Yorick Hardy

Finite State Machines

Finite state machines [49, 67] provide a visual representation of algorithms. Algorithms are implemented on a machine with a finite number of states representing the state of the algorithm. This provides an abstract way of designing algorithms. The chapter will only cover deterministic machines (the actions of the machines are determined uniquely).

GENE EXPRESSION PROGRAMMING AND ONE-DIMENSIONAL CHAOTIC MAPS

Gene expression programming is applied to find one-dimensional maps. A survey on gene expression programming is also given.

Entangled Quantum States and the Kronecker Product

C Entangled quantum states are an important component of quantum computing techniques such as quantum error-correction, dense coding and quantum teleportation. We determine the requirements for a state in the Hilbert space ⊗ Cnfor m, n ∈ N to be entangled and a solution to the corresponding “factorization” problem if this is not the case.We consider the implications of these criteria for computer algebra applications.

Exceptional points, non-normal matrices, hierarchy of spin matrices and an eigenvalue problem

Exceptional points of a class of non-hermitian Hamilton operators Ĥ of the form Ĥ = Ĥ0 + iĤ1 are studied, where Ĥ0 and Ĥ1 are hermitian operators. Finite dimensional Hilbert spaces are considered. The linear operators Ĥ0 and Ĥ1 are given by spin matrices for spin s = 1/2, 1, 3/2, …. Since the linear operators studied are non-normal, properties of such operators are described.

Entanglement, Kronecker Product, Pauli Spin Matrices and a Nonlinear Eigenvalue Problem

Entanglement for the eigenvectors of a nonlinear eigenvalue problem given by the Kronecker product of the Pauli spin matrices is investigated. Fully and partially entangled eigenvectors are found.

A SEQUENCE OF QUANTUM GATES

We study a sequence of quantum gates in finite-dimensional Hilbert spaces given by the normalized eigenvectors of the unitary operators. The corresponding sequence of the Hamilton operators is also given. From the Hamilton operators we construct another hierarchy of quantum gates via the Cayley transform.

Chaotic Maps, Invariants, Bose Operators, and Coherent States

We first show how an autonomous system of ordinary first-order difference equations can be embedded into a Hilbert space description by using Bose operators and coherent states. Then we describe how an invariant can be expressed using Bose operators. Two examples are given.

Fermi Systems, Hubbard Model And A Symbolicc++ Implementation

We show how the anticommutation relations for Fermi operators can be implemented with computer algebra using SymbolicC++. We describe applications to the Hubbard model. An important identity for Fermi operators is proved. Then, we test for higher order constants of motion for the Hubbard model. Finally, the matrix representation for the four point Hubbard model is calculated.

Entangled Quantum States

Entangled quantum states are an important component of quantum computingtechniques such as quantum error correction, dense coding, and quantumteleportation. We determine the requirements for a state in the Hilbert space C9to be entangled and a solution to the corresponding factorization problem if thisis not the case.

Exponential of a matrix, a nonlinear problem, and quantum gates

We describe solutions of the matrix equation exp(z(A − In)) = A, where z ∈ ℂ. Applications in quantum computing are given. Both normal and non-normal matrices are studied. For normal matrices, the Lambert W-function plays a central role.