Techniques algbriques en calcul quantique E. Jeandel Laboratoire de - - PowerPoint PPT Presentation

SLIDE 1

Techniques algébriques en calcul quantique

• E. Jeandel

Laboratoire de l’Informatique du Parallélisme LIP , ENS Lyon, CNRS, INRIA, UCB Lyon

8 Avril 2005

• E. Jeandel, LIP

, ENS Lyon Techniques algébriques en calcul quantique 1/54

SLIDE 2

Algebraic Techniques in Quantum Computing

• E. Jeandel

Laboratoire de l’Informatique du Parallélisme LIP , ENS Lyon, CNRS, INRIA, UCB Lyon

April 8th, 2005

• E. Jeandel, LIP

, ENS Lyon Algebraic Techniques in Quantum Computing 2/54

SLIDE 3

Outline

1

Combinatorial setting: Quantum gates Definitions Completeness and Universality

2

Algebraic setting Quantum gates are unitary matrices Computing the group Density

3

Conclusion Automata Conclusion

• E. Jeandel, LIP

, ENS Lyon Algebraic Techniques in Quantum Computing 3/54

SLIDE 4

Introduction

Classical Quantum State q αiqi The system may be in all states simultaneously Operators Maps Unitary (hence reversible) maps

• E. Jeandel, LIP

, ENS Lyon Algebraic Techniques in Quantum Computing 4/54

SLIDE 5

Outline

1

Combinatorial setting: Quantum gates Definitions Completeness and Universality

2

Algebraic setting Quantum gates are unitary matrices Computing the group Density

3

Conclusion Automata Conclusion

• E. Jeandel, LIP

, ENS Lyon Algebraic Techniques in Quantum Computing 5/54

SLIDE 6

What is a quantum gate ?

✐

. . . . . .

✐ ✐

M

✐ ✐ ✐

. . . . . .

• E. Jeandel, LIP

, ENS Lyon Algebraic Techniques in Quantum Computing 6/54

SLIDE 7

What is a quantum gate ?

✐

. . . . . .

✐ ✐

1 M

✐ ✐ ✐

. . . . . . 1 β0 + iα1 α0 + β1

• E. Jeandel, LIP

, ENS Lyon Algebraic Techniques in Quantum Computing 6/54

SLIDE 8

What is a quantum gate ?

✐

. . . . . .

✐ ✐

β0 − δ1 γ1 + α0 M

✐ ✐ ✐

. . . . . . β0 + α1 γ0 + δ1

• E. Jeandel, LIP

, ENS Lyon Algebraic Techniques in Quantum Computing 6/54

SLIDE 9

What can we do with quantum gates ?

❜

. . . . . .

❜ ❜

N . . . . . . M

❜

. . . . . .

❜ ❜

(a) The multiplication MN

❜ ❜ ❜ ❜ ✁ ✁ ❅ ❅

M

❆ ❆

• ❜

❜ ❜ ❜ ❜

. . . . . .

❜ ❜

M

❜ ❜ ❜

. . . . . .

❜ ❜ ❜ ❜

(b) M[σ] (c) The operation M ⊗ I A quantum circuit is everything we can obtain by applying these constructions.

• E. Jeandel, LIP

, ENS Lyon Algebraic Techniques in Quantum Computing 7/54

SLIDE 10

What we cannot do

✐

x

❍❍❍ ❍ ✟✟✟ ✟ ✐ ✐

x x

✟✟✟✟✟✟✟✟✟✟✟✟ ✟ ✟✟✟✟✟✟✟✟✟✟✟✟ ✟ ✟✟✟✟✟✟✟✟✟✟✟✟ ✟ ❍❍❍❍❍❍❍❍❍❍❍❍ ❍ ❍❍❍❍❍❍❍❍❍❍❍❍ ❍ ❍❍❍❍❍❍❍❍❍❍❍❍ ❍

Quantum mechanics implies no-cloning.

• E. Jeandel, LIP

, ENS Lyon Algebraic Techniques in Quantum Computing 8/54

SLIDE 11

Outline

1

Combinatorial setting: Quantum gates Definitions Completeness and Universality

2

Algebraic setting Quantum gates are unitary matrices Computing the group Density

3

Conclusion Automata Conclusion

• E. Jeandel, LIP

, ENS Lyon Algebraic Techniques in Quantum Computing 9/54

SLIDE 12

Completeness

A (finite) set of gates is complete if every quantum gate can be

• btained by a quantum circuit built on these gates.

How to show that some set of gates is complete ?

• E. Jeandel, LIP

, ENS Lyon Algebraic Techniques in Quantum Computing 10/54

SLIDE 13

Completeness

A (finite) set of gates is complete if every quantum gate can be

• btained by a quantum circuit built on these gates.

How to show that some set of gates is complete ?

• E. Jeandel, LIP

, ENS Lyon Algebraic Techniques in Quantum Computing 10/54

SLIDE 14

Game: Design this gate

✐

R G G

✐ ✐

G B G

✐ ✐

B R G

✐

• E. Jeandel, LIP

, ENS Lyon Algebraic Techniques in Quantum Computing 11/54

SLIDE 15

Toolkit 1

❞ ❞ ❞

R 1 1 M

❞ ❞ ❞

G 1 1

❞ ❞ ❞

R 1 M

❞ ❞ ❞

R 1

❞ ❞ ❞

R 1 M

❞ ❞ ❞

R 1

❞ ❞ ❞

R M

❞ ❞ ❞

R

• E. Jeandel, LIP

, ENS Lyon Algebraic Techniques in Quantum Computing 12/54

SLIDE 16

Toolkit 1: Universality

Fact

If there are two wires set to 1, we can make the gate G. This is called universality with ancillas.

