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IEEJ Transactions on Electronics Information and Systems

Simplification of the Gram Matrix Eigenvalue Problem for Quantum Signals Formed by Rotating Signal Points in a Circular Sector Region

Tiancheng Wang, Ryusuke Miyazaki, Souichi Takahira, Tsuyoshi Sasaki Usuda

Published Nov 30, 2022• 1 citation
Publisher

Abstract

Efficient computation of eigenvalues and eigenvectors of the Gram matrix for quantum signals is desirable in the field of quantum communication theory. Because various quantities such as the error probability, mutual information, channel capacity, and upper and lower bounds of the reliability function can be obtained by the eigenvalues and eigenvectors. Moreover, solving the eigenvalue problem also provides a matrix representation of quantum signals, which is useful for simulating quantum systems. In the case of symmetric signals, analytic solutions to the eigenvalue problem of the Gram matrix have been obtained and efficient computations are possible. However, for asymmetric signals, there is no analytic solution and universal numerical algorithms must be used. Recently, we have shown that, for certain asymmetric signals, the Gram matrix eigenvalue problem can be simplified and it is sufficient to consider the eigenvalue problem of smaller matrices at half or quarter size. In this paper, we consider nm-ary quantum signals formed by rotating signal points in a circular sector region and show that solving the 1/n-size matrix eigenvalue problem is sufficient to give the eigenvalues and eigenvectors of the Gram matrix.

