Quantum lower bounds by polynomials

Tight Quantum Bounds by Polynomials

We examine the number T of oracle calls that a quantum network requires to compute some Boolean function on {0, 1} in the so-called black-box model, where the input is given as an oracle. We show that the acceptance probability of a network can be written as an N-variate polynomial of the input, having degree at most 2T . Using lower bounds on the degrees of polynomials that equal or approximat...

Communication Complexity Lower Bounds by Polynomials

The quantum version of communication complexity allows Alice and Bob to communicate qubits and/or to make use of prior entanglement (shared EPR-pairs). Some lower bound techniques are available for qubit communication [17, 11, 2], but except for the inner product function [11], no bounds are known for the model with unlimited prior entanglement. We show that the “log rank” lower bound extends t...

Quantum Lower Bounds by Quantum Arguments1

Lecture 11: Quantum Query Lower Bounds Using Polynomials

Let’s review some some algorithms we’ve learned so far, and revisit their query complexity. Recall Grover’s algorithm for unstructured search: Given access to a database of size N and some function f : [N ] → {0, 1} along with a promise that f(x) = 1 for a unique x∗, find x∗. Using a quantum computer, we find O( √ N) queries sufficient to find x∗. Now, we compare this result with query bounds o...

Lower Bounds Using Dual Polynomials ∗

Representations of Boolean functions by real polynomials play an important role in complexity theory. Typically, one is interested in the least degree of a polynomial p(x 1 ,. .. , x n) that approximates or sign-represents a given Boolean function f (x 1 ,. .. , x n). This article surveys a new and growing body of work in communication complexity that centers around the dual objects, i.e., poly...