❞ ❞ ❞ ❞ ❞

R 1 1

✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆

M

❞ ❞ ❞ ❞ ❞

G

• E. Jeandel, LIP

, ENS Lyon Algebraic Techniques in Quantum Computing 13/54

SLIDE 17

Toolkit 1: Non-completeness

Fact

If among the additional wires, strictly less than 2 are set to 1, the gate G cannot be made. Any circuit, even the most intricate, cannot produce any 1 using only the gate M.

❞ ❞ ❞ ❞ ❞

R

✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ✁ ✁ ✁ ✁ ❅ ❅

• ❆

❆ ❆ ❆

M R M R

❞ ❞ ❞ ❞ ❞

• E. Jeandel, LIP

, ENS Lyon Algebraic Techniques in Quantum Computing 14/54

SLIDE 18

Toolkit 1: Summary

Theorem (8.7)

There exists a set of gates Bi such that Bi is 2-universal but neither 1-universal nor k-complete.

• E. Jeandel, LIP

, ENS Lyon Algebraic Techniques in Quantum Computing 15/54

SLIDE 19

Toolkit 2

❞ ❞ ❞ ❞

R 1 1 1 M

❞ ❞ ❞ ❞

G 1 1 1

❞ ❞ ❞ ❞

R M

❞ ❞ ❞ ❞

G

❞ ❞ ❞ ❞

R x y z M

❞ ❞ ❞ ❞

R x y z

• therwise
• E. Jeandel, LIP

, ENS Lyon Algebraic Techniques in Quantum Computing 16/54

SLIDE 20

Toolkit 2: Non-completeness

Fact

Without any additional wire, we cannot realise the gate G. If the three given wires are set to 1, 1 and 0 there is no mean to have three 1 or three 0.

❞ ❞ ❞ ❞

R 1 1 M

❞ ❞ ❞ ❞

R 1 1

• E. Jeandel, LIP

, ENS Lyon Algebraic Techniques in Quantum Computing 17/54

SLIDE 21

Toolkit 2: 2 additional wires

We are given two additional 0/1-wires. We have now five 0/1-wires. 3 of them must be equal !

❞ ❞ ❞ ❞ ❞ ❞

R 1 1 1

✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆

M

❞ ❞ ❞ ❞ ❞ ❞

G Problem: The wiring depends on the 3 equal wires.

• E. Jeandel, LIP

, ENS Lyon Algebraic Techniques in Quantum Computing 18/54

SLIDE 22

Toolkit 2: 2 additional wires

We are given two additional 0/1-wires. We have now five 0/1-wires. 3 of them must be equal !

❞ ❞ ❞ ❞ ❞ ❞

R 1 1 1

✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆

M

❞ ❞ ❞ ❞ ❞ ❞

G Problem: The wiring depends on the 3 equal wires.

• E. Jeandel, LIP

, ENS Lyon Algebraic Techniques in Quantum Computing 18/54

SLIDE 23

Toolkit 2: Solution

Consider the following circuit:

❜ ❜ ❜ ❜ ❜ ❜

M

❅

• M

❆ ❆ ✁ ✁

M

❆ ❆ ✁ ✁

M

❆ ❆ ✁ ✁

M

❇ ❇ ❇ ❇

• ✁

✁

M

❇ ❇ ❇ ❇ ✂ ✂ ✂ ✂

M

❇ ❇ ❇ ❇ ✂ ✂ ✂ ✂

M

❆ ❆ ✁ ✁

M

❈ ❈ ❈ ❈

• ✂

✂ ✂ ✂

M

❜ ❜ ❜ ❜ ❜ ❜

• E. Jeandel, LIP

, ENS Lyon Algebraic Techniques in Quantum Computing 19/54

SLIDE 24

Toolkit 2: Solution

If 4 bits are equal:

❜ ❜ ❜ ❜ ❜ ❜ R

1 1 1 1

M

R

1 1

❅

1

• 1

M

G

1 1

❆ ❆

1

✁ ✁

1

M

B

1

❆ ❆

1

✁ ✁

1 1

M

B

1

❆ ❆

1 1

✁ ✁

1

M

B

1

❇ ❇ ❇ ❇

1

• 1

1

✁ ✁

M

R ❇ ❇ ❇ ❇

1 1 1

✂ ✂ ✂ ✂

1

M

G

1

❇ ❇ ❇ ❇

1 1

✂ ✂ ✂ ✂

1

M

G

1

❆ ❆

1 1

✁ ✁

1

M

G ❈ ❈ ❈ ❈

• 1

1 1

✂ ✂ ✂ ✂

1

M

G ❜ ❜ ❜ ❜ ❜ ❜

• E. Jeandel, LIP

, ENS Lyon Algebraic Techniques in Quantum Computing 20/54

SLIDE 25

Toolkit 2: Solution

If 4 bits are equal:

