Related papers: On the Computing Power of $+$, $-$, and $\times$
We show that the notion of generalized Berry phase i.e., non-abelian holonomy, can be used for enabling quantum computation. The computational space is realized by a $n$-fold degenerate eigenspace of a family of Hamiltonians parametrized by…
The computational study of elections generally assumes that the preferences of the electorate come in as a list of votes. Depending on the context, it may be much more natural to represent the list succinctly, as the distinct votes of the…
In this paper we study the complexity of quantum query algorithms computing the value of Boolean function and its relation to the degree of algebraic polynomial representing this function. We pay special attention to Boolean functions with…
In this article, we will show that uncomputability is a relative property not only of oracle Turing machines, but also of subrecursive classes. We will define the concept of a Turing submachine, and a recursive relative version for the Busy…
Let a polynomial $f \in \mathbb{Z}[X_1,\ldots,X_n]$ be given. The square sieve can provide an upper bound for the number of integral $\mathbf{x} \in [-B,B]^n$ such that $f(\mathbf{x})$ is a perfect square. Recently this has been generalized…
We develop algorithms for writing a polynomial as sums of powers of low degree polynomials. Consider an $n$-variate degree-$d$ polynomial $f$ which can be written as $$f = c_1Q_1^{m} + \ldots + c_s Q_s^{m},$$ where each $c_i\in…
LS is a particular type of computational processes simulating living tissue. They use an unlimited branching process arising from the simultaneous substitutions of some words instead of letters in some initial word. This combines the…
What is computable with limited resources? How can we verify the correctness of computations? How to measure computational power with precision? Despite the immense scientific and engineering progress in computing, we still have only…
We consider the concept of a local set of inference rules. A local rule set can be automatically transformed into a rule set for which bottom-up evaluation terminates in polynomial time. The local-rule-set transformation gives…
This paper deals with simultaneously fast and in-place algorithms for formulae where the result has to be linearly accumulated: some of the output variables are also input variables, linked by a linear dependency. Fundamental examples…
This paper shows that, if we could examine the entire history of a hidden variable, then we could efficiently solve problems that are believed to be intractable even for quantum computers. In particular, under any hidden-variable theory…
This paper furthers existing evidence that quantum computers are capable of computations beyond classical computers. Specifically, we strengthen the collapse of the polynomial hierarchy to the second level if: (i) Quantum computers with…
Semigroup theory is a branch of abstract algebra, and it provides mathematical tools for the theory of computation. Finite semigroups can describe state transition systems and thus they model physically realizable computers. Engineering…
Deterministic models are approximations of reality that are easy to interpret and often easier to build than stochastic alternatives. Unfortunately, as nature is capricious, observational data can never be fully explained by deterministic…
Symbolic powers are a classical commutative algebra topic that relates to primary decomposition, consisting, in some circumstances, of the functions that vanish up to a certain order on a given variety. However, these are notoriously…
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…
Given $(a_1, \dots, a_n, t) \in \mathbb{Z}_{\geq 0}^{n + 1}$, the Subset Sum problem ($\mathsf{SSUM}$) is to decide whether there exists $S \subseteq [n]$ such that $\sum_{i \in S} a_i = t$. There is a close variant of the $\mathsf{SSUM}$,…
We develop a theory of complexity for numerical computations that takes into account the condition of the input data and allows for roundoff in the computations. We follow the lines of the theory developed by Blum, Shub, and Smale for…
Tailoring many-body interactions among a proper quantum system endows it with computing ability by means of static quantum computation in the sense that some of the physical degrees of freedom can be used to store binary information and the…
We study a class of functional problems reducible to computing $f^{(n)}(x)$ for inputs $n$ and $x$, where $f$ is a polynomial-time bijection. As we prove, the definition is robust against variations in the type of reduction used in its…