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arXiv:2310.19504 · International Conference on Communication Systems & Networks (COMSNETS)

Novel Objective Function and Expectation Value Estimation Method for the Variational Quantum Singular Value Decomposition Algorithm

Jezer Jojo, Ankit Khandelwal, M Girish Chandra

Published Jan 3, 2024• 2 citations
arXivHTMLPDF

Abstract

In this work, we discuss two modifications that can be made to a known variational quantum singular value decomposition algorithm popular in the literature. The first is a change to the objective function which hints at improved performance of the algorithm and decreases the depth of the circuits. The second modification introduces a new way of computing expectation values of general matrices, which is a key step in the algorithm. We then benchmark this modified algorithm and compare the performance of our new objective function with the existing one.

DOI
10.1109/comsnets59351.2024.10427083

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In this work, we discuss two modifications that can be made to a known variational quantum singular value decomposition algorithm popular in the literature. The first is a change to the objective function which hints at improved performance of the algorithm and decreases the depth of the circuits. The second modification introduces a new way of computing expectation values of general matrices, which is a key step in the algorithm. We then benchmark this modified algorithm and compare the performance of our new objective function with the existing one.
Abstract We propose the variational quantum singular value decomposition based on encoding the elements of the considered N × N matrix into the state of a quantum system of appropriate dimension. This method doesn’t use the expansion of this matrix in terms of the unitary matrices. Controlled measurement is involved to avoid small success probability in ancilla measurement. The objective function for maximization algorithm can be obtained probabilistically via measurement of the states of two one-qubit subsystems. The circuit requires O (log N ) qubits for realization of this algorithm whose depths is proportional to log N / ε , where ε is the precision required for calculation of singular values.
Singular value decomposition (SVD) is fundamentally important and broadly useful in both quantum and classical computing. There are several quantum algorithms known for SVD. Variational quantum algorithms with parametric quantum circuits (PQC) are the most promising approach to find SVD with near-term quantum computers. This paper reports a new method for quantum SVD (QSVD) that finds the singular vectors ordered by the magnitude of singular values by adding extra CNOT gates and a new local cost function design. This ordering is important because the singular vectors with the larger singular values is more representative for the given information. The ordering property of this approach is investigated with the standard Iris dataset by numerical simulations of 4-qubit states. Methods for choosing appropriate choices for the cost function hyperparameter and cost function terms as well as an application example of a quantum encoder are also discussed.
Singular value decomposition is central to many problems in engineering and scientific fields. Several quantum algorithms have been proposed to determine the singular values and their associated singular vectors of a given matrix. Although these algorithms are promising, the required quantum subroutines and resources are too costly on near-term quantum devices. In this work, we propose a variational quantum algorithm for singular value decomposition (VQSVD). By exploiting the variational principles for singular values and the Ky Fan Theorem, we design a novel loss function such that two quantum neural networks (or parameterized quantum circuits) could be trained to learn the singular vectors and output the corresponding singular values. Furthermore, we conduct numerical simulations of VQSVD for random matrices as well as its applications in image compression of handwritten digits. Finally, we discuss the applications of our algorithm in recommendation systems and polar decomposition. Our work explores new avenues for quantum information processing beyond the conventional protocols that only works for Hermitian data, and reveals the capability of matrix decomposition on near-term quantum devices.
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Singular Value Decomposition (SVD) is one of the most useful techniques for analyzing data in linear algebra. SVD decomposes a rectangular real or complex matrix into two orthogonal matrices and one diagonal matrix. In this work we introduce a new approach to improve the preciseness of the standard Quantum Fourier Transform. The presented Quantum-SVD algorithm is based on the singular value decomposition mechanism. While the complexity of the proposed scheme is the same as the standard Quantum Fourier Transform, the precision of the Quantum-SVD approach is some orders higher. The Quantum-SVD approach also exploits the benefits of quantum searching.
Hybrid quantum-classical variational algorithms such as the variational quantum eigensolver (VQE) and the quantum approximate optimization algorithm (QAOA) are promising applications for noisy, intermediate-scale quantum computers. Both VQE and QAOA variationally extremize the expectation value of a Hamiltonian. All work to date on VQE and QAOA has been limited to Pauli representations of Hamiltonians. However, many cases exist in which a sparse representation of the Hamiltonian is known but there is no efficient Pauli representation. We extend VQE to general sparse Hamiltonians. We provide a decomposition of a fermionic second-quantized Hamiltonian into a number of one-sparse, self-inverse, Hermitian terms linear in the number of ladder operator monomials in the second-quantized representation. We provide a decomposition of a general d-sparse Hamiltonian into O(d^{2}) such terms. In both cases, a single sample of any term can be obtained using two ansatz state preparations and at most six oracle queries. The number of samples required to estimate the expectation value to precision ε scales as ε^{-2} as for Pauli-based VQE. This widens the domain of applicability of VQE to systems whose Hamiltonian and other observables are most efficiently described in terms of sparse matrices.