❜ ❜ ❜ ❜ ❜ ❜ R

1 1 1 1

M

R

1 1

❅

1

• 1

M

G

1 1

❆ ❆

1

✁ ✁

1

M

B

1

❆ ❆

1

✁ ✁

1 1

M

B

1

❆ ❆

1 1

✁ ✁

1

M

B

1

❇ ❇ ❇ ❇

1

• 1

1

✁ ✁

M

R ❇ ❇ ❇ ❇

1 1 1

✂ ✂ ✂ ✂

1

M

G

1

❇ ❇ ❇ ❇

1 1

✂ ✂ ✂ ✂

1

M

G

1

❆ ❆

1 1

✁ ✁

1

M

G ❈ ❈ ❈ ❈

• 1

1 1

✂ ✂ ✂ ✂

1

M

G ❜ ❜ ❜ ❜ ❜ ❜

• E. Jeandel, LIP

, ENS Lyon Algebraic Techniques in Quantum Computing 20/54

SLIDE 26

Toolkit 2: Solution

If 4 bits are equal:

❜ ❜ ❜ ❜ ❜ ❜ R

1 1 1 1

M

R

1 1

❅

1

• 1

M

G

1 1

❆ ❆

1

✁ ✁

1

M

B

1

❆ ❆

1

✁ ✁

1 1

M

B

1

❆ ❆

1 1

✁ ✁

1

M

B

1

❇ ❇ ❇ ❇

1

• 1

1

✁ ✁

M

R ❇ ❇ ❇ ❇

1 1 1

✂ ✂ ✂ ✂

1

M

G

1

❇ ❇ ❇ ❇

1 1

✂ ✂ ✂ ✂

1

M

G

1

❆ ❆

1 1

✁ ✁

1

M

G ❈ ❈ ❈ ❈

• 1

1 1

✂ ✂ ✂ ✂

1

M

G ❜ ❜ ❜ ❜ ❜ ❜

• E. Jeandel, LIP

, ENS Lyon Algebraic Techniques in Quantum Computing 20/54

SLIDE 27

Toolkit 2: Solution

If 4 bits are equal:

❜ ❜ ❜ ❜ ❜ ❜ R

1 1 1 1

M

R

1 1

❅

1

• 1

M

G

1 1

❆ ❆

1

✁ ✁

1

M

B

1

❆ ❆

1

✁ ✁

1 1

M

B

1

❆ ❆

1 1

✁ ✁

1

M

B

1

❇ ❇ ❇ ❇

1

• 1

1

✁ ✁

M

R ❇ ❇ ❇ ❇

1 1 1

✂ ✂ ✂ ✂

1

M

G

1

❇ ❇ ❇ ❇

1 1

✂ ✂ ✂ ✂

1

M

G

1

❆ ❆

1 1

✁ ✁

1

M

G ❈ ❈ ❈ ❈

• 1

1 1

✂ ✂ ✂ ✂

1

M

G ❜ ❜ ❜ ❜ ❜ ❜

• E. Jeandel, LIP

, ENS Lyon Algebraic Techniques in Quantum Computing 20/54

SLIDE 28

Toolkit 2: Solution

If 4 bits are equal:

❜ ❜ ❜ ❜ ❜ ❜ R

1 1 1 1

M

R

1 1

❅

1

• 1

M

G

1 1

❆ ❆

1

✁ ✁

1

M

B

1

❆ ❆

1

✁ ✁

1 1

M

B

1

❆ ❆

1 1

✁ ✁

1

M

B

1

❇ ❇ ❇ ❇

1

• 1

1

✁ ✁

M

R ❇ ❇ ❇ ❇

1 1 1

✂ ✂ ✂ ✂

1

M

G

1

❇ ❇ ❇ ❇

1 1

✂ ✂ ✂ ✂

1

M

G

1

❆ ❆

1 1

✁ ✁

1

M

G ❈ ❈ ❈ ❈

• 1

1 1

✂ ✂ ✂ ✂

1

M

G ❜ ❜ ❜ ❜ ❜ ❜

• E. Jeandel, LIP

, ENS Lyon Algebraic Techniques in Quantum Computing 20/54

SLIDE 29

Toolkit 2: Solution

If 4 bits are equal:

❜ ❜ ❜ ❜ ❜ ❜ R

1 1 1 1

M

R

1 1

❅

1

• 1

M

G

1 1

❆ ❆

1

✁ ✁

1

M

B

1

❆ ❆

1

✁ ✁

1 1

M

B

1

❆ ❆

1 1

✁ ✁

1

M

B

1

❇ ❇ ❇ ❇

1

• 1

1

✁ ✁

M

R ❇ ❇ ❇ ❇

1 1 1

✂ ✂ ✂ ✂

1

M

G

1

❇ ❇ ❇ ❇

1 1

✂ ✂ ✂ ✂

1

M

G

1

❆ ❆

1 1

✁ ✁

1

M

G ❈ ❈ ❈ ❈

• 1

1 1

✂ ✂ ✂ ✂

1

M

G ❜ ❜ ❜ ❜ ❜ ❜

• E. Jeandel, LIP

, ENS Lyon Algebraic Techniques in Quantum Computing 20/54

SLIDE 30

Toolkit 2: Solution

If 3 bits are equal:

