Natural Numbers and Quantum States in Fock Space (original) (raw)

Representation of natural numbers in quantum mechanics

Physical Review A, 2001

This paper represents one approach to making explicit some of the assumptions and conditions implied in the widespread representation of numbers by composite quantum systems. Any nonempty set and associated operations is a set of natural numbers or a model of arithmetic if the set and operations satisfy the axioms of number theory or arithmetic. This work is limited to k − ary representations of length L and to the axioms for arithmetic modulo k L . A model of the axioms is described based on an abstract L fold tensor product Hilbert space H arith . Unitary maps of this space onto a physical parameter based product space H phy are then described. Each of these maps makes states in H phy , and the induced operators, a model of the axioms. Consequences of the existence of many of these maps are discussed along with the dependence of Grover's and Shor's Algorithms on these maps. The importance of the main physical requirement, that the basic arithmetic operations are efficiently implementable, is discussed. This condition states that there exist physically realizable Hamiltonians that can implement the basic arithmetic operations and that the space-time and thermodynamic resources required are polynomial in L.

The Representation of Numbers in Quantum Mechanics 1

Earlier work on modular arithmetic of k-ary representations of length L of the natural numbers in quantum mechanics is extended here to k-ary representations of all natural numbers, and to integers and rational numbers. Since the length L is indeterminate, representations of states and operators using creation and annihilation operators for bosons and fermions are defined. Emphasis is on definitions and properties of operators corresponding to the basic operations whose properties are given by the axioms for each type of number. The importance of the requirement of efficient implementability for physical models of the axioms is emphasized. Based on this, successor operations for each value of j corresponding to +k j−1 are defined. It follows from the efficient implementability of these successors, which is the case for all computers, that implementation of the addition and multiplication operators, which are defined in terms of polynomially many iterations of the successors, should be efficient. This is not the case for definitions based on just the successor for j = 1. This is the only successor defined in the usual axioms of arithmetic.

The Representation of Numbers in Quantum Mechanics

Algorithmica, 2002

Earlier work on modular arithmetic of k-ary representations of length L of the natural numbers in quantum mechanics is extended here to k-ary representations of all natural numbers, and to integers and rational numbers. Since the length L is indeterminate, representations of states and operators using creation and annihilation operators for bosons and fermions are defined. Emphasis is on definitions and properties of operators corresponding to the basic operations whose properties are given by the axioms for each type of number. The importance of the requirement of efficient implementability for physical models of the axioms is emphasized. Based on this, successor operations for each value of j corresponding to +k j−1 are defined. It follows from the efficient implementability of these successors, which is the case for all computers, that implementation of the addition and multiplication operators, which are defined in terms of polynomially many iterations of the successors, should be efficient. This is not the case for definitions based on just the successor for j = 1. This is the only successor defined in the usual axioms of arithmetic.

Space of Quantum Theory Representations of Natural Numbers, Integers, and Rational Numbers

This paper extends earlier work on quantum theory representations of natural numbers N, integers I, and rational numbers Ra to describe a space of these representations and transformations on the space. The space is parameterized by 4-tuple points in a parameter set. Each point, (k,m,h,g), labels a specific representation of X = N, I, Ra as a Fock space F^{X}_{k,m,h} of states of finite length strings of qukits q and a string state basis B^{X}_{k,m,h,g}. The pair (m,h) locates the q string in a square integer lattice I \times I, k is the q base, and the function g fixes the gauge or basis states for each q. Maps on the parameter set induce transformations on on the representation space. There are two shifts, a base change operator W_{k',k}, and a basis or gauge transformation function U_{k}. The invariance of the axioms and theorems for N, I, and Ra under any transformation is discussed along with the dependence of the properties of W_{k',k} on the prime factors of k' an...

The Representation of Numbers by States in Quantum Mechanics

Quantum Communication, Computing, and Measurement 3, 2002

The representation of numbers by tensor product states of composite quantum systems is examined. Consideration is limited to k − ary representations of length L and arithmetic modk L . An abstract representation on an L fold tensor product Hilbert space H arith of number states and operators for the basic arithmetic operations is described. Unitary maps onto a physical parameter based tensor product space H phy are defined and the relations between these two spaces and the dependence of algorithm dynamics on the unitary maps is discussed. The important condition of efficient implementation by physically realizable Hamiltonians of the basic arithmetic operations is also discussed.

Quantum Computational Semantics on Fock Space

International Journal of Theoretical Physics, 2005

In the Fock space semantics, meanings of sentences are identified with density operators of the (unsymmetrized) Fock space F based on the Hilbert space C 2 . Generally, the meaning of a sentence is smeared over different sectors of F . The standard quantum computational semantics is a limit case of the Fock space semantics, where the meaning of any sentence α only "lives" in one sector of F , which is determined by the logical complexity of α. We prove that the global Fock space semantics and the standard quantum computational semantics characterize the same logic.

Multibosons in quantum computation

Laser Physics, 2006

The role of the three-boson algebra in the theoretical construction of both logical states and operators for quantum computing is investigated. The computational basis consists of three orthonormal codewords obtained from the coherent states of the algebra. The gate and phase operators acting on the codewords are found, and the relation between their matrix representations in the Fock space and the fundamental representation of su (3) is discussed.

Quantum computational structures

Mathematica Slovaca, 2004

Quantum computation has suggested new forms of quantum logic, called quantum computational logics ([CDCGL01]). The basic semantic idea is the following: the meaning of a sentence is identified with a quregister, representing a possible pure state of a compound physical system, whose associated Hilbert space is an n-fold tensor product ⊗ n C 2 . The generalization to density operators, which might be useful to analyse entanglement-phenomena, is due to Gudder [Gu03]. In this paper we study structural properties of density operators systems, where some basic quantum logical gates are defined. We introduce the notions of standard reversible and standard irreversible quantum computational structure. We prove that the second structure is isomorphic to an algebra based on a particular set of complex numbers.

Matula numbers, Gödel numbering and Fock space

Journal of Mathematical Chemistry, 2013

By making use of Matula numbers, which give a 1-1 correspondence between rooted trees and natural numbers, and a Gödel type relabelling of quantum states, we construct a bijection between rooted trees and vectors in the Fock space. As a by product of the aforementioned correspondence (rooted trees ↔ Fock space) we show that the fundamental theorem of arithmetic is related to the grafting operator, a basic construction in many Hopf algebras. Also, we introduce the Heisenberg-Weyl algebra built in the vector space of rooted trees rather than the usual Fock space. This work is a cross-fertilization of concepts from combinatorics (Matula numbers), number theory (Gödel numbering) and quantum mechanics (Fock space).

Quantum Logic as a Basis for Computations

International Journal of Theoretical Physics - INT J THEOR PHYS, 2000

It is shown that computations can be founded on the laws of the genuine(Birkhoff—nvon Neumann) quantum logic treated as a particular version ofLukasiewicz infinite-valued logic. A new way of encoding nonexact data whichencodes both the value of a number and its “fuzziness” is introduced. A simpleexample of a full adder that works in the proposed way is given and it is comparedwith other designs of quantum adders existing in the literature. A controversybetween the meaning of the very term “quantum logic” as used recently withinthe theory of quantum computations and the traditional meaning of this term isbriefly discussed.