Hidden Subgroup Hidden Subgroup Def. A Map is - PDF document

Continuous Quantum Continuous Quantum Hidden Subgroup Hidden Subgroup Algorithms Algorithms This work is in collaboration with This work is in collaboration with Samuel J. Lomonaco, Jr. Louis H. Kauffman Louis H. Kauffman Dept. of Comp. Sci. & Electrical Engineering University of Maryland Baltimore County Baltimore, MD 21250 Email: Lomonaco@UMBC.EDU WebPage: http://www.csee.umbc.edu/~lomonaco Defense Advanced Research Projects Agency (DARPA) & Defense Advanced Research Projects Agency (DARPA) & Air Force Research Laboratory, Air Force Materiel Command, USAF Air Force Research Laboratory, Air Force Materiel Command, USAF Agreement Number F30602- -01 01- -2 2- -0522 0522 Agreement Number F30602 This work is supported by: This work is supported by: Lomonaco, Entanglement Entanglement Kauffman & Lomonaco, • Kauffman & • Criteria – Criteria – Quantum and Topological Quantum and Topological, , http://xxx.lanl.gov/abs/quant- -ph/0304091 ph/0304091 http://xxx.lanl.gov/abs/quant The Defense Advance Research Projects • Agency (DARPA) & Air Force Research Laboratory (AFRL), Air Force Materiel Command, • Lomonaco & Kauffman, Lomonaco & Kauffman, Continuous Continuous • USAF Agreement Number F30602-01-2-0522. Quantum Hidden Subgroup Algorithms, Quantum Hidden Subgroup Algorithms, http://xxx.lanl.gov/abs/quant http://xxx.lanl.gov/abs/quant- -ph/0304084 ph/0304084 The National Institute for Standards • and Technology (NIST) • The Mathematical Sciences Research Institute (MSRI). • The L-O-O-P Fund. L- -O O- -O O- -P P L Hidden Subgroup Subgroup Hidden Ambient Ambient Target Target Structure Structure Group Group Set Set Hidden Subgroup Hidden Subgroup ϕ → Def. A Map is said to have A Map is said to have hidden : A hidden S Def. subgroup structure subgroup structure if there exist if there exist Hidden Hidden Algorithms Algorithms Subgroup Subgroup • A subgroup of , and • A subgroup of , and K ϕ A • An injection • An injection ι → : / A K S ϕ ϕ ϕ ϕ Set of Right Set of Right Cosets Cosets ϕ s. t. the diagram s. t. the diagram → A S Hidden Natural Hidden Natural Surjection Surjection ν ↑ ι � ϕ is commutative. is commutative. / A K ϕ 1

Hidden Subgroup Hidden Subgroup Structure Structure (Cont.) (Cont.) Origin of QHS Algorithms Origin of QHS Algorithms ϕ Shor’ ’s s Quantum factoring algorithm Quantum factoring algorithm Shor → Hidden Hidden A S reduces the task of factoring an integer reduces the task of factoring an integer Epimorphism Epimorphism ν ↑ ι � to the task of finding the period to the task of finding the period N P ϕ of a function of a function / A K ϕ Hidden Quotient Hidden Quotient ϕ Group Group Z   → Z mod N If is an invariant invariant subgroup subgroup of , then of , then If is an K ϕ A � n n a mod N = / H A K ϕ ϕ ϕ ϕ Kitaev observed that finding the period Kitaev observed that finding the period P ν → is a group, and is an is a group, and is an : / A A K ϕ is equivalent to finding the subgroup , is equivalent to finding the subgroup , P ⊂ Z Z epimorphism epimorphism ϕ i.e., the kernel of . i.e., the kernel of . Two Ways to Create New Quantum Algorithms Two Ways to Create New Quantum Algorithms Lifting and Pushing Lifting and Pushing ϕ → Given Given Three Methods for Three Methods for : A S Creating New Quantum Creating New Quantum Algorithms � Algorithms Lifted Lifted Gp Gp ϕ L Lift Lift η ϕ Amb Amb. . Gp Gp Target Set Target Set A S ν ι Push Push � ϕ ϕ = ϕ ϕ ι � Approx Gp Approx Gp H A 3rd Way to Create New Quantum Algorithms A 3rd Way to Create New Quantum Algorithms Summary Summary Duality Duality 3 Ways to create New Quantum Algorithms 3 Ways to create New Quantum Algorithms QHS QHS Alg Alg ϕ Amb. . Gp Gp Amb • • → A S Lifting Lifting • • Pushing Pushing Φ � S ′ → • • A Duality Duality Dual Dual Dual Gp Dual Gp QHS Alg QHS Alg 2