DOI
10.1541/ieejeiss.142.1253

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Efficient computation of eigenvalues and eigenvectors of the Gram matrix for quantum signals is desirable in the field of quantum communication theory. Because various quantities such as the error probability, mutual information, channel capacity, and upper and lower bounds of the reliability function can be obtained by the eigenvalues and eigenvectors. Moreover, solving the eigenvalue problem also provides a matrix representation of quantum signals, which is useful for simulating quantum systems. In the case of symmetric signals, analytic solutions to the eigenvalue problem of the Gram matrix have been obtained and efficient computations are possible. However, for asymmetric signals, there is no analytic solution and universal numerical algorithms must be used. Recently, we have shown that, for certain asymmetric signals, the Gram matrix eigenvalue problem can be simplified and it is sufficient to consider the eigenvalue problem of smaller matrices at half or quarter size. In this paper, we consider nm-ary quantum signals formed by rotating signal points in a circular sector region and show that solving the 1/n-size matrix eigenvalue problem is sufficient to give the eigenvalues and eigenvectors of the Gram matrix.
In quantum information science, it is very important to solve the eigenvalue problem of the Gram matrix for quantum signals. This allows various quantities to be calculated, such as the error probability, mutual information, channel capacity, and the upper and lower bounds of the reliability function. Solving the eigenvalue problem also provides a matrix representation of quantum signals, which is useful for simulating quantum systems. In the case of symmetric signals, analytic solutions to the eigenvalue problem of the Gram matrix have been obtained, and efficient computations are possible. However, for asymmetric signals, there is no analytic solution and universal numerical algorithms that must be used, rendering the computations inefficient. Recently, we have shown that, for asymmetric signals such as amplitude-shift keying coherent-state signals, the Gram matrix eigenvalue problem can be simplified by exploiting its partial symmetry. In this paper, we clarify a method for simplifying the eigenvalue problem of the Gram matrix for quadrature amplitude modulation (QAM) signals, which are extremely important for applications in quantum communication and quantum ciphers. The results presented in this paper are applicable to ordinary QAM signals as well as modified QAM signals, which enhance the security of quantum cryptography.
Classical physics began to be superseded by quantum physics because it could not explain microscopic phenomena. Quantum physics employs mathematical approaches such as wave mechanics and matrix mechanics, one of which is the eigenvalue problem involving matrix operators. There are many types of matrix operators; however, not all are suitable for measuring physical quantities in quantum physics. Operators in quantum physics must be Hermitian to ensure that their eigenvalues are real. Eigenvalue problems for matrix operators can be solved either analytically or numerically. Therefore, the objective of this research is to analyze symmetric and antisymmetric matrix operators in the measurement of physical quantities through eigenvalue problems. The research method utilized is a theoretical computational approach. The results demonstrate that symmetric matrix operators are Hermitian and yield real eigenvalues. In contrast, antisymmetric matrix operators are non Hermitian, yielding imaginary or complex eigenvalues. The results of solving eigen problems using Python obtained relative errors between 0 and orders., while the absolute error remained below 10%, thus confirming the validity of the results. Based on the findings, symmetric matrix operators satisfy the requirements for use in physical problems through eigenvalue analysis. The eigenvalues of antisymmetric matrices cannot represent the measurement of physical quantities directly and serve only as rotation operators. However, antisymmetric matrices can be utilized in quantum physics for physical measurements if they are transformed into Hermitian matrices. Furthermore, solving eigenvalue problems with Python is effective due to its high accuracy and small error margins.
Solving linear systems and computing eigenvalues are two fundamental problems in linear algebra. For solving linear systems, many efficient quantum algorithms have been discovered. For computing eigenvalues, currently, we have efficient quantum algorithms for Hermitian and unitary matrices. However, the general case is far from fully understood. Combining quantum phase estimation, quantum algorithm to solve linear differential equations and quantum singular value estimation, we propose two quantum algorithms to compute the eigenvalues of diagonalizable matrices that only have real eigenvalues and normal matrices. The output of the quantum algorithms is a superposition of the eigenvalues and the corresponding eigenvectors. The complexities are dominated by solving a linear system of ODEs and performing quantum singular value estimation, which usually can be solved efficiently in a quantum computer. In the special case when the matrix MMM is sss-sparse, the complexity is O~(sρ2κ2/ε2)\widetilde{O}(sρ^2 κ^2/ε^2)O(sρ2κ2/ε2) for diagonalizable matrices that only have real eigenvalues, and O~(sρ∥M∥max⁡/ε2)\widetilde{O}(sρ\|M\|_{\max} /ε^2)O(sρ∥M∥max​/ε2) for normal matrices. Here ρρρ is an upper bound of the eigenvalues, κκκ is the conditioning of the eigenvalue problem, and εεε is the precision to approximate the eigenvalues. We also extend the quantum algorithm to diagonalizable matrices with complex eigenvalues under an extra assumption.
Eigenproblem arises in a large number of disciplines of sciences and engineering. Combining quantum algorithms to solve linear differential equations, quantum singular value estimation and quantum phase estimation, we propose quantum algorithms to compute the eigenvalues of diagonalizable matrices of two types: matrices that only have real eigenvalues and normal matrices. The quantum algorithms return a quantum state such that the first register stores eigenvalues and the second register stores the corresponding eigenvectors. For normal matrices, the complexity to obtain this quantum state is dominated by the complexity to perform quantum singular value estimation. For the other case, the complexity is determined by the solving of a linear system of differential equations. Under certain conditions (e.g. sparse), the complexities are polylog on the size nnn of the given matrix. If we perform measurements, then we can obtain all the eigenvalues classically. If MMM is sss sparse, then the complexity is O~(sn∥M∥max⁡κ2/ϵ2)\widetilde{O}(sn\|M\|_{\max}\kappa^2/\epsilon^2)O(sn∥M∥max​κ2/ϵ2), where κ\kappaκ is the condition number of the matrix of eigenvectors.
K. Nakahira et al. introduced iterative algorithms to determine the optimal quantum detection for general quantum state signals under the minimum-error and minimax criteria. In this paper, we aim to modify these algorithms, assuming that the signal states are pure and linearly independent. We demonstrate that the algorithms can be modified using the properties of the Gram matrix of linearly independent pure states. This modification eliminates the need to compute the inverse of the square root of the weighted Gram matrix, which is required in the original algorithms, thereby simplifying their implementation.
The eigenvalue density of a matrix plays an important role in various types of scientific computing such as electronic-structure calculations. In this paper, we propose a quantum algorithm for computing the eigenvalue density in a given interval. Our quantum algorithm is based on a method that approximates the eigenvalue counts by applying the numerical contour integral and the stochastic trace estimator applied to a matrix involving resolvent matrices. As components of our algorithm, the HHL solver is applied to an augmented linear system of the resolvent matrices, and the quantum Fourier transform (QFT) is adopted to represent the operation of the numerical contour integral. To reduce the size of the augmented system, we exploit a certain symmetry of the numerical integration. We also introduce a permutation formed by CNOT gates to make the augmented system solution consistent with the QFT input. The eigenvalue count in a given interval is derived as the probability of observing a bit pattern in a fraction of the qubits of the output state.