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

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

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

Outline Combinatorial setting: Quantum gates 1 Definitions Completeness and Universality Algebraic setting 2 Quantum gates are unitary matrices Computing the group Density Conclusion 3 Automata Conclusion E. Jeandel, LIP , ENS Lyon Algebraic Techniques in Quantum Computing 3/54

Introduction Classical Quantum � α i q i State q 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

Outline Combinatorial setting: Quantum gates 1 Definitions Completeness and Universality Algebraic setting 2 Quantum gates are unitary matrices Computing the group Density Conclusion 3 Automata Conclusion E. Jeandel, LIP , ENS Lyon Algebraic Techniques in Quantum Computing 5/54

What is a quantum gate ? ✐ ✐ ✐ ✐ M . . . . . . . . . . . . ✐ ✐ E. Jeandel, LIP , ENS Lyon Algebraic Techniques in Quantum Computing 6/54

What is a quantum gate ? 0 1 ✐ ✐ 0 β 0 + i α 1 ✐ ✐ M . . . . . . . . . . . . 1 α 0 + β 1 ✐ ✐ E. Jeandel, LIP , ENS Lyon Algebraic Techniques in Quantum Computing 6/54

What is a quantum gate ? β 0 + α 1 0 ✐ ✐ β 0 − δ 1 0 ✐ ✐ M . . . . . . . . . . . . γ 1 + α 0 γ 0 + δ 1 ✐ ✐ E. Jeandel, LIP , ENS Lyon Algebraic Techniques in Quantum Computing 6/54

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

What we cannot do ❍❍❍❍❍❍❍❍❍❍❍❍ ❍❍❍❍❍❍❍❍❍❍❍❍ ❍❍❍❍❍❍❍❍❍❍❍❍ ✟ ✟ ✟ ✟✟✟✟✟✟✟✟✟✟✟✟ ✟✟✟✟✟✟✟✟✟✟✟✟ ✟✟✟✟✟✟✟✟✟✟✟✟ x ✟ ✐ ✟✟✟ x ✐ ❍❍❍ x ❍ ✐ ❍ ❍ ❍ Quantum mechanics implies no-cloning. E. Jeandel, LIP , ENS Lyon Algebraic Techniques in Quantum Computing 8/54

Outline Combinatorial setting: Quantum gates 1 Definitions Completeness and Universality Algebraic setting 2 Quantum gates are unitary matrices Computing the group Density Conclusion 3 Automata Conclusion E. Jeandel, LIP , ENS Lyon Algebraic Techniques in Quantum Computing 9/54

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

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

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

Toolkit 1 R G R R ❞ ❞ ❞ ❞ 1 1 0 0 ❞ ❞ ❞ ❞ M M 1 1 1 1 ❞ ❞ ❞ ❞ R R R R ❞ ❞ ❞ ❞ 1 1 0 0 ❞ ❞ ❞ ❞ M M 0 0 0 0 ❞ ❞ ❞ ❞ E. Jeandel, LIP , ENS Lyon Algebraic Techniques in Quantum Computing 12/54

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 G ❞ ❞ ❞ M ❞ ❆ ✁ ❆ ✁ ✁ ✁ ❆ ❆ ❞ ❞ ✁ ✁ ❆ ✁ ❆ ✁ ✁ ✁ ❆ ❆ ✁ ❆ ✁ ❆ 1 ❆ ❆ ❞ ❞ ✁ ✁ ❆ ❆ ✁ ❆ ✁ ❆ 1 ❞ ❞ E. Jeandel, LIP , ENS Lyon Algebraic Techniques in Quantum Computing 13/54

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 R R ❞ ❞ 0 ❞ ❞ M M ❆ ✁ ❆ � ✁ � 0 ❆ ❆ ❞ ❞ ✁ ❆ ✁ ✁ ✁ ✁ ❆ ❆ ✁ ❆ ❆ 0 ❆ ❞ ❞ ✁ ✁ ❅ ❆ ✁ ❆ ✁ ❅ 0 ❞ ❞ E. Jeandel, LIP , ENS Lyon Algebraic Techniques in Quantum Computing 14/54

Toolkit 1: Summary Theorem (8.7) There exists a set of gates B i such that B i is 2 -universal but neither 1 -universal nor k-complete. E. Jeandel, LIP , ENS Lyon Algebraic Techniques in Quantum Computing 15/54

Toolkit 2 R G R G ❞ ❞ ❞ ❞ 1 1 0 0 ❞ ❞ ❞ ❞ M M 1 1 0 0 ❞ ❞ ❞ ❞ 1 1 0 0 ❞ ❞ ❞ ❞ R R ❞ ❞ x x otherwise ❞ ❞ M y y ❞ ❞ z z ❞ ❞ E. Jeandel, LIP , ENS Lyon Algebraic Techniques in Quantum Computing 16/54

