The Quantum Computer

An Introduction by Jacob West
 (back to paper)

Glossary of Terms

 

 Boolean, a system of logical thought developed by English mathematician George Boole (1815-64) that defines variables with the value of either true or false, represented in a classical computer as a 1 or 0 respectively, for which simple logical constraints can be applied that have a true or false result.  For example, the AND and OR operators are common Boolean logic gates that are defined as follows:
     AND, returns true if and only if the compared variables have the same values.  Ex.  (1 AND 1) = 1, (1 AND 0) = 0
     OR, returns true if any of the compared variables have the value true.  Ex.  (1 OR 0) = 1, (0 OR 0) = 0

exponential function, a mathematical function of the form f(x) = ax where a is constant and x is a variable.  The most common exponential function is ex where e is approximately equal to 2.718.
     1.  Exponentially large implies that the size of an object or number increases like the slope of an exponential function, or in
          other words, very very quickly.

Hilbert space, a vector space over the complex numbers C with an inner product in which sequences that should converge actually do converge to points in the space.  (for a more in depth definition)

macroscopic, visible to the naked eye:  opposed to microscopic.

matrix, pl. matrices, in mathematics, any rectangular arrangement of symbols or numbers into columns and rows that can be used to represent the state of an object.

qubit, (derived from quantum bit) a ket (state) in a two dimensional Hilbert space.

superposition, in wave mechanics, the interference of waves, or summing of amplitude's at the point of intersection;  similarly, in quantum mechanics, the summation of probability distributions that represent possible states of a system.

vector space*, a nonempty set of objects, called elements, that satisfy the following ten axioms: 
Let V denote a vector space,
     Closure axioms
     Axiom 1.  Closure Under Addition.  For every pair of elements x and y in V there corresponds a unique element in V
                     called the sum of x and y, denoted by x + y.
     Axiom 2.  Closure Under Multiplication by Real Numbers.  For every x in V and every real number a there 
                     corresponds an element in V called the product of a and x, denoted by ax.

     Axioms for Addition
     Axiom 3.  Commutative Law.  For all x and y in V, we have x + y = y + x.
     Axiom 4.  Associative Law.  For all x, y, and z in V, we have (x + y) + z = x + (y + z).
     Axiom 5.  Existence of Zero Element.  There is an element in V, denoted by 0, such that x + 0 = x for all x in V.
     Axiom 6.  Existence of Negatives.  For every x in V, the elements (-1)x has the property x + (-1)x = 0.

     Axioms for Multiplication by Numbers
     Axiom 7.  Associative Law.  For every x in V and all real numbers a and b, we have a(bx) = (ab)x.
     Axiom 8.  Distributive Law for Addition in V.  For all x and y in V and all real a, we have a(x + y) = ax + ay.
     Axiom 9.  Distributive Law for Addition of Numbers.  For all x in V and all real a and b, we have (a + b)x = ax + bx.
     Axiom 10.  Existence of Identity.  For every x in V, we have 1x = x.
*Note:  definition take from "Calculus" Volume II, Second Edition by Tom M. Apostol.

Last Modified:  04/28/00