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.
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.
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 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.
Current NIQST machines have a relatively small number of qubits and a limited circuit depth. In this regard, a class of quantum variational algorithms (VQAs) have been developed to be able to split the computational cost between classical and quantum hardware. In this study, the VQE algorithm has been used in order to solve the problem of propagation modes in rectangular waveguides described analytically by the Helmholtz partial differential equation. The IBM quantum device called manila was used in these calculations to provide noise present in physical hardware. In order to implement the VQE, IBM's proprietary framework called Qiskit has been used. The results show good agreement between calculations run on simulated quantum hardware and those run on real quantum hardware. This agreement between theory and experiment is evidence that high fidelity modeling of electromagnetic fields utilizing quantum hardware is possible.
The efficient computation of higher-order modes in multiconductor transmission lines is crucial, as these modes alter the distribution of TEM modes and increase cross-talk, affecting electromagnetic compatibility and signal integrity in high-frequency circuits. Traditional numerical methods face challenges in handling large-scale eigenvalue problems due to increasing computational complexity. Quantum computing offers a promising alternative by leveraging quantum principles such as superposition and entanglement to solve large eigenvalue problems more efficiently than classical solvers. In this work, we explore the variational quantum eigensolver as a quantum-assisted method for waveguide modal analysis. Starting from the Helmholtz equation for TM modes, we discretize the system using the finite difference method, map the Hamiltonian onto the Pauli basis, and implement the VQE with a hardware-efficient ansatz optimized via BFGS on the Qiskit statevector simulator of IBM [1]. As a test case, we analyze a shielded stripline. The quantum eigensolver successfully computes the first two TM modes and their cutoff frequencies while reconstructing the Ezand Exfield distributions at 1 GHz. This preliminary study shows the feasibility of quantum algorithms for solving large eigenvalue problems in computational electromagnetics where classical computing can fail, opening new possibilities for the efficient analysis of shielded multiconductor transmission lines, where higher-order modes significantly impact cross-talk and signal integrity. Future work will focus on scaling this approach to analyze multiconductor propagation in complex transmission-line structures.
We propose a divide-and-conquer method for the quantum-classical hybrid algorithm to solve larger problems with small-scale quantum computers.Specifically, we concatenate a variational quantum eigensolver (VQE) with a reduction in the system dimension, where the interactions between divided subsystems are taken as an effective Hamiltonian expanded by the reduced basis.Then the effective Hamiltonian is further solved by the VQE, which we call deep VQE.Deep VQE allows us to apply quantum-classical hybrid algorithms on small-scale quantum computers to large systems with strong intrasubsystem interactions and weak intersubsystem interactions, or strongly correlated spin models on large regular lattices.As proof-of-principle numerical demonstrations, we use the proposed method for quasi-one-dimensional models, including one-dimensionally coupled 12-qubit Heisenberg antiferromagnetic models on kagome lattices as well as two-dimensional Heisenberg antiferromagnetic models on square lattices.The largest problem size of 64 qubits is solved by simulating 20-qubit quantum computers with a reasonably good accuracy approximately a few %.The proposed scheme enables us to handle the problems of > 1000 qubits by concatenating VQEs with a few tens of qubits.While it is unclear how accurate ground-state energy can be obtained for such a large system, our numerical results on a 64-qubit system suggest that deep VQE provides a good approximation (discrepancy within a few percent) and has room for further improvement.Therefore, deep VQE provides us a promising pathway to solve practically important problems on noisy intermediate-scale quantum computers.
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.