Categorical Quantum Computing

Categorical Quantum Computing

Alex Lang

Department of Computer ScienceUniversity of Oxford

July 31 2011 / SuperHappyDevHouse 45.5

Alex Lang Categorical Quantum Computing

Outline


Who am I?

I study Category theoryI study Quantum computing


Who am I?

I study Category theoryI study Quantum computing


What I want to talk about

Quantum computingWithout assuming (much) QM


What I want to talk about

Quantum computingWithout assuming (much) QM


What is QC?

Like classical computingBut faster! (potentially)By taking advantage of quantum mechanics


What is QC?



What is QC?



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)


What I do in QC



What I do in QC



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}


What does this represent?

Think of the wires as circuit wiresThink of the dots as circuit gates


What does this represent?

Think of the wires as circuit wiresThink of the dots as circuit gates


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/)


What do the rules mean?

Tells you what programs are “equivalent”Tells you how to execute the program


What do the rules mean?

Tells you what programs are “equivalent”Tells you how to execute the program


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


What’s the point?

It’s simple!It’s universal


What’s the point?

It’s simple!It’s universal


The End

Ask me questions!


Technology

Categorical Quantum Computing