❜ ❜ ❜ ❜ ❜ ❜ R

1 1 1

M

R

1

❅

1

• 1

M

R

1

❆ ❆

1

✁ ✁

1

M

R

1

❆ ❆

1

✁ ✁

1

M

R

1

❆ ❆

1

✁ ✁

1

M

R

1

❇ ❇ ❇ ❇

1

• 1

✁ ✁

M

G ❇ ❇ ❇ ❇

1 1

✂ ✂ ✂ ✂

1

M

G ❇ ❇ ❇ ❇

1 1

✂ ✂ ✂ ✂

1

M

G ❆ ❆

1 1

✁ ✁

1

M

G ❈ ❈ ❈ ❈

• 1

1 1

✂ ✂ ✂ ✂

M

G ❜ ❜ ❜ ❜ ❜ ❜

• E. Jeandel, LIP

, ENS Lyon Algebraic Techniques in Quantum Computing 21/54

SLIDE 31

Toolkit 2: Solution

If all 5 bits are equal:

❜ ❜ ❜ ❜ ❜ ❜ R

1 1 1 1 1

M

G

1 1

❅

1

• 1

1

M

B

1 1

❆ ❆

1 1

✁ ✁

1

M

R

1

❆ ❆

1 1

✁ ✁

1 1

M

G

1 1

❆ ❆

1 1

✁ ✁

1

M

B

1

❇ ❇ ❇ ❇

1

• 1

1

✁ ✁

1

M

R ❇ ❇ ❇ ❇

1 1 1

✂ ✂ ✂ ✂

1 1

M

G

1

❇ ❇ ❇ ❇

1 1 1

✂ ✂ ✂ ✂

1

M

B

1 1

❆ ❆

1 1

✁ ✁

1

M

R ❈ ❈ ❈ ❈

1

• 1

1 1

✂ ✂ ✂ ✂

1

M

G ❜ ❜ ❜ ❜ ❜ ❜

• E. Jeandel, LIP

, ENS Lyon Algebraic Techniques in Quantum Computing 22/54

SLIDE 32

Toolkit 2: Summary

Fact

The previous circuit simulates the gate G whatever the bits on the wires are. This is called 2-completeness (since we use 2 additional wires). Up to some technical details, we obtain:

Theorem (8.8)

There exists a set of gates Bi such that Bi is 3-complete but not complete.

• E. Jeandel, LIP

, ENS Lyon Algebraic Techniques in Quantum Computing 23/54

SLIDE 33

Toolkit 2: Summary

Fact

The previous circuit simulates the gate G whatever the bits on the wires are. This is called 2-completeness (since we use 2 additional wires). Up to some technical details, we obtain:

Theorem (8.8)

There exists a set of gates Bi such that Bi is 3-complete but not complete.

• E. Jeandel, LIP

, ENS Lyon Algebraic Techniques in Quantum Computing 23/54

SLIDE 34

Outline

1

Combinatorial setting: Quantum gates Definitions Completeness and Universality

2

Algebraic setting Quantum gates are unitary matrices Computing the group Density

3

Conclusion Automata Conclusion

• E. Jeandel, LIP

, ENS Lyon Algebraic Techniques in Quantum Computing 24/54

SLIDE 35

What is a quantum gate ?

✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐

M

• E. Jeandel, LIP

, ENS Lyon Algebraic Techniques in Quantum Computing 25/54

SLIDE 36

What is a quantum gate ?

✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐

M

• E. Jeandel, LIP

, ENS Lyon Algebraic Techniques in Quantum Computing 25/54

SLIDE 37

What is a quantum gate ?

✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐

M

• E. Jeandel, LIP

, ENS Lyon Algebraic Techniques in Quantum Computing 25/54

SLIDE 38

What is a quantum gate ?

M

• E. Jeandel, LIP

, ENS Lyon Algebraic Techniques in Quantum Computing 25/54

SLIDE 39

What is a quantum gate ?

M A quantum gate over n qubits is a 2n × 2n unitary matrix

• E. Jeandel, LIP

, ENS Lyon Algebraic Techniques in Quantum Computing 25/54

SLIDE 40

Approximating Quantum Circuits

Problem

Given unitary matrices X1 . . . Xn and a unitary matrix M, is M in the group generated by the Xi ? In the real life, we do not try to obtain quantum gates, but rather to approximate them.

Problem

Given unitary matrices X1 . . . Xn and a unitary matrix M, is M in the euclidean closure of the group generated by the Xi ? (More generally, investigate finitely generated compact groups)

• E. Jeandel, LIP

, ENS Lyon Algebraic Techniques in Quantum Computing 26/54

SLIDE 41

Approximating Quantum Circuits

Problem

Given unitary matrices X1 . . . Xn and a unitary matrix M, is M in the group generated by the Xi ? In the real life, we do not try to obtain quantum gates, but rather to approximate them.

Problem

Given unitary matrices X1 . . . Xn and a unitary matrix M, is M in the euclidean closure of the group generated by the Xi ? (More generally, investigate finitely generated compact groups)

• E. Jeandel, LIP

, ENS Lyon Algebraic Techniques in Quantum Computing 26/54

SLIDE 42

Why compact groups ?

Property

A compact group G of Mn(R) is algebraic. That is there exists polynomials p1 . . . pk such that X ∈ G ⇐ ⇒ ∀i, pi(X) = 0 For instance, if G = O2(R), then G =

• X =

a b c d

• : XX T = I
• =

   a b c d

• :

   a2 + b2 − 1 = c2 + d2 − 1 = ac + bd =    We can compute things ! Now we focus on algebraic groups.

