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Exploring Optimal and Suboptimal Beamforming Solutions Using Weighted Subspace-Search-VQE

Bidisha Dhara, Monika Agrawal, Sumantra Dutta Roy

Published Jan 6, 2025• 0 citations
Publisher

Abstract

This work explores the application of a Quantum algorithm, namely Weighted Subspace-Search-Variational-Quantum-Eigensolver (SSVQE), a variant of the traditional VQE algorithm, to predict optimal as well as sub-optimal sparse array configurations for the problem of beamforming. The optimization formulation is designed to maximize the Signal-to-Interference-plus-Noise Ratio (SINR), which is then reformulated into Quadratic Unconstrained Binary Optimization (QUBO) representation to serve as the Hamiltonian for the weighted SSVQE algorithm. The results obtained are presented along with classically obtained SINR plots to facilitate a comparison of the sparse array configurations. Our findings indicate that the output from the quantum algorithm precisely match with the results obtained from the classical method.

DOI
10.1109/comsnets63942.2025.10885773

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This work explores the application of a Quantum algorithm, namely Weighted Subspace-Search-Variational-Quantum-Eigensolver (SSVQE), a variant of the traditional VQE algorithm, to predict optimal as well as sub-optimal sparse array configurations for the problem of beamforming. The optimization formulation is designed to maximize the Signal-to-Interference-plus-Noise Ratio (SINR), which is then reformulated into Quadratic Unconstrained Binary Optimization (QUBO) representation to serve as the Hamiltonian for the weighted SSVQE algorithm. The results obtained are presented along with classically obtained SINR plots to facilitate a comparison of the sparse array configurations. Our findings indicate that the output from the quantum algorithm precisely match with the results obtained from the classical method.
Abstract This study investigates the application of quantum algorithms, specifically the Variational Quantum Eigensolver (VQE) and the Quantum Approximate Optimization Algorithm (QAOA), to design optimal sensor configurations for beamforming, enhancing signal quality and overall system performance. We propose two distinct optimization formulations: one aimed at maximising array gain while the other aimed at maximising signal‐to‐noise‐interference ratio (SINR). Our findings show that the outputs obtained from quantum algorithms are consistent with those derived from classical methods.
The Quantum Approximate Optimization Algorithm (QAOA) is a promising quantum algorithm for solving combinatorial optimization problems. In this work, we explore the Recursive QAOA (RQAOA) approach, which improves upon the standard QAOA by successively reducing the problem size, to optimize sensor configuration for beamforming with the objective of maximizing the Signal-to-Interference-plus-Noise Ratio (SINR). We present a series of experiments across various setups, including different numbers of sensors and a range of optimizers. Additionally, we compare the obtained SINR with the maximum achievable SINR to assess the efficiency of the algorithm.
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The weighted subspace-search variational quantum eigensolver (SSVQE) is a prominent algorithm for calculating excited-state properties of molecular quantum systems. In this work, we elaborate on some of its fundamental features with the aim of improving its practical realization. First, we demonstrate that the initial ansätze for various excited states could be prepared into a single pure state through a minimal number of ancilla qubits, followed by the optimization of a subsequent global unitary rotation in the targeted subspace. Since the ancillas' sole purpose is to purify an underlying ensemble ρwρ_{\boldsymbol{w}}ρw​ state with spectral weights w\boldsymbol{w}w, their measurement would just collapse ρwρ_{\boldsymbol{w}}ρw​ with probabilities wjw_jwj​ to one of its eigenstates ∣Ψj⟩|Ψ_j \rangle∣Ψj​⟩. We thus observe that our realization of SSVQE is equivalent to the original SSVQE improved by importance sampling. Then, we elaborate by numerical means on the potential influence of the auxiliary weights w\boldsymbol{w}w on the accuracy of the sought-after eigenstates and eigenenergies. Clear trends are discovered which are contrasted with some recent mathematical results concerning the ensemble variational principle that underlies SSVQE.
Abstract The use of the variational quantum eigensolver (VQE) on near-term quantum computers for solving Kohn–Sham Hamiltonian is still underexplored. In this study, we apply the weighted subspace search variational quantum eigensolver (weighted SSVQE) to obtain the occupied KS orbitals, their corresponding energies, and the total energy, exploring the effects of varying entanglement unitary blocks, circuit depths, and optimization strategies. While the choice of optimization algorithm has minimal impact on the total energy, it significantly influences convergence behavior, stability, and computational time. Our results further reveal the existence of an optimal weight configuration that achieves both faster convergence and improved energy accuracy. Furthermore, our findings highlight the critical role of ansatz depth and entanglement unitary blocks on total energy calculations across molecular geometries. The most effective design begins with a low-expressibility, low-entangling ansatz, such as linear entanglement unitary blocks, and repeats it until a sufficient number of layers is achieved. Using homoatomic and heteroatomic molecule as test sets, the approach remains valid and effective across systems exhibiting different types of electron correlation. This approach proves highly efficient, enabling shallow circuits that avoid convergence issues and closely match standard density functional calculations. This work thus offers a design strategy for shallow, convergence-friendly VQE circuits that match standard density functional energies.
The variational quantum eigensolver (VQE) is an algorithm for finding the ground states of a given Hamiltonian. Its application to binary-formulated combinatorial optimization (CO) has been widely studied in recent years. However, typical VQE approaches for CO problems often suffer from local minima or barren plateaus, limiting their ability to achieve optimal solutions. The quantum imaginary time evolution (QITE) offers an alternative approach for effective ground-state preparation but requires large circuits to approximate non-unitary operations. Although compressed QITE (cQITE) reduces circuit depth, accumulated errors eventually cause energy increases. To address these challenges, we propose an Adaptive Weighted QITE-VQE (AWQV) algorithm that integrates the VQE gradients with the cQITE updates through an adaptive weighting scheme during optimization. In numerical simulations for MaxCut on unweighted regular graphs, AWQV achieves near-optimal approximation ratios, while for weighted Erdős-Rényi instances, it outperforms the classical Goemans-Williamson algorithm.