Quantum Computation∗
Lecture 5: Quantum Computation and Computer Science I
Notes taken by Robert Stevens
October 2007
Summary: This class helps to integrate the big picture ideas of Computer Science with Quantum Com-
putation. These ideas include Turing Machine, Halting Problem, Computational Complexity Theory,
Polynomial Time, and the Strong Church-Turing Thesis.
1 Turing Machine
The Turing Machine is an idealized and rigorously defined mathematical model of computing device. In our case we
will be using a model slightly different.
The programs input and output, for this case a Universal Computer, can be represented using a pair of uniquely
encoded pair of positive integers.
1. Kolomogorov Complexity can be described using as follows
•θrepresents a universal Turing machine
•p represents a program
•x represents a string
Proof.
Kφ(x) = min
φ(p) = xI(p)
2. Universal Turing Machine
•Turing was the first person to create a Universal Turing Machine
•Marvin Minksy in the early 1960s created a 7 state TM
•Sequences that have small complexity tend to have regular patterns
Proof. K(xn)∝O(log n)
•Sequences that have large complexity typically have random patterns
(a) Typically these patterns are incompressible
(b) A fair coin being flipped can be represented as:
∗Lecture Notes for a course given by Stephen F. Bush at RPI.
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