Classical and Quantum Complexity of the Sturm-Liouville Eigenvalue Problem

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 C2([0, 1]) and study the minimal number n(ε) of function evaluations or queries that are necessary to compute an ε-approximation of the smallest eigenvalue. We prove that n(ε) = Θ(ε−1/2) in the (deterministic) worst case setting, and n(ε) = Θ(ε−2/5) in the randomized setting. The quantum setting offers a polynomial speedup with bit queries and an exponential speedup with 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(12 iM), where M is an n × 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. In particular, we show how to compute an ε-approximation with probability 3 4 using n(ε) = Θ(ε−1/3) bit queries. For power queries, we use the phase estimation algorithm as a basic tool and present the algorithm that solves the problem using n(ε) = Θ(log ε−1) power queries, log ε−1 quantum operations, and 3 2 log ε −1 quantum bits. We also prove that the minimal number of qubits needed for this problem (regardless of the kind of queries used) is at least roughly 1 2 log ε −1. The lower bound on the number of quantum queries is proven in [5]. We derive a formula that relates the Sturm-Liouville eigenvalue problem to a weighted integration problem. Many computational problems may be recast as this weighted integration problem, which allows us to solve them with a polylog number of power queries. Examples include Grover’s search, the approximation of the Boolean mean, NP-complete problems, and many multivariate integration problems. In this paper we only provide the relationship formula. The implications are covered in [25].