Post on 01-Dec-2014
description
Categorical Quantum Computing
Alex Lang
Department of Computer ScienceUniversity of Oxford
July 31 2011 / SuperHappyDevHouse 45.5
Alex Lang Categorical Quantum Computing
Outline
Alex Lang Categorical Quantum Computing
Who am I?
I study Category theoryI study Quantum computing
Alex Lang Categorical Quantum Computing
Who am I?
I study Category theoryI study Quantum computing
Alex Lang Categorical Quantum Computing
What I want to talk about
Quantum computingWithout assuming (much) QM
Alex Lang Categorical Quantum Computing
What I want to talk about
Quantum computingWithout assuming (much) QM
Alex Lang Categorical Quantum Computing
What is QC?
Like classical computingBut faster! (potentially)By taking advantage of quantum mechanics
Alex Lang Categorical Quantum Computing
What is QC?
Like classical computingBut faster! (potentially)By taking advantage of quantum mechanics
Alex Lang Categorical Quantum Computing
What is QC?
Like classical computingBut faster! (potentially)By taking advantage of quantum mechanics
Alex Lang Categorical Quantum Computing
What I do in QC
We haven’t really built a quantum computer yetBut I don’t really care!Once (if) we can a quantum computer, what will quantumprograms mean? (semantics)
Alex Lang Categorical Quantum Computing
What I do in QC
We haven’t really built a quantum computer yetBut I don’t really care!Once (if) we can a quantum computer, what will quantumprograms mean? (semantics)
Alex Lang Categorical Quantum Computing
What I do in QC
We haven’t really built a quantum computer yetBut I don’t really care!Once (if) we can a quantum computer, what will quantumprograms mean? (semantics)
Alex Lang Categorical Quantum Computing
A proposal
We can program with red and green dots and wires.
H
π
H
H
π π
π
{v2} {v
2}{v
3}
{v3}
Alex Lang Categorical Quantum Computing
What does this represent?
Think of the wires as circuit wiresThink of the dots as circuit gates
Alex Lang Categorical Quantum Computing
What does this represent?
Think of the wires as circuit wiresThink of the dots as circuit gates
Alex Lang Categorical Quantum Computing
Some rules11
Figure 1. Rules for the ZX-calculus
2.2. The ZX-equational rules
In addition to the rules for constructing diagrams, the calculus consists of a set of equations thatspecify how one diagram may be transformed into another. These rules are presented in figure 1.We now expand on these rules and give some examples of their use.
2.2.1. The T-rule. The informally stated T-rule will be made more precise in sections 4.3–4.5.For practical purposes, the intuitive reading of ‘only the topology matters’ suffices: the wiresof the diagram may be arbitrarily stretched, bent, twisted, tied in knots, etc, without alteringthe meaning of the diagram, provided the connections are maintained. More precisely, afteridentifying (e.g. by enumerating) the inputs and the outputs, any topological deformation of theinternal structure of the network yields a network that is equal to the given one.
Two important examples of such ‘homotopic rewrites’ are
In fact, these two rules can also be seen as consequences of the S-rules, when introducing agreen dot on the caps and cups as in (S2); see example 2.4 below. The reason for consideringthem within the T-rule will become clear in section 4.5.
New Journal of Physics 13 (2011) 043016 (http://www.njp.org/)
Alex Lang Categorical Quantum Computing
What do the rules mean?
Tells you what programs are “equivalent”Tells you how to execute the program
Alex Lang Categorical Quantum Computing
What do the rules mean?
Tells you what programs are “equivalent”Tells you how to execute the program
Alex Lang Categorical Quantum Computing
What can you do with this?
Quantum Teleportationin
out
Bob
H
Ơ ơ
Alice
ơƠ
in
out
BobAlice
プブ
ブプ
in
out
BobAlice
ブ
ブ
in
out
BobAlice
Alex Lang Categorical Quantum Computing
What’s the point?
It’s simple!It’s universal
Alex Lang Categorical Quantum Computing
What’s the point?
It’s simple!It’s universal
Alex Lang Categorical Quantum Computing
The End
Ask me questions!
Alex Lang Categorical Quantum Computing