Quantum lower bounds by entropy numbers

Quantum lower bounds by entropy numbers

We use entropy numbers in combination with the polynomial method to derive a new general lower bound for the n-th minimal error in the quantum setting of information-based complexity. As an application, we improve some lower bounds on quantum approximation of embeddings between finite dimensional Lp spaces and of Sobolev embeddings.

Quantum Lower Bounds by Quantum Arguments1

Entropy lower bounds of quantum decision tree complexity

We prove a general lower bound of quantum decision tree complexity in terms of some entropy notion. We regard decision tree computation as a communication process in which the oracle and the computer exchange several rounds of messages, each round consisting of O(log n) bits. Let E(f) be the Shannon entropy of the random variable f(X), where X is taken uniformly random in f ’s domain. Our main ...

Entropy lower bounds for quantum decision tree complexity

In this note we address the problem of quantum decision tree complexity lower bounds for computing functions that have large image size. We regard the computation as a communication process in which the oracle and the computer exchange messages, each of lg n + 1 bits. Let E(f) be the Shannon entropy of f(X) when X is uniformly random in f ’s domain. Our main result is, to compute any total func...

Lower bounds for Z-numbers

Let p/q be a rational noninteger number with p > q ≥ 2. A real number λ > 0 is a Zp/q-number if {λ(p/q)n} < 1/q for every nonnegative integer n, where {x} denotes the fractional part of x. We develop several algorithms to search for Zp/q-numbers, and use them to determine lower bounds on such numbers for several p and q. It is shown, for instance, that if there is a Z3/2-number, then it is grea...