Related papers: Quantum integers and cyclotomy

For positive integers d, r, and M, we consider the class of rational functions on real d-dimensional space whose denominators are products of at most r functions of the form 1+Q(x) where each Q is a quadratic form with eigenvalues bounded…

While quantum computing provides an exponential advantage in solving system of linear equations, there is little work to solve system of nonlinear equations with quantum computing. We propose quantum Newton's method (QNM) for solving…

Quantum mechanics is formulated as a geometric theory on a Hilbert manifold. Images of charts on the manifold are allowed to belong to arbitrary Hilbert spaces of functions including spaces of generalized functions. Tensor equations in this…

This work deals with function theory on quantum complex hyperbolic spaces. The principal notions are expounded. We obtain explicit formulas for invariant integrals on `finite' functions on a quantum hyperbolic space and on the associated…

This paper initiates a systematic study of quantum functions, which are (partial) functions defined in terms of quantum mechanical computations. Of all quantum functions, we focus on resource-bounded quantum functions whose inputs are…

Let $\mathbb{F}_q$ denote the finite field of order $q,$ $n$ be a positive integer coprime to $q$ and $t \geq 2$ be an integer. In this paper, we enumerate all the complementary-dual cyclic $\mathbb{F}_q$-linear $\mathbb{F}_{q^t}$-codes of…

We give the q-analogue of the sums of the n-th powers of positive integers up to k-1.

In this paper, a novel class of quantum fractal functions is introduced based on the Meyer-K\"onig-Zeller operator $M_{q,n}$. These quantum Meyer-K\"onig-Zeller (MKZ) fractal functions employ $M_{q,n} f$ as the base function in the iterated…

The Quantum Fourier Transformation (QFT) is a well-known subroutine for algorithms on qubit-based universal quantum computers. In this work, the known QFT circuit is used to derive an efficient circuit for the multidimensional QFT. The…

We present a quite curious generalization of multi-step Fibonacci numbers. For any positive rational $q$, we enumerate binary words of length $n$ whose maximal factors of the form $0^a1^b$ satisfy $a = 0$ or $aq > b$. When $q$ is an integer…

For any finite field ${\mathbb F}_q$ with $q$ elements, we study the set ${\mathcal F}_{(q,m)}$ of functions from ${\mathbb F}_q^m$ into ${\mathbb F}^q$. We introduce a transformation that allows us to determine a linear system of $q^{m+1}$…

Quantum physics frequently involves a need to count the states, subspaces, measurement outcomes, and other elements of quantum dynamics. However, with quantum mechanics assigning probabilities to such objects, it is often desirable to work…

Let $k$ be an integer greater than or equal $4$. We show that if a multiplicative function $f$ satisfies \[ f(x_1^2 + x_2^2 + \dots + x_k^2) = f(x_1)^2 + f(x_2)^2 + \dots + f(x_k)^2 \] for all positive integers $x_i$'s, then $f$ is the…

We consider the question of approximating any real number $\alpha$ by sums of $n$ rational numbers $\frac{a_1}{q_1} + \frac{a_2}{q_2} + ... + \frac{a_n}{q_n}$ with denominators $1 \leq q_1, q_2, ..., q_n \leq N$. This leads to an inquiry on…

We determine the {\em real} counting function $N(q)$ ($q\in [1,\infty)$) for the hypothetical "curve" $C=\overline {\Sp \Z}$ over $\F_1$, whose corresponding zeta function is the complete Riemann zeta function. Then, we develop a theory of…

The Hilbert space formalism of quantum mechanics is reviewed with emphasis on applications to quantum computing. Standard interferomeric techniques are used to construct a physical device capable of universal quantum computation. Some…

We propose a theory of characterizing quantum circuits with qubit functional configurations. Any quantum circuit can be decomposed into alternating sequences of 1-qubit unitary gates and CNOT gates. Each CNOT sequence prepares the current…

We show by a dynamical argument that there is a positive integer valued function $q$ defined on positive integer set $\mathbb N$ such that $q([\log n]+1)$ is a super-polynomial with respect to positive $n$ and \[\liminf_{n\rightarrow\infty}…

We say a power series $\sum_{n=0}^\infty a_n q^n$ is multiplicative if the function $n\mapsto a_n/a_1$ ($n\ge 1$) is so. In this paper, we consider multiplicative power series $f$ such that $f^2$ is also multiplicative. We find various…

Quantum computation has suggested new forms of quantum logic, called quantum computational logics. The basic semantic idea is the following: the meaning of a sentence is identified with a quregister, a system of qubits, representing a…