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 R ❞ ❞ 1 1 ❞ ❞ M 1 1 ❞ ❞ 0 0 ❞ ❞ E. Jeandel, LIP , ENS Lyon Algebraic Techniques in Quantum Computing 17/54

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 G ❞ ❞ 1 ❞ ❞ M 0 ❞ ❞ ❆ ✁ ❆ ✁ ✁ ✁ 0 ❆ ❆ ❞ ❞ ✁ ✁ ❆ ✁ ❆ ✁ ✁ ✁ ❆ ❆ ✁ ❆ ✁ ❆ 1 ❆ ❆ ❞ ❞ ✁ ✁ ❆ ❆ ✁ ❆ ✁ ❆ 1 ❞ ❞ Problem: The wiring depends on the 3 equal wires. E. Jeandel, LIP , ENS Lyon Algebraic Techniques in Quantum Computing 18/54

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 G ❞ ❞ 1 ❞ ❞ M 0 ❞ ❞ ❆ ✁ ❆ ✁ ✁ ✁ 0 ❆ ❆ ❞ ❞ ✁ ✁ ❆ ✁ ❆ ✁ ✁ ✁ ❆ ❆ ✁ ❆ ✁ ❆ 1 ❆ ❆ ❞ ❞ ✁ ✁ ❆ ❆ ✁ ❆ ✁ ❆ 1 ❞ ❞ Problem: The wiring depends on the 3 equal wires. E. Jeandel, LIP , ENS Lyon Algebraic Techniques in Quantum Computing 18/54

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

Toolkit 2: Solution If 4 bits are equal: R R G B B ❜ ❜ 1 1 1 1 1 M M M M M ❜ 1 1 1 ❆ ✁ 0 0 ❜ 0 � ❅ 1 ❆ ✁ 1 ✁ ❆ 1 ❆ ✁ 1 ❜ 1 0 ✁ ❆ 0 1 ✁ ❆ 1 ❜ 1 1 1 1 1 B R G G G G ❜ ❜ 1 ❇ ✂ ✂ 1 1 1 ❈ � 0 M M M M M ❜ 1 1 0 0 1 � ❇ ✂ ❇ ❇ ✂ ✂ ❈ ✂ ✂ ❜ 1 1 1 1 1 ❇ ✁ ✂ ❇ ❇ ❇ ✂ ❆ ✁ ✂ ❈ ❜ ✁ ❇ ❇ 1 1 ✂ ❇ ❇ 1 ✁ ❆ 1 ✂ ❈ 1 ❜ 0 0 1 1 1 E. Jeandel, LIP , ENS Lyon Algebraic Techniques in Quantum Computing 20/54

Toolkit 2: Solution If 4 bits are equal: R R G B B ❜ ❜ 1 1 1 1 1 M M M M M ❜ 1 1 1 ❆ ✁ 0 0 ❜ 0 � ❅ 1 ❆ ✁ 1 ✁ ❆ 1 ❆ ✁ 1 ❜ 1 0 ✁ ❆ 0 1 ✁ ❆ 1 ❜ 1 1 1 1 1 B R G G G G ❜ ❜ 1 ❇ ✂ ✂ 1 1 1 ❈ � 0 M M M M M ❜ 1 1 0 0 1 � ❇ ✂ ❇ ❇ ✂ ✂ ❈ ✂ ✂ ❜ 1 1 1 1 1 ❇ ✁ ✂ ❇ ❇ ❇ ✂ ❆ ✁ ✂ ❈ ❜ ✁ ❇ ❇ 1 1 ✂ ❇ ❇ 1 ✁ ❆ 1 ✂ ❈ 1 ❜ 0 0 1 1 1 E. Jeandel, LIP , ENS Lyon Algebraic Techniques in Quantum Computing 20/54

Toolkit 2: Solution If 4 bits are equal: R R G B B ❜ ❜ 1 1 1 1 1 M M M M M ❜ 1 1 1 ❆ ✁ 0 0 ❜ 0 � ❅ 1 ❆ ✁ 1 ✁ ❆ 1 ❆ ✁ 1 ❜ 1 0 ✁ ❆ 0 1 ✁ ❆ 1 ❜ 1 1 1 1 1 B R G G G G ❜ ❜ 1 ❇ ✂ ✂ 1 1 1 ❈ � 0 M M M M M ❜ 1 1 0 0 1 � ❇ ✂ ❇ ❇ ✂ ✂ ❈ ✂ ✂ ❜ 1 1 1 1 1 ❇ ✁ ✂ ❇ ❇ ❇ ✂ ❆ ✁ ✂ ❈ ❜ ✁ ❇ ❇ 1 1 ✂ ❇ ❇ 1 ✁ ❆ 1 ✂ ❈ 1 ❜ 0 0 1 1 1 E. Jeandel, LIP , ENS Lyon Algebraic Techniques in Quantum Computing 20/54