The input for the quantum convolution layer is a \(3\)D tensor input given as \(X^{\ell} \in \mathbb{R}^{H^{\ell} \times W^{\ell} \times D^{\ell}}\). The weights layer, filter layer, or \(4\)D tensor kernel layer is denoted as \(K^{\ell} \in \mathbb{R}^{H \times W \times D^{\ell} \times D^{\ell+1}}\). The input and kernel layer are both stored in QRAM. There are precision parameters \(\epsilon\) and \(\Delta>0\)... read more
Quantum Principal Component Analysis identifies large eigenvalues of unknown density matrices utilizing corresponding eigenvectors in \(O(\log d)\). Where principal component analysis analyzes positive semi-definite Hermitian matrices by decomposing eigenvectors in relation to the largest eigenvalues in the matrix for dimensionality reduction. Improved computational complexity will hopefully allow new methods for... read more
Using quantum computing, the authors exploit quantum mechanics for the algorithmic complexity optimization of a Support Vector Machine with high-dimensional feature space. Where the high-dimensional classical data is mapped non-linearly to Hilbert Space and a hyperplane in quantum space is used to separate and label the data. By using the... read more
In biological intelliegent systems there are multiple mechanisms working in congruence on multiple levels, both at the structural and neurobiological level to develop complex cognitive abilities. What remains unknown is which mechanisms are necessary and sufficent to synthetically replicate these cognitive abilities for artificial intelligence. A neurocomputational model is offered... read more
Three properties were required by Shannon: \(I(p) \geq 0\), i.e. information is a real non-negative measure. \(I(p_{1},p_{2})=I(p_{1})+I(p_{2})\) for independent events. \(I(p)\) is a continous function of \(p\). The mathematical function that satisfies these requirements is: \(I(p)=k\;log(p)\) In the equation, the value of \(k\) is arbitrary... read more
Quantum Computing Theory is a field of computer science that uses the principles of quantum mechanics, mathematics, and computer science. By borrowing concepts from each field scientists can rigorously define both a broad and narrow theoretical model of a quantum computer and later apply it to the real world. These... read more
\(G = (V, E)\) \(V\) is a set of vertices \(E \subseteq \left\{\left\{x, y\right\}\;|\;x, y \in V\;and\;x \neq y\right\}\) A simple undirected graph \(G\) is an ordered pair or tuple \((V, E)\) where \(V\) and \(E\) are finite sets. \(E \subseteq \left\{\left\{x, y\right\}\;|\;x, y \in V\;and\;x \neq y\right\}\)... read more
A deterministic finite automaton (DFA) is a 5-tuple: \((Q, \Sigma, \delta, q_{0}, F)\) where: \(Q\) is a finite set of states \(\Sigma\) is an alphabet \(\delta\) is a transition function described as \(\delta : Q \times \Sigma \rightarrow Q\) \(q_{0} \in Q\) is the initial state \(F \subseteq Q\) is a... read more
Algorithmic analysis is used to help computer scientists understand the resources required by an algorithm for time, storage, and other uses. Algorithmic anlysis must analyze algorithms in a methodical, universal, and fair way. To do this computer scientist implement mathematical models that describe the resources used by algorithms. This work... read more