in terms of 2-D eigenmode field distributions, are close to ideal values with a few number of qubits, confirming the possibility to exploit the quantum advantage to formulate innovative eigensolvers.
Quantum computing offers a promising paradigm for electromagnetic eigenmode analysis, enabling compact representations of complex field interactions and potential exponential speedup over classical numerical solvers. Recent efforts have applied variational quantum eigensolver (VQE) based methods to compute waveguide modes, demonstrating the feasibility of simulating TE and TM field distributions on noisy intermediate-scale quantum (NISQ) hardware. However, these studies typically employ manually designed, fixed-depth parameterized quantum circuits and uniform measurement-shot strategies, resulting in excessive quantum resource consumption, limited circuit expressivity, and reduced robustness under realistic noise conditions. To address these limitations, we propose an architecture and shot adaptive subspace variational quantum eigensolver for efficient microwave waveguide eigenmode simulation on NISQ devices. The proposed framework integrates a reinforcement learning (RL) based circuit design strategy and an adaptive shot allocation mechanism to jointly reduce quantum resource overhead. Specifically, the RL agent autonomously explores the quantum circuit space to generate hardware-efficient parameterized quantum circuits, while the adaptive measurement scheme allocates sampling resources according to Hamiltonian term weights. Numerical experiments on three- and five-qubit systems demonstrate that the proposed framework achieves accurate estimation of TE and TM mode eigenvalues, with a minimum absolute error down to 10−810^{-8}10−8 and reconstructed field distributions under noiseless conditions in excellent agreement with classical electromagnetic solutions.
Quantum computers have the potential to deliver speed-ups for solving certain important problems that are intractable for classical counterparts, making them a promising avenue for advancing modern computation. However, many quantum algorithms require deep quantum circuits, which are challenging to implement on current noisy devices. To address this limitation, variational quantum algorithms (VQAs) have been actively developed, enabling practical quantum computing in the noisy intermediate-scale quantum (NISQ) era. Among them, the variational quantum eigensolver (VQE) stands out as a leading approach for solving problems in quantum chemistry, many-body physics, and even integer factorization. The VQE algorithm can be implemented on various quantum hardware platforms, including photonic systems, quantum dots, trapped ions, neutral atoms, and superconducting circuits. In particular, photonic platforms offer several advantages: they operate at room temperature, exhibit low decoherence, and support multiple degrees of freedom, making them suitable for scalable, high-dimensional quantum computation. Here we present methodologies for realizing VQE on photonic systems, highlighting their potential for practical quantum computing. We first provide a theoretical overview of the VQE framework, focusing on the procedure for variationally estimating ground state energies. We then explore how photonic systems can implement these processes, showing that a wide variety of problems can be addressed using either multiple qubit states or a single qudit state.
We describe the contextual subspace variational quantum eigensolver (CS-VQE), a hybrid quantum-classical algorithm for approximating the ground state energy of a Hamiltonian. The approximation to the ground state energy is obtained as the sum of two contributions. The first contribution comes from a noncontextual approximation to the Hamiltonian, and is computed classically. The second contribution is obtained by using the variational quantum eigensolver (VQE) technique to compute a contextual correction on a quantum processor. In general the VQE computation of the contextual correction uses fewer qubits and measurements than the VQE computation of the original problem. Varying the number of qubits used for the contextual correction adjusts the quality of the approximation. We simulate CS-VQE on tapered Hamiltonians for small molecules, and find that the number of qubits required to reach chemical accuracy can be reduced by more than a factor of two. The number of terms required to compute the contextual correction can be reduced by more than a factor of ten, without the use of other measurement reduction schemes. This indicates that CS-VQE is a promising approach for eigenvalue computations on noisy intermediate-scale quantum devices.
The contextual subspace variational quantum eigensolver (CS-VQE) is a hybrid quantum-classical algorithm that approximates the ground-state energy of a given qubit Hamiltonian. It achieves this by separating the Hamiltonian into contextual and noncontextual parts. The ground-state energy is approximated by classically solving the noncontextual problem, followed by solving the contextual problem using VQE, constrained by the noncontextual solution. In general, computation of the contextual correction needs fewer qubits and measurements compared with solving the full Hamiltonian via traditional VQE. We simulate CS-VQE on different tapered molecular Hamiltonians and apply the unitary partitioning measurement reduction strategy to further reduce the number of measurements required to obtain the contextual correction. Our results indicate that CS-VQE combined with measurement reduction is a promising approach to allow feasible eigenvalue computations on noisy intermediate-scale quantum devices. We also provide a modification to the CS-VQE algorithm; the CS-VQE algorithm previously could cause an exponential increase in Hamiltonian terms but with this modification now at worst will scale quadratically.
The advent of noisy intermediate-scale quantum (NISQ) systems signifies an important stage in quantum computing development. Despite the constraints due to their limited qubit numbers and noise susceptibility, NISQ devices exhibit substantial potential to tackle complex computational challenges via hybrid classical-quantum algorithms. Among the various hybrid algorithms, variational quantum algorithms (VQAs) are gaining increasing attention due to their ability to solve highly complex, large-scale problems where classical algorithms fail. In particular, the variational quantum eigensolver (VQE) shows its potential in calculating the energies and ground states of large systems, where the complexity of solving such problems grows exponentially and becomes intractable for classical computers. At this regard, the aim of this paper is to extend the use of VQE for solving circular waveguide modes to verify their applicability to mathematically complex EM problems. In particular, we propose to calculate the fundamental and the some higher order modes for both transverse electric and transverse magnetic cases in circular waveguides. This is mathematically challenging due to the nature of geometry and the associated boundary conditions of circular structures. The results confirm the possibility of applying VQE for mathematically complex EM problems, announcing its potential to scale up and solve high-dimensional, large-scale EM problems where classical algorithms can fail.