Hidden Subgroup Algorithms Hidden Subgroup Algorithms Free Abel Free Abel Wandering Wandering Shor Shor Target Target Finite Finite Rk Rk Some Past Algorithms Some Past Algorithms Set Set Amb GP GP Amb • Wandering • ϕ Wandering Shor Shor ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ → � � � � � � � � � � � � � � � � � � A S • • Lomonaco & Kauffman, Push Push Lomonaco & Kauffman, Quantum Hidden Quantum Hidden ι ↑↓ ↑↓ ν Shor Shor Subgroup Algorithms: A Mathematical A Mathematical Subgroup Algorithms: � Transv Transv ϕ ϕ = ϕ ι ϕ � Perspective, AMS, CONM/305, (2002). AMS, CONM/305, (2002). Perspective, • Continuous � • http://xxx.lanl.gov/abs/quant http://xxx.lanl.gov/abs/quant- -ph/0201095 ph/0201095 Q Approx Approx Map Map Continuous Shor Shor Approx Gp Gp Approx • Lomonaco & Kauffman, • Lomonaco & Kauffman, A Continuous A Continuous Variable Shor Variable Shor Algorithm Algorithm, , http://xxx.lanl.gov/abs/quant http://xxx.lanl.gov/abs/quant- -ph/0210141 ph/0210141 Continuous Shor Continuous Shor Three Recent QHS Algorithms Three Recent QHS Algorithms Ambient Group Ambient Group ϕ Add. Gp Gp of of Reals Reals Add. → • A quantum algorithm on the → � A quantum algorithm on the Circle Circle A S S • A quantum algorithm to A quantum algorithm to Shor dual dual Shor’ ’s s algorithm algorithm Key Idea: Lifting of discrete algorithms to Key Idea: Lifting of discrete algorithms to a continuous groups a continuous groups • A highly speculative quantum algorithm for A highly speculative quantum algorithm for functional integrals functional integrals ⇒ ? Quantum algorithm for ? Quantum algorithm for the Jones polynomial the Jones polynomial Road Map Road Map Lifting Lifting & & Duality Duality Shor’ Shor ’s s Alg Alg ϕ � S Lift of Lift of Shor Shor Lifting Lifting Algorithm Algorithm QHS Algs QHS Algs for for ~ QHS Alg Alg on on � ϕ QHS Shor Shor Functional Functional � Algorithm Algorithm Integrals Integrals Duality Duality Q Dual Dual ϕ � � Dual Lifted Dual Lifted QHS Alg Alg on on QHS / � � / S Algorithm Algorithm Pushing Pushing ~ ϕ Dual Shor Shor Dual � Dual of Shor Dual of Shor’ ’s s Alg Alg Algorithm Algorithm Q 3

A Momentary Digression A Momentary Digression Fourier Analysis A Lifting of Shor A Lifting of Shor’ ’s s Fourier Analysis Quantum Factoring Quantum Factoring on the on the Algorithm to Algorithm to Integers � Integers Circle Circle The Circle as a Group The Circle as a Group The Circle as a Group The Circle as a Group can also also be viewed as The circle group circle group can be viewed as The The The circle circle group group can be viewed as can be viewed as • An • An additive additive group group, i.e., as , i.e., as • A • A multiplicative multiplicative group group, i.e., as the unit , i.e., as the unit = circle in the complex plane � circle in the complex plane � � / mod1 reals { } π ∈ � 2 ix : + e x mod 1 x y ( ( ) ) π π π π = π + 2 2 ix i 2 iy i x y e e e where denotes the additive group of where denotes the additive group of � where denotes the additive group of where denotes the additive group of � reals reals. . integers. integers. The Character Groups of The Character Groups of and and � � The Character Group The Character Group / � • The The character group character group of is of is � � The character group character group of an abelian abelian group group The of an A A is defined as { } is defined as � χ π = = ∈ ∈ = � � 2 inx � � � : : / x n e x � ( ( ) ) = , A Hom A Circle { } • • The The character group character group of is of is � � / = = χ χ → → χ χ : A Circle : a morphism with group operation (in multiplicative notation), with group operation (in multiplicative notation), { } � ≅ ≅ χ π ∈ ∈ � � � 2 inx � ( ( ) ( ) ) / : : ( ) ( ) ( ) ( ) ( ) x e n χ χ χ χ = χ χ i i a a a n 1 2 1 2 { { } } ≅ ≅ χ ∈ ∈ = � � � : mod 1: x nx n or (in additive notation) as or (in additive notation) as n ( ( ) ( ) ) ( ) ( ) ( ) ( ) ( ) ⇔ χ χ + + χ χ = = χ + χ � � � / a a a 1 2 1 2 4