Related papers: Quantum Computers Can Find Quadratic Nonresidues i…

Grover's algorithm solves the unstructured search problem. Grover's algorithm can find the target state with certainty only if searching one out of four. Designing the deterministic search algorithm can avoid any repetition of the…

We consider the quantum complexity of estimating matrix elements of unitary irreducible representations of groups. For several finite groups including the symmetric group, quantum Fourier transforms yield efficient solutions to this…

In this paper, we begin by reviewing some of the known properties of QQR codes and proved that $PSL_2(p)$ acts on the extended QQR code when $p \equiv 3 \pmod 4$. Using this discovery, we then showed their weight polynomials satisfy a…

If P is a rational polytope in R^d, then $i_P(t):=#(tP\cap Z^d)$ is a quasi-polynomial in t, called the Ehrhart quasi-polynomial of P. A period of i_P(t) is D(P), the smallest positive integer D such that D*P has integral vertices. Often,…

Given a quadratic map Q : K^n -> K^k defined over a computable subring D of a real closed field K, and a polynomial p(Y_1,...,Y_k) of degree d, we consider the zero set Z=Z(p(Q(X)),K^n) of the polynomial p(Q(X_1,...,X_n)). We present a…

Given a finite set of vectors spanning a lattice and lying in a halfspace of a real vector space, to each vector $a$ in this vector space one can associate a polytope consisting of nonnegative linear combinations of the vectors in the set…

Within context of quantum logic, it is possible to assign dispersion-free probabilities to experimental propositions pertaining to qubits. This makes qubits distinct from the rest of quantum systems since the latter do not admit…

Approximation algorithms for classical constraint satisfaction problems are one of the main research areas in theoretical computer science. Here we define a natural approximation version of the QMA-complete local Hamiltonian problem and…

We show that nonlinear problems including nonlinear partial differential equations can be efficiently solved by variational quantum computing. We achieve this by utilizing multiple copies of variational quantum states to treat…

Utilization of a quantum system whose time-development is described by the nonlinear Schrodinger equation in the transformation of qubits would make it possible to construct quantum algorithms which would be useful in a large class of…

This paper presents in detail the originally developed Quadratic Point Estimate Method (QPEM), aimed at efficiently and accurately computing the first four output moments of probabilistic distributions, using 2n^2+1 sample (or sigma)…

In this paper, we construct a special family of cyclic codes, known as quadratic residue codes of prime length \( p \equiv \pm 1 \pmod{44} ,\) \( p \equiv \pm 5 \pmod{44} ,\) \( p \equiv \pm 7 \pmod{44} ,\) \( p \equiv \pm 9 \pmod{44} \)…

In a 2006 paper, Damg{\aa}rd and Frandsen designed a faster version of the Quadratic Frobenius Test. This test assumes the Extended Riemann Hypothesis in order to find small nonresidues, which allow construction of quadratic extensions with…

We consider the inverse problem of determining a time-dependent damping coefficient $a$ and a time-dependent potential $q$, appearing in the wave equation $\partial_t^2u-\Delta_x u+a(t,x)\partial_tu+q(t,x)u=0$ in $Q=(0,T)\times\Omega$, with…

We show that $n$-bit integers can be factorized by independently running a quantum circuit with $\tilde{O}(n^{3/2})$ gates for $\sqrt{n}+4$ times, and then using polynomial-time classical post-processing. The correctness of the algorithm…

Primitive polynomials over finite fields are crucial for various domains of computer science, including classical pseudo-random number generation, coding theory and post-quantum cryptography. Nevertheless, the pursuit of an efficient…

Let $X=X_1\sqcup X_2\sqcup\ldots\sqcup X_k$ be a partitioned set of variables such that the variables in each part $X_i$ are noncommuting but for any $i\neq j$, the variables $x\in X_i$ commute with the variables $x'\in X_j$. Given as input…

We propose quantum algorithms, purely quantum in nature, for calculating the determinant and inverse of an $(N-1)\times (N-1)$ matrix (depth is $O(N^2\log N)$) which is a simple modification of the algorithm for calculating the determinant…

We consider the fundamental problem of solving quadratic systems of equations in $n$ variables, where $y_i = |\langle \boldsymbol{a}_i, \boldsymbol{x} \rangle|^2$, $i = 1, \ldots, m$ and $\boldsymbol{x} \in \mathbb{R}^n$ is unknown. We…

Pollard's Rho is a method for solving the integer factorization problem. The strategy searches for a suitable pair of elements belonging to a sequence of natural numbers that given suitable conditions yields a nontrivial factor. In…