• E. Jeandel, LIP

, ENS Lyon Algebraic Techniques in Quantum Computing 27/54

SLIDE 43

Outline

1

Combinatorial setting: Quantum gates Definitions Completeness and Universality

2

Algebraic setting Quantum gates are unitary matrices Computing the group Density

3

Conclusion Automata Conclusion

• E. Jeandel, LIP

, ENS Lyon Algebraic Techniques in Quantum Computing 28/54

SLIDE 44

Question

Problem

Given matrices X1 . . . Xn, compute the algebraic group generated by the matrices Xi. Computing the group means finding polynomials pi such that X ∈ G ⇐ ⇒ ∀i, pi(X) = 0 Algebraic sets (defined by polynomials) are the closed sets of a topology called the Zariski topology.

• E. Jeandel, LIP

, ENS Lyon Algebraic Techniques in Quantum Computing 29/54

SLIDE 45

Irreducible groups

Theorem

If G1 and G2 are irreducible algebraic groups given by polynomials,

• ne may obtain polynomials for G1, G2 by the following algorithm:

1

H := G1 · G2

2

While H · H = H do H := H · H (A is the Zariski-closure of A, the smallest algebraic set containing A. A · B may be obtained by using Groebner basis techniques)

• E. Jeandel, LIP

, ENS Lyon Algebraic Techniques in Quantum Computing 30/54

SLIDE 46

Irreducible groups

Theorem

If G1 and G2 are irreducible algebraic groups given by polynomials,

• ne may obtain polynomials for G1, G2 by the following algorithm:

1

H := G1 · G2

2

While H · H = H do H := H · H Sketch of proof: At each step H is an irreducible algebraic variety. If H · H = H, H · H is of a greater dimension, which proves that the algorithm terminates.

• E. Jeandel, LIP

, ENS Lyon Algebraic Techniques in Quantum Computing 30/54

SLIDE 47

General groups

Fact

Let G be an algebraic group generated by X1 . . . Xk. Then G = S · H with

1

∀i, Xi ∈ S · H

2

H is an irreducible algebraic group

3

S · H · S · H = S · H

4

H is normal in G : S · H · S−1 = H

5

S is finite Furthermore, if the conditions are satisfied by some S and H, then G = S · H is the algebraic group generated by the Xi.

• E. Jeandel, LIP

, ENS Lyon Algebraic Techniques in Quantum Computing 31/54

SLIDE 48

General groups

Fact

Let G be an algebraic group generated by X1 . . . Xk. Then G = S · H with

1

∀i, Xi ∈ S · H

2

H is an irreducible algebraic group

3

S · S ⊆ S · H

4

H is normal in G : S · H · S−1 = H

5

S is finite Furthermore, if the conditions are satisfied by some S and H, then G = S · H is the algebraic group generated by the Xi.

• E. Jeandel, LIP

, ENS Lyon Algebraic Techniques in Quantum Computing 31/54

SLIDE 49

Sketch of an algorithm

Define by induction

1

S0 = {Xi}, H0 = {I}

2

Hn+1 := Hn · Hn

3

Sn+1 := Sn. For X, Y in Sn, if X · Y ∈ SnHn then Sn+1 := Sn+1 ∪ {X · Y}

4

For X in Sn do Hn+1 := X · Hn+1 · X −1 · Hn+1 Then the limit S = Sn, H = Hn satisfies all conditions of the previous fact. . . except perhaps the last one.

• E. Jeandel, LIP

, ENS Lyon Algebraic Techniques in Quantum Computing 32/54

SLIDE 50

Sketch of an algorithm

Define by induction

1

S0 = {Xi}, H0 = {I}

2

Hn+1 := Hn · Hn

3

Sn+1 := Sn. For X, Y in Sn, if X · Y ∈ SnHn then Sn+1 := Sn+1 ∪ {X · Y}

4

For X in Sn do Hn+1 := X · Hn+1 · X −1 · Hn+1 Then the limit S = Sn, H = Hn satisfies all conditions of the previous fact. . . except perhaps the last one.

• E. Jeandel, LIP

, ENS Lyon Algebraic Techniques in Quantum Computing 32/54

SLIDE 51

General groups revisited

Fact

Let G be an algebraic group generated by X1 . . . Xk. Then G = S · H with

1

∀i, Xi ∈ S · H

2

H is an irreducible algebraic group

3

S · S ⊆ S · H

4

H is normal in G : S · H · S−1 = H

5

S is finite Furthermore, if the conditions are satisfied by some S and H, then S is finite and G = S · H is the algebraic group generated by the Xi.

• E. Jeandel, LIP

, ENS Lyon Algebraic Techniques in Quantum Computing 33/54

SLIDE 52

General groups revisited

Fact

Let G be an algebraic group generated by X1 . . . Xk. Then G = S · H with

1

∀i, Xi ∈ S · H

2

H is an irreducible algebraic group

3

S · S ⊆ S · H

4

H is normal in G : S · H · S−1 = H

5

∀X ∈ S there exists n > 0 such that X n ∈ H. Furthermore, if the conditions are satisfied by some S and H, then S is finite and G = S · H is the algebraic group generated by the Xi.

