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(ɛ) of function evaluations or queries that are necessary to compute an ɛ-approximation of the smallest eigenvalue. We prove that n(ɛ)=Θ(ɛ−1/2) in the (deterministic) worst case setting, and n(ɛ)=Θ(ɛ−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((1/2) 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 ɛ-approximation with probability (3/4) using n(ɛ)=Θ(ɛ−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(ɛ)=Θ(log ɛ−1) power queries, log 2ɛ−1 quantum operations, and (3/2) log ɛ−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 ɛ−1. The lower bound on the number of quantum queries is proven in Bessen (in preparation). 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 a forthcoming paper (in preparation).
Quantum Information Processing – Springer Journals
Published: Feb 9, 2005
It’s your single place to instantly
discover and read the research
that matters to you.
Enjoy affordable access to
over 18 million articles from more than
15,000 peer-reviewed journals.
All for just $49/month
Query the DeepDyve database, plus search all of PubMed and Google Scholar seamlessly
Save any article or search result from DeepDyve, PubMed, and Google Scholar... all in one place.
Get unlimited, online access to over 18 million full-text articles from more than 15,000 scientific journals.
Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more.
All the latest content is available, no embargo periods.
“Hi guys, I cannot tell you how much I love this resource. Incredible. I really believe you've hit the nail on the head with this site in regards to solving the research-purchase issue.”Daniel C.
“Whoa! It’s like Spotify but for academic articles.”@Phil_Robichaud
“I must say, @deepdyve is a fabulous solution to the independent researcher's problem of #access to #information.”@deepthiw
“My last article couldn't be possible without the platform @deepdyve that makes journal papers cheaper.”@JoseServera