In this work, we propose quantum assisted eigenvalue estimation and target detection algorithms for a large sensor array via Hamiltonian simulation. Quantum algorithms provide complexity advantage of a certain class of problems on a quantum computer with fewer physical resources as compared to their classical counterparts. The proposed algorithms make use of the quantum phase estimation (QPE) as its core computing component. We have introduced an analytical quantum framework to map from classical to quantum in the context of target detection. Target detection involves an appropriate choice of threshold based on the probability of detection or false alarm. We exploited the massive sensor array structure and invoked the random matrix theory to propose an optimal threshold. It also takes into account the quantum measurement noise in the framework. Numerical simulations are performed to ascertain the efficacy of the proposed framework. The results suggest near term applications of the quantum algorithm for large-scale linear systems.
In this work, we propose quantum assisted eigenvalue estimation and target detection algorithms for a large sensor array via Hamiltonian simulation. Quantum algorithms provide complexity advantage of a certain class of problems on a quantum computer with fewer physical resources as compared to their classical counterparts. The proposed algorithms make use of the quantum phase estimation (QPE) as its core computing component. We have introduced an analytical quantum framework to map from classical to quantum in the context of target detection. Target detection involves an appropriate choice of threshold based on the probability of detection or false alarm. We exploited the massive sensor array structure and invoked the random matrix theory to propose an optimal threshold. It also takes into account the quantum measurement noise in the framework. Numerical simulations are performed to ascertain the efficacy of the proposed framework. The results suggest near term applications of the quantum algorithm for large-scale linear systems.
Vandermonde matrix finds a wide range of applications in the digital signal processing domain including antenna array processing. Eigenspectrum estimation remains a challenge in such a system when the matrix dimension increases. This precise problem can be well addressed in a quantum simulation framework based on quantum Hamiltonian simulation (QHS). However, the structural exploitation of the matrix is never considered, which can give benefits in terms of quantum gate complexity. In this work, we exploit the special structural property of a dense Vandermonde matrix by decomposing it into several sparse matrices and then use them for a complexity-efficient eigenvalue spectrum estimation in a quantum framework. In this context, we propose an iterative quantum simulation framework for a Vandermonde-structured Hamiltonian called quantum Vandermonde simulation (QVS). The proposed approach requires fewer quantum gate operations compared to the standard QHS of a dense Vandermonde matrix. In this work, we have considered a large antenna array system, that exploits a delay Vandermonde matrix (DVM) to create a multi-beam beamformer using the proposed low complex quantum architecture. Extensive theoretical and numerical analysis in terms of complexity and other quality parameters have been conducted to show the benefit of the proposed method.
Toeplitz matrix reconstruction algorithms (TMRAs) are one of the central subroutines in array processing for wireless communication applications. The classical TMRAs have shown excellent accuracy in the spectral estimation for both uncorrelated and coherence sources in the recent era. However, TMRAs incorporate the classical eigenvalue decomposition technique for estimating the eigenvalues of the Toeplitz-structured covariance matrices that demand very high computational complexity for large arrays. We demonstrate a low-complexity quantum simulation framework exploiting the structured Hamiltonian of Toeplitz and circulant variants. In this framework, we consider two approaches for the estimation of the eigenvalue spectrum of a given Toeplitz-structured matrix: first, an analytical framework with Jordan form-based sparse-decomposition of a dense-Toeplitz matrix, and second, an approximation method for the conversion of a Toeplitz matrix into a circulant matrix embedding quantum subroutines. We have also compared the efficacy of the proposed algorithms with standard Hamiltonian simulation and quantum phase estimation techniques for different quantum time resolutions and gate complexities. We show quantum gate-complexity analysis for our proposed algorithms. Considering the large dimensions of the Toeplitz matrix, we have employed random matrix theory in deriving the error bounds for the estimated eigenvalues. The numerical results are obtained partly in a classical computer and in an IBM quantum simulator.