• E. Jeandel, LIP

, ENS Lyon Algebraic Techniques in Quantum Computing 33/54

SLIDE 53

Sketch of an algorithm, revisited

Define by induction

1

S0 = {Xi}, H = {I}

2

Hn+1 := Hn · Hn

3

Sn+1 := Sn. For X, Y in Sn, if X · Y ∈ SnHn then Sn+1 := Sn+1 ∪ {X · Y}

4

For X in Sn do Hn+1 := X · Hn+1 · X −1 · Hn+1

5

For X in Sn, compute the group GX = SXHX generated by X and add HX to Hn+1 : Hn+1 := HX · Hn+1 Then the limit S = Sn, H = Hn satisfies all conditions of the previous fact. In particular, S is finite.

• E. Jeandel, LIP

, ENS Lyon Algebraic Techniques in Quantum Computing 34/54

SLIDE 54

The new algorithm works

Theorem

The previous algorithm terminates and gives sets S, H such that G = S · H is the algebraic group generated by the Xi. We need only to know how to compute the group generated by one matrix.

• E. Jeandel, LIP

, ENS Lyon Algebraic Techniques in Quantum Computing 35/54

SLIDE 55

Group generated by one matrix : example

X =     β2 β βγ−3 γ     The group generated by X is X =            β2k βk βkγ−3k γk     , k ∈ Z        The algebraic group generated by X is            a b c d     , ab−2 = 1, b−1d3c = 1       

• E. Jeandel, LIP

, ENS Lyon Algebraic Techniques in Quantum Computing 36/54

SLIDE 56

Group generated by one matrix

A unitary matrix, up to a change of basis is of the form    α1 ... αn    (Multiplicative) relationships between the αi is the key point: (m1, . . . , mn) ∈ Γ ⇐ ⇒

i αmi i

= 1 The algebraic group generated by X is then         λ1 ... λn    :

i λmi i

= 1 ∀(m1, . . . , mn) ∈ Γ      To find Γ, we must find bounds for the mi.

• E. Jeandel, LIP

, ENS Lyon Algebraic Techniques in Quantum Computing 37/54

SLIDE 57

Group generated by one matrix

Theorem (Ge)

There exists a polynomial-time algorithm which given the αi computes the multiplicative relations between the αi.

Corollary

There exists an algorithm which computes the compact group generated by a unitary matrix X.

Theorem

There exists an algorithm which computes the algebraic group generated by a matrix X.

• E. Jeandel, LIP

, ENS Lyon Algebraic Techniques in Quantum Computing 38/54

SLIDE 58

Group generated by one matrix

Theorem (Ge)

There exists a polynomial-time algorithm which given the αi computes the multiplicative relations between the αi.

Corollary

There exists an algorithm which computes the compact group generated by a unitary matrix X.

Theorem

There exists an algorithm which computes the algebraic group generated by a matrix X.

• E. Jeandel, LIP

, ENS Lyon Algebraic Techniques in Quantum Computing 38/54

SLIDE 59

Group generated by one matrix

Theorem (Ge)

There exists a polynomial-time algorithm which given the αi computes the multiplicative relations between the αi.

Corollary

There exists an algorithm which computes the compact group generated by a unitary matrix X.

Theorem

There exists an algorithm which computes the algebraic group generated by a matrix X.

• E. Jeandel, LIP

, ENS Lyon Algebraic Techniques in Quantum Computing 38/54

SLIDE 60

Summary

Theorem (3.3)

There exists an algorithm which given matrices Xi computes the algebraic group generated by the Xi. Due to the method (keep going until it stabilises), there is absolutely no bound of complexity for the algorithm.

Theorem

There exists an algorithm which given unitary matrices Xi computes the compact group generated by the Xi.

• E. Jeandel, LIP

, ENS Lyon Algebraic Techniques in Quantum Computing 39/54

SLIDE 61

Summary

Theorem (3.3)

There exists an algorithm which given matrices Xi computes the algebraic group generated by the Xi. Due to the method (keep going until it stabilises), there is absolutely no bound of complexity for the algorithm.

Theorem

There exists an algorithm which given unitary matrices Xi computes the compact group generated by the Xi.

• E. Jeandel, LIP

, ENS Lyon Algebraic Techniques in Quantum Computing 39/54

SLIDE 62

Outline

1

Combinatorial setting: Quantum gates Definitions Completeness and Universality

2

Algebraic setting Quantum gates are unitary matrices Computing the group Density

3

Conclusion Automata Conclusion

• E. Jeandel, LIP

, ENS Lyon Algebraic Techniques in Quantum Computing 40/54

SLIDE 63

Question

Problem

Given matrices X1 . . . Xk, decide if the group generated by the matrices Xi is dense in the algebraic group G. The good notion of “density” for an algebraic group is the Zariski-density.

Problem

Given unitary matrices X1 . . . Xk of dimension n, decide if the group generated by the matrices Xi is dense in Un

• E. Jeandel, LIP

, ENS Lyon Algebraic Techniques in Quantum Computing 41/54

SLIDE 64

Question

Problem

Given matrices X1 . . . Xk, decide if the group generated by the matrices Xi is dense in the algebraic group G. The good notion of “density” for an algebraic group is the Zariski-density.

Problem

Given unitary matrices X1 . . . Xk of dimension n, decide if the group generated by the matrices Xi is dense in Un

• E. Jeandel, LIP

, ENS Lyon Algebraic Techniques in Quantum Computing 41/54

SLIDE 65

Simple groups

A simple group has no non-trivial normal irreducible subgroups. This gives an algorithm for a simple group:

Theorem

H is dense in a simple group G iff H is infinite and H is normal in G. There exists an algorithm from Babai, Beals and Rockmore to test if a finitely generated group is finite. We only have to find a way to show that H is normal in G.

