Quantum phase estimation provides a path to quantum computation of solutions to Hermitian eigenvalue problems $Hv = λv$, such as those occurring in quantum chemistry. It is natural to ask whether the same technique can be applied to generalized eigenvalue problems $Av = λB v$, which arise in many areas of science and engineering. We answer this question affirmatively. A restricted class of generalized eigenvalue problems could be solved as efficiently as standard eigenvalue problems. A paradigmatic example is provided by Sturm--Liouville problems. Another example comes from linear ideal magnetohydrodynamics, where phase estimation could be used to determine the stability of magnetically confined plasmas in fusion reactors.
Based on an iterative method for solving the goundstate of Schroedinger equation, it is found that a kind of generalized Sombrero-shaped potentials in N-dimensional space has groundstates with zero eigenvalue. The restrictions on the parameters in the potential are discussed.
Three-Dimensional Eigenvalues of Harmonic Oscillator- and Coulomb-type Potentials from One-Dimensional Generalized Morse Potential: Perturbative Analyse based on Generalized Laguerre Polynomials
The generalized eigenvalue (GE) problems are of particular importance in various areas of science engineering and machine learning. We present a variational quantum algorithm for finding the desired generalized eigenvalue of the GE problem, $\mathcal{A}|ψ\rangle=λ\mathcal{B}|ψ\rangle$, by choosing suitable loss functions. Our approach imposes the superposition of the trial state and the obtained eigenvectors with respect to the weighting matrix $\mathcal{B}$ on the Rayleigh-quotient. Furthermore, both the values and derivatives of the loss functions can be calculated on near-term quantum devic…
We propose a quantum algorithm for finding eigenvalues of non-unitary matrices. We show how to construct, through interactions in a quantum system and projective measurements, a non-Hermitian or non-unitary matrix and obtain its eigenvalues and eigenvectors. This proposal combines ideas of frequent measurement, measured quantum Fourier transform, and quantum state tomography. It provides a generalization of the conventional phase estimation algorithm, which is limited to Hermitian or unitary matrices.
We consider several problems that involve finding the eigenvalues and generating the eigenstates of unknown unitary gates. We first examine Controlled-U gates that act on qubits, and assume that we know the eigenvalues. It is then shown how to use singlet states to produce qubits in the eigenstates of the gate. We then remove the assumption that we know the eigenvalues and show how to both find the eigenvalues and produce qubits in the eigenstates. Finally, we look at the case where the unitary operator acts on qutrits and has eigenvalues of 1 and -1, where the eigenvalue 1 is doubly degenerat…
Matrix encoding method in variational algorithm of calculating eigenvalues and generalized eigenvalues
Federal Research Center of Problems of Chemical Physics and Medicinal Chemistry RAS（俄罗斯科学院化学物理与医药化学问题联邦研究中心）；School of Mathematical Sciences, Zhejiang University（浙江大学数学科学学院）
We present an efficient method for estimating the eigenvalues of a Hamiltonian $H$ from the expectation values of the evolution operator for various times. For a given quantum state $ρ$, our method outputs a list of eigenvalue estimates and approximate probabilities. Each probability depends on the support of $ρ$ in those eigenstates of $H$ associated with eigenvalues within an arbitrarily small range. The complexity of our method is polynomial in the inverse of a given precision parameter $ε$, which is the gap between eigenvalue estimates. Unlike the well-known quantum phase estimation algori…
A quantum computer can efficiently find the order of an element in a group, factors of composite integers, discrete logarithms, stabilisers in Abelian groups, and `hidden' or `unknown' subgroups of Abelian groups. It is already known how to phrase the first four problems as the estimation of eigenvalues of certain unitary operators. Here we show how the solution to the more general Abelian `hidden subgroup problem' can also be described and analysed as such. We then point out how certain instances of these problems can be solved with only one control qubit, or `flying qubits', instead of entir…
Quantum phase estimation is the workhorse behind any quantum algorithm and a promising method for determining ground state energies of strongly correlated quantum systems. Low-cost quantum phase estimation techniques make use of circuits which only use a single ancilla qubit, requiring classical post-processing to extract eigenvalue details of the system. We investigate choices for phase estimation for a unitary matrix with low-depth noise-free or noisy circuits, varying both the phase estimation circuits themselves as well as the classical post-processing to determine the eigenvalue phases. W…
Many eigenvalue problems arising in practice are often of the generalized form $A\x=λB\x$. One particularly important case is symmetric, namely $A, B$ are Hermitian and $B$ is positive definite. The standard algorithm for solving this class of eigenvalue problems is to reduce them to Hermitian eigenvalue problems. For a quantum computer, quantum phase estimation is a useful technique to solve Hermitian eigenvalue problems. In this work, we propose a new quantum algorithm for symmetric generalized eigenvalue problems using ordinary differential equations. The algorithm has lower complexity than…
Eigenvalue problems for linear differential equations, such as time-independent Schrödinger equations, can be generalized to eigenvalue problems for nonlinear differential equations. In the nonlinear context a separatrix plays the role of an eigenfunction and the initial conditions that give rise to the separatrix play the role of eigenvalues. Previously studied examples of nonlinear differential equations that possess discrete eigenvalue spectra are the first-order equation $y'(x)=\cos[πxy(x)]$ and the first, second, and fourth Painlevé transcendents. It is shown here that the differential eq…
The eigenvalue of a Hamiltonian, $\mathcal{H}$, can be estimated through the phase estimation algorithm given the matrix exponential of the Hamiltonian, $exp(-i\mathcal{H})$. The difficulty of this exponentiation impedes the applications of the phase estimation algorithm particularly when $\mathcal{H}$ is composed of non-commuting terms. In this paper, we present a method to use the Hamiltonian matrix directly in the phase estimation algorithm by using an ancilla based framework: In this framework, we also show how to find the power of the Hamiltonian matrix-which is necessary in the phase est…
So far, topological band theory is discussed mainly for systems described by eigenvalue problems. Here, we develop a topological band theory described by a generalized eigenvalue problem (GEVP). Our analysis elucidates that non-Hermitian topological band structures may emerge for systems described by a GEVP with Hermitian matrices. The above result is verified by analyzing a two-dimensional toy model where symmetry-protected exceptional rings (SPERs) emerge although the matrices involved are Hermitian. Remarkably, these SPERs are protected by emergent symmetry, which is unique to the systems d…