Quantum computing is certainly one of the greatest advances in the computer science field. Thanks to the parallelism and entanglement properties, it has proved to offer several advantages compared to the classical algorithms especially in the great reduction of the processing time. Quantum phase estimation (QPE) is one of the most important algorithms for quantum computing. It is known as the eigenvalue finding module for unitary operators. The Fourier transform is the key to this procedure. It has been researched and used to solve many problems such as the order finding problem, and the factoring problem. It was also applied for quantum sampling algorithms and the calculation of the eigenvalues of unitary matrices. In this paper, we study three important quantum algorithms for machine learning that use the QPE algorithm as a subroutine: the quantum principal components analysis (PCA) for data visualization, the Harrow-Hassidim-Lloyd (HHL) algorithm for solving linear systems, and the quantum singular value thresholding (SVT) for matrix completion in recommender systems. We also discuss the advantages and limits of such algorithms compared to their classical versions. Then we discuss potential ways of amelioration of such algorithms, and end with a proposed approach for further improvement.
This paper introduces a pioneering method in the realm of signal processing for Massive Multiple-Input Multiple-Output (MIMO) systems, leveraging the synergistic potential of hybrid quantum-classical computing paradigms. The escalating demand for high data rate communications necessitates the advancement of MIMO technologies. However, the computational complexity associated with signal processing in such systems, especially in scenarios involving large array sizes, presents a significant challenge. This study proposes an innovative framework that integrates quantum computing algorithms with traditional classical methods to address this challenge. The quantum algorithms are specifically designed to expedite matrix operations and optimization tasks that are central to MIMO signal processing, such as channel estimation and precoding. A comparative analysis with purely classical approaches demonstrates a substantial improvement in computational efficiency and latency reduction, while maintaining, and in certain aspects enhancing, the accuracy of signal detection and system throughput. Furthermore, this hybrid approach offers scalability and adaptability to varying channel conditions and system configurations, a crucial feature for next-generation wireless communication systems. The theoretical foundation is complemented by simulation results, showcasing the feasibility and effectiveness of this approach in practical scenarios. This research not only paves the way for advanced signal processing techniques in Massive MIMO systems but also opens new horizons in the application of quantum computing in telecommunications.
Accurate computation of multiple eigenvalues of quantum Hamiltonians is essential in quantum chemistry, materials science, and molecular spectroscopy. Estimating excited-state energies is challenging for classical algorithms due to exponential scaling with system size, posing an even harder problem than ground-state calculations. We present a quantum algorithm for estimating eigenvalues and singular values of parametrized matrix families, including solving generalized eigenvalue problems that frequently arise in quantum simulations. Our method uses quantum amplitude amplification and phase estimation to identify matrix eigenvalues by locating minima in the singular value spectrum. We demonstrate our algorithm by proposing a quantum computing formulation of the pseudospectral collocation method for the Schrödinger equation. We estimate fault-tolerant quantum resource requirements for the quantum collocation method, showing favorable scaling in the size of the problem N [up to O ̃ ( N ) ] compared to classical implementations with O ̃ ( N ) , for certain well-behaved potentials. Additionally, unlike the standard collocation method, which results in a generalized eigenvalue problem requiring matrix inversion, our algorithm circumvents the associated numerical instability by scanning a parametrized matrix family and detecting eigenvalues through singular value minimization. This approach is particularly effective when multiple eigenvalues are needed or when the generalized eigenvalue problem involves a high condition number. In the fault-tolerant era, our method may thus be useful for simulating high-dimensional molecular systems with dense spectra involving highly excited states, such as those encountered in molecular photodynamics or quasicontinuum regimes in many-body and solid-state systems.
Quantum phase estimation is a central primitive in quantum algorithms and sensing, where performance is governed by the sensitivity of measurement signals to the target parameter. While existing methods have developed increasingly sophisticated inference and adaptive design strategies, the signal family used for phase learning is often largely pre-specified. Here we propose a programmable signal design framework for quantum phase estimation based on quantum signal processing, which enables the measurement signal to be tailored to the current uncertainty region. We cast phase estimation as a max-min optimization problem over admissible signals and introduce a sensitivity efficiency parameter that quantifies information gain per query depth. The resulting iterative algorithm combines optimized quantum signal transformations with structured classical inference, retaining Heisenberg-limited scaling while improving sensitivity efficiency and practical resource prefactors. Numerical results show reduced estimation variance compared with standard protocols such as robust phase estimation. Our framework also extends to Hamiltonian eigenvalue estimation in higher dimensions and establishes a quantum-classical co-design paradigm through programmable signal shaping.