• E. Jeandel, LIP

, ENS Lyon Algebraic Techniques in Quantum Computing 42/54

SLIDE 66

Normal groups

H is normal in G ⇐ ⇒ ∀X ∈ G, XHX −1 = H Denote by KG the set {M → XMX −1, X ∈ G}. KG is a set (in fact a group) of endomorphisms of Mn. H is normal in G ⇐ ⇒ ∀φ ∈ KG, φ(H) = H

Fact

∀φ ∈ KH, φ(H) = H.

Corollary

If KH = KG then H is normal in G. Testing KH = KG is not that easy..

• E. Jeandel, LIP

, ENS Lyon Algebraic Techniques in Quantum Computing 43/54

SLIDE 67

Normal groups

H is normal in G ⇐ ⇒ ∀X ∈ G, XHX −1 = H Denote by KG the set {M → XMX −1, X ∈ G}. KG is a set (in fact a group) of endomorphisms of Mn. H is normal in G ⇐ ⇒ ∀φ ∈ KG, φ(H) = H

Fact

∀φ ∈ KH, φ(H) = H.

Corollary

If KH = KG then H is normal in G. Testing KH = KG is not that easy..

• E. Jeandel, LIP

, ENS Lyon Algebraic Techniques in Quantum Computing 43/54

SLIDE 68

Normal groups

H is normal in G ⇐ ⇒ ∀X ∈ G, XHX −1 = H Denote by KG the set {M → XMX −1, X ∈ G}. KG is a set (in fact a group) of endomorphisms of Mn. H is normal in G ⇐ ⇒ ∀φ ∈ KG, φ(H) = H

Fact

∀φ ∈ KH, φ(H) = H.

Corollary

If KH = KG then H is normal in G. Testing KH = KG is not that easy..

• E. Jeandel, LIP

, ENS Lyon Algebraic Techniques in Quantum Computing 43/54

SLIDE 69

Normal groups

H is normal in G ⇐ ⇒ ∀X ∈ G, XHX −1 = H Denote by KG the set {M → XMX −1, X ∈ G}. KG is a set (in fact a group) of endomorphisms of Mn. H is normal in G ⇐ ⇒ ∀φ ∈ KG, φ(H) = H

Fact

∀φ ∈ KH, φ(H) = H.

Corollary

If KH = KG then H is normal in G. Testing KH = KG is not that easy..

• E. Jeandel, LIP

, ENS Lyon Algebraic Techniques in Quantum Computing 43/54

SLIDE 70

Normal groups

Denote by Span(S) the vector space generated by S.

Theorem (2.5)

If Span(KH) = Span(KG), then H is normal in G.

Proof.

We use Lie algebras techniques. The condition implies that the Lie algebra of H is an ideal of the Lie algebra of G.

Fact

Testing whether Span(KH) = Span(KG) is easy.

• E. Jeandel, LIP

, ENS Lyon Algebraic Techniques in Quantum Computing 44/54

SLIDE 71

Normal groups

Denote by Span(S) the vector space generated by S.

Theorem (2.5)

If Span(KH) = Span(KG), then H is normal in G.

Proof.

We use Lie algebras techniques. The condition implies that the Lie algebra of H is an ideal of the Lie algebra of G.

Fact

Testing whether Span(KH) = Span(KG) is easy.

• E. Jeandel, LIP

, ENS Lyon Algebraic Techniques in Quantum Computing 44/54

SLIDE 72

Normal groups

Denote by Span(S) the vector space generated by S.

Theorem (2.5)

If Span(KH) = Span(KG), then H is normal in G.

Proof.

We use Lie algebras techniques. The condition implies that the Lie algebra of H is an ideal of the Lie algebra of G.

Fact

Testing whether Span(KH) = Span(KG) is easy.

• E. Jeandel, LIP

, ENS Lyon Algebraic Techniques in Quantum Computing 44/54

SLIDE 73

Computing Span(KH)

Let E be the vector space generated by the morphisms M → XiMX −1

i

While E is not stable by multiplication (composition), let E := EE = {φ ◦ ψ : φ ∈ E, ψ ∈ E}

Theorem

For every simple group G, there exists a polynomial time algorithm which decides if a finitely generated subgroup H is dense in G.

• E. Jeandel, LIP

, ENS Lyon Algebraic Techniques in Quantum Computing 45/54

SLIDE 74

Computing Span(KH)

Let E be the vector space generated by the morphisms M → XiMX −1

i

While E is not stable by multiplication (composition), let E := EE = {φ ◦ ψ : φ ∈ E, ψ ∈ E}

Theorem

For every simple group G, there exists a polynomial time algorithm which decides if a finitely generated subgroup H is dense in G.

• E. Jeandel, LIP

, ENS Lyon Algebraic Techniques in Quantum Computing 45/54

SLIDE 75

Generalisation

Theorem (2.26)

For every reductive group G, there exists a polynomial time algorithm which decides if a finitely generated subgroup H is Zariski-dense in G.

Theorem (2.27)

For every compact group G, there exists a polynomial time algorithm which decides if a finitely generated subgroup H is dense in G.

• E. Jeandel, LIP

, ENS Lyon Algebraic Techniques in Quantum Computing 46/54

SLIDE 76

Generalisation

Theorem (2.26)

For every reductive group G, there exists a polynomial time algorithm which decides if a finitely generated subgroup H is Zariski-dense in G.

