We study the approximation of the smallest eigenvalue of a Sturm-Liouville problem in the classical and quantum settings. We consider a univariate Sturm-Liouville eigenvalue problem with a nonnegative function $q$ from the class $C^2([0,1])$ and study the minimal number $n(\e)$ of function evaluations or queries that are necessary to compute an $\e$-approximation of the smallest eigenvalue. We prove that $n(\e)=Θ(\e^{-1/2})$ in the (deterministic) worst case setting, and $n(\e)=Θ(\e^{-2/5})$ in the randomized setting. The quantum setting offers a polynomial speedup with {\it bit} queries and an exponential speedup with {\it power} queries. Bit queries are similar to the oracle calls used in Grover's algorithm appropriately extended to real valued functions. Power queries are used for a number of problems including phase estimation. They are obtained by considering the propagator of the discretized system at a number of different time moments. They allow us to use powers of the unitary matrix $\exp(\tfrac12 {\rm i}M)$ , where $M$ is an $n\times n$ matrix obtained from the standard discretization of the Sturm-Liouville differential operator. The quantum implementation of power queries by a number of elementary quantum gates that is polylog in $n$ is an open issue.We study the approximation of the smallest eigenvalue of a Sturm-Liouville problem in the classical and quantum settings. We consider a univariate Sturm-Liouville eigenvalue problem with a nonnegative function $q$ from the class $C^2([0,1])$ and study the minimal number $n(\e)$ of function evaluations or queries that are necessary to compute an $\e$-approximation of the smallest eigenvalue. We prove that $n(\e)=Θ(\e^{-1/2})$ in the (deterministic) worst case setting, and $n(\e)=Θ(\e^{-2/5})$ in the randomized setting. The quantum setting offers a polynomial speedup with {\it bit} queries and an exponential speedup with {\it power} queries. Bit queries are similar to the oracle calls used in Grover's algorithm appropriately extended to real valued functions. Power queries are used for a number of problems including phase estimation. They are obtained by considering the propagator of the discretized system at a number of different time moments. They allow us to use powers of the unitary matrix $\exp(\tfrac12 {\rm i}M)$, where $M$ is an $n\times n$ matrix obtained from the standard discretization of the Sturm-Liouville differential operator. The quantum implementation of power queries by a number of elementary quantum gates that is polylog in $n$ is an open issue.

arXiv:quant-ph/0502054Feb 9, 20056 citations

WKB analysis of $\mathcal P$ $\mathcal T$ -symmetric Sturm–Liouville problemsWKB analysis of $\mathcal P$$\mathcal T$-symmetric Sturm–Liouville problems

Carl M. Bender, Hugh F Jones

Most studies of -symmetric quantum-mechanical Hamiltonians have considered the Schrödinger eigenvalue problem on an infinite domain. This paper examines the consequences of imposing the boundary conditions on a finite domain. As is the case with regular Hermitian Sturm–Liouville problems, the eigenvalues of the -symmetric Sturm–Liouville problem grow like n 2 for large n . However, the novelty is that a eigenvalue problem on a finite domain typically exhibits a sequence of critical points at which pairs of eigenvalues cease to be real and become complex conjugates of one another. For the potentials considered here this sequence of critical points is associated with a turning point on the imaginary axis in the complex plane. WKB analysis is used to calculate the asymptotic behaviours of the real eigenvalues and the locations of the critical points. The method turns out to be surprisingly accurate even at low energies. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Quantum physics with non-Hermitian operators’.Most studies of -symmetric quantum-mechanical Hamiltonians have considered the Schrödinger eigenvalue problem on an infinite domain. This paper examines the consequences of imposing the boundary conditions on a finite domain. As is the case with regular Hermitian Sturm–Liouville problems, the eigenvalues of the -symmetric Sturm–Liouville problem grow like n 2 for large n . However, the novelty is that a eigenvalue problem on a finite domain typically exhibits a sequence of critical points at which pairs of eigenvalues cease to be real and become complex conjugates of one another. For the potentials considered here this sequence of critical points is associated with a turning point on the imaginary axis in the complex plane. WKB analysis is used to calculate the asymptotic behaviours of the real eigenvalues and the locations of the critical points. The method turns out to be surprisingly accurate even at low energies. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Quantum physics with non-Hermitian operators’.

arXiv:1201.1234Journal of Physics A Mathematical and TheoreticalOct 23, 201213 citations

q

from the class

C^2([0,1])

and study the minimal number $n(\e)$ of function evaluations or queries that are necessary to compute an $\e$-approximation of the smallest eigenvalue. We prove that $n(\e)=Θ(\e^{-1/2})$ in the (deterministic) worst case setting, and $n(\e)=Θ(\e^{-2/5})$ in the randomized setting. The quantum setting offers a polynomial speedup with {\it bit} queries and an exponential speedup with {\it power} queries. Bit queries are similar to the oracle calls used in Grover's algorithm appropriately extended to real valued functions. Power queries are used for a number of problems including phase estimation. They are obtained by considering the propagator of the discretized system at a number of different time moments. They allow us to use powers of the unitary matrix

\exp(\tfrac12 {\rm i}M)

, where

M

is an

n\times n

matrix obtained from the standard discretization of the Sturm-Liouville differential operator. The quantum implementation of power queries by a number of elementary quantum gates that is polylog in

n

is an open issue.

WKB analysis of

\mathcal P

\mathcal T

-symmetric Sturm–Liouville problems