Related papers: An Exponential Separation Between Quantum Query Co…
We show that almost all n-bit Boolean functions have bounded-error quantum query complexity at least n/2, up to lower-order terms. This improves over an earlier n/4 lower bound of Ambainis, and shows that van Dam's oracle interrogation is…
We shed new light on entanglement measures in multipartite quantum systems by taking a computational-complexity approach toward quantifying quantum entanglement with two familiar notions--approximability and distinguishability. Built upon…
The relationship between quantum physics and discrete mathematics is reviewed in this article. The Boolean functions unitary representation is considered. The relationship between Zhegalkin polynomial, which defines the algebraic normal…
We exhibit a Boolean function for which the quantum communication complexity is exponentially larger than the classical information complexity. An exponential separation in the other direction was already known from the work of Kerenidis…
We show that a certain tensor norm, the completely bounded norm, can be expressed by a semidefinite program. This tensor norm recently attracted attention in the field of quantum computing, where it was used by Arunachalam, Bri\"{e}t and…
Recent breakthroughs in quantum query complexity have shown that any formula of size n can be evaluated with O(sqrt(n)log(n)/log log(n)) many quantum queries in the bounded-error setting [FGG08, ACRSZ07, RS08b, Rei09]. In particular, this…
Chattopadhyay, Mande and Sherif (ECCC 2018) recently exhibited a total Boolean function, the sink function, that has polynomial approximate rank and polynomial randomized communication complexity. This gives an exponential separation…
In this note we investigate the relationship between worst-case quantum query complexity and average-case classical query complexity. Specifically, we show that if a quantum computer can evaluate a total Boolean function f with bounded…
We consider quantum versions of two well-studied classical learning models: Angluin's model of exact learning from membership queries and Valiant's Probably Approximately Correct (PAC) model of learning from random examples. We give…
This paper gives a nearly tight characterization of the quantum communication complexity of the permutation-invariant Boolean functions. With such a characterization, we show that the quantum and randomized communication complexity of the…
We show that there exists a Boolean function $F$ which observes the following separations among deterministic query complexity $(D(F))$, randomized zero error query complexity $(R_0(F))$ and randomized one-sided error query complexity…
It is proved that the universal degree bound for separating polynomial invariants of a finite abelian group (in non-modular characteristic) is strictly smaller than the universal degree bound for generators of polynomial invariants, unless…
We give an explicit upper bound for the algebraic degree and an explicit lower bound for the absolute value of the minimum of a polynomial function on a compact connected component of a basic closed semialgebraic set when this minimum is…
Quantum computers can sometimes exponentially outperform classical ones, but only for problems with sufficient structure. While it is well known that query problems with full permutation symmetry can have at most polynomial quantum speedup…
We present a number of results related to quantum algorithms with small error probability and quantum algorithms that are zero-error. First, we give a tight analysis of the trade-offs between the number of queries of quantum search…
While in general there is no one-to-one correspondence between complex and quaternion quantum mechanics (QQM), there exists at least one version of QQM in which a {\em partial} set of {\em translations} may be made. We define these…
The hidden shift problem is a natural place to look for new separations between classical and quantum models of computation. One advantage of this problem is its flexibility, since it can be defined for a whole range of functions and a…
Relations between the decision tree complexity and various other complexity measures of Boolean functions is a thriving topic of research in computational complexity. It is known that decision tree complexity is bounded above by the cube of…
We study when partial Boolean functions can (and cannot) exhibit superpolynomial quantum query speedups, and develop a general framework for ruling out such speedups via two complementary lenses: promise-aware complexity measures and…
We present a new example of a partial boolean function whose one-way quantum communication complexity is exponentially lower than its one-way classical communication complexity. The problem is a natural generalisation of the previously…