Theorem (2.27)

For every compact group G, there exists a polynomial time algorithm which decides if a finitely generated subgroup H is dense in G.

• E. Jeandel, LIP

, ENS Lyon Algebraic Techniques in Quantum Computing 46/54

SLIDE 77

Back to circuits

Theorem (8.5)

There exists a polynomial time algorithm which decides if a set of gates is complete.

Theorem (8.4)

There exists an algorithm which decides if a set of gates is universal.

• E. Jeandel, LIP

, ENS Lyon Algebraic Techniques in Quantum Computing 47/54

SLIDE 78

Back to circuits

Theorem (8.5)

There exists a polynomial time algorithm which decides if a set of gates is complete.

Theorem (8.4)

There exists an algorithm which decides if a set of gates is universal.

• E. Jeandel, LIP

, ENS Lyon Algebraic Techniques in Quantum Computing 47/54

SLIDE 79

Outline

1

Combinatorial setting: Quantum gates Definitions Completeness and Universality

2

Algebraic setting Quantum gates are unitary matrices Computing the group Density

3

Conclusion Automata Conclusion

• E. Jeandel, LIP

, ENS Lyon Algebraic Techniques in Quantum Computing 48/54

SLIDE 80

Automata (Sketch)

We are given a gate for each letter a, b, c . . . .

❜

. . . . . .

❜ ❜

a

❜ ❜ ❜

. . . . . .

❜

. . . . . .

❜ ❜

b

❜ ❜ ❜

. . . . . .

❜

. . . . . .

❜ ❜

c

❜ ❜ ❜

. . . . . . The value (or probability) of a word ω is function of the result of the circuit corresponding to ω.

• E. Jeandel, LIP

, ENS Lyon Algebraic Techniques in Quantum Computing 49/54

SLIDE 81

Automata (Sketch)

❜

. . . . . .

❜ ❜

a . . . . . . c . . . . . . c . . . . . . ❜

❜ ❜α0 + β1

acc is accepted with probability |α|2.

❜

. . . . . .

❜ ❜

b . . . . . . b . . . . . . a . . . . . . b . . . . . . ❜

❜ ❜δ0 + ǫ1

bbab is accepted with probability |δ|2.

• E. Jeandel, LIP

, ENS Lyon Algebraic Techniques in Quantum Computing 50/54

SLIDE 82

Theorems

Some theorems about quantum automata :

Theorem (5.4)

We can decide given an automaton A and a threshold λ if there exists a word accepted with a probability strictly greater than λ. We use the algorithm which computes the group generated by some matrices.

Theorem (7.1)

Non-deterministic quantum automata with an isolated threshold recognise only regular languages. The proof introduces a new model of automata, called topological automata.

• E. Jeandel, LIP

, ENS Lyon Algebraic Techniques in Quantum Computing 51/54

SLIDE 83

Theorems

Some theorems about quantum automata :

Theorem (5.4)

We can decide given an automaton A and a threshold λ if there exists a word accepted with a probability strictly greater than λ. We use the algorithm which computes the group generated by some matrices.

Theorem (7.1)

Non-deterministic quantum automata with an isolated threshold recognise only regular languages. The proof introduces a new model of automata, called topological automata.

• E. Jeandel, LIP

, ENS Lyon Algebraic Techniques in Quantum Computing 51/54

SLIDE 84

Outline

1

Combinatorial setting: Quantum gates Definitions Completeness and Universality

2

Algebraic setting Quantum gates are unitary matrices Computing the group Density

3

Conclusion Automata Conclusion

• E. Jeandel, LIP

, ENS Lyon Algebraic Techniques in Quantum Computing 52/54

SLIDE 85

Conclusion

Study of quantum objects using algebraic groups techniques. New algorithms about algebraic groups. Many other potentially interesting things.

• E. Jeandel, LIP

, ENS Lyon Algebraic Techniques in Quantum Computing 53/54

SLIDE 86

Conclusion

Study of quantum objects using algebraic groups techniques. New algorithms about algebraic groups. Many other potentially interesting things.

• E. Jeandel, LIP

, ENS Lyon Algebraic Techniques in Quantum Computing 53/54

SLIDE 87

Conclusion

Study of quantum objects using algebraic groups techniques. New algorithms about algebraic groups. Many other potentially interesting things.

• E. Jeandel, LIP

, ENS Lyon Algebraic Techniques in Quantum Computing 53/54

SLIDE 88

Perspectives and open problems

Problem

What if the number of auxiliary wires depends on the gate to realise (∞-universality) ? Is it equivalent to m-universality for some m ?

Problem

Find an efficient algorithm to decide whether some matrix X is in the algebraic group generated by the matrices Xi. More generally, use the structure of the algebraic groups more efficiently.

• E. Jeandel, LIP

, ENS Lyon Algebraic Techniques in Quantum Computing 54/54

SLIDE 89

Perspectives and open problems

Problem

What if the number of auxiliary wires depends on the gate to realise (∞-universality) ? Is it equivalent to m-universality for some m ?

Problem

Find an efficient algorithm to decide whether some matrix X is in the algebraic group generated by the matrices Xi. More generally, use the structure of the algebraic groups more efficiently.

• E. Jeandel, LIP

, ENS Lyon Algebraic Techniques in Quantum Computing 54/54