Related papers: Quadratic addition rules for quantum integers

We present a formulation of quantum mechanics based on orthogonal polynomials. The wavefunction is expanded over a complete set of square integrable basis in configuration space where the expansion coefficients are orthogonal polynomials in…

We propose a new circuit for in-place addition of a classical $n$-bit constant to a quantum $n$-qubit integer modulo $2^n$. Our circuit uses $n-3$ ancilla qubits and has a T-count of $4n-5$. We also propose controlled version of this…

Quantum algorithm is an algorithm for solving mathematical problems using quantum systems encoded as information, which is found to outperform classical algorithms in some specific cases. The objective of this study is to develop a quantum…

We prove that quantum computation is polynomially equivalent to classical probabilistic computation with an oracle for estimating the value of simple sums, quadratically signed weight enumerators. The problem of estimating these sums can be…

We prove that for every positive integer $k$ there exist an inclusion-exclusion polynomial $Q_{\{q_1,q_2,...,q_k\}}$ with the height at least $c^{2^k}\prod_{j=1}^{k-2}q_j^{2^{k-j-1}-1}$, where $c$ is a positive constant and…

Let $a_k(n)$ denotes the number of representations of a non-negative integer $n$ as sum of $k$ quadratic forms of the type $x^2+xy+y^2$ and $a_{\lambda_1,\lambda_2,\lambda_3\dots\lambda_k}(n)$ denotes the number of representations $n$ as a…

The Quantum Fourier Transform offers an interesting way to perform arithmetic operations on a quantum computer. We review existing Quantum Fourier Transform adders and multipliers and propose some modifications that extend their…

We give an optimal necessary and sufficient condition for the quotient polynomial and remainder in the division algorithm to have positive coefficients.

Let $n\ge1$, $r\ge0$ and $s\ge0$ be integers satisfying $4+r+3 s\le3^{n+1}$. Given linear polynomials $f_{i}(x)=m_{i} x+n_{i}$ for $1 \le i \le r+s$, where the coefficients $m_{i} , n_{i}$ are positive integers satisfying certain…

Let $\mathbb{F}_q$ be the finite field of $q$ elements. In this paper we obtain bounds on the following counting problem: given a polynomial $f(x)\in \mathbb{F}_q[x]$ of degree $k+m$ and a non-negative integer $r$, count the number of…

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

In previous work we computed the number $C_n(q)$ of ideals of codimension $n$ of the algebra ${\mathbb{F}}_q[x,y,x^{-1}, y^{-1}]$ of two-variable Laurent polynomials over a finite field: it turned out that $C_n(q)$ is a palindromic…

In this paper, we study the binomial sum $S_{n}(q):=% \overset{n}{\underset{k=0}{\sum }}a_{k}\binom{n}{k}\left( 1-q\right) ^{k}q^{n-k}$ for a given sequence $\left( a_{n}\right) $ of real or complex numbers. We express $S_{n}(q)$ in…

We propose a method for determining which integers can be written as a sum of two integral squares for quadratic fields $\Q(\sqrt{\pm p})$, where $p$ is a prime.

We start with a (q,t)-generalization of a binomial coefficient. It can be viewed as a polynomial in t that depends upon an integer q, with combinatorial interpretations when q is a positive integer, and algebraic interpretations when q is…

In Parts I-III we showed that the number of ways to place $q$ nonattacking queens or similar chess pieces on an $n\times n$ chessboard is a quasipolynomial function of $n$ whose coefficients are essentially polynomials in $q$. In this part…

In this paper, we answer the question: "what is the qth Fibonacci number, where q is a positive rational?". The answer is the codenominator function, which is an integral-valued map. It is defined via a pair of functional equations. Many…

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…

Quadrature rules estimate the value of an integral when the function is given by a table of values. Every binary string defines a quadrature rule by choosing which endpoint of each interval represents the interval. The standard rules, such…

Quantum algorithms are known for presenting more efficient solutions to certain computational tasks than any corresponding classical algorithm. It has been thought that the origin of the power of quantum computation has its roots in…