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Quantum Kan Extensions and Applications Quantum Kan Extensions and their Applications IARPA QCS PI Meeting z 16–17 July 2012 0 | i F ϕ ψ A B | i ǫ =⇒ X Lan (X ) F θ y Sets x 1 | i BakeÖ ÅÓÙÒØaiÒ Science Technology Service Dr.RalphL.Wojtowicz Dr.NosonS.Yanofsky Baker Mountain Research Corporation Department of Computer Yellow Spring, WV and Information Science Brooklyn College Contract: D11PC20232 Background Right Kan Extensions Left Kan Extensions Theory Plans Project Overview Goals: Implement and analyze classical Kan extensions algorithms Research and implement quantum algorithms for Kan extensions Research Kan liftings and homotopy Kan extensions and their applications to quantum computing Performance period: 26 September 2011 – 25 September 2012 Progress: Implemented Carmody-Walters classical Kan extensions algorithm Surveyed complexity of Todd-Coxeter coset enumeration algorithm Proved that hidden subgroups are examples of Kan extensions Found quantum algorithms for: (1) products, (2) pullbacks and (3) equalizers [(1) and (3) give all right Kan extensions] Found quantum algorithm for (1) coproducts and made progress on (2) coequalizers [(1) and (2) give all left Kan extensions] Implemented quantum algorithm for coproducts Report on Kan liftings in progress ÅÓÙÒØaiÒ BakeÖ Quantum Kan Extensions — 17 July 2012 1/34 Background Right Kan Extensions Left Kan Extensions Theory Plans Outline 1 Background Definitions Examples and Applications The Carmody-Walters Kan Extension Algorithm 2 Right Kan Extensions Products Pullbacks Equalizers 3 Left Kan Extensions Coproducts Coequalizers 4 Theory The Hidden Subgroup Problem Kan Liftings Homotopy Kan Extensions and Liftings 5 Plans ÅÓÙÒØaiÒ BakeÖ Quantum Kan Extensions — 17 July 2012 2/34 Background Right Kan Extensions Left Kan Extensions Theory Plans Outline 1 Background Definitions Examples and Applications The Carmody-Walters Kan Extension Algorithm 2 Right Kan Extensions Products Pullbacks Equalizers 3 Left Kan Extensions Coproducts Coequalizers 4 Theory The Hidden Subgroup Problem Kan Liftings Homotopy Kan Extensions and Liftings 5 Plans ÅÓÙÒØaiÒ BakeÖ Quantum Kan Extensions — 17 July 2012 3/34 Background Right Kan Extensions Left Kan Extensions Theory Plans Kan Extensions: Definitions Given: Categories and (presented as directed graphs with commutativity constraints)A B A functor F : (assigning a -path to each -edge) An action X :A→BSets (assigningB a set to each A-vertex and a function to eachA →-edge) A A left Kan extensionAof X along F consists of: An action L : Sets B → A natural transformation ǫA : X (A) L(F (A)) → F Aǫ B =⇒ X L Sets These ingredients satisfy a universal mapping property. A right Kan extension has ǫ going the other way ǫA : L(F (A)) X (A). → ÅÓÙÒØaiÒ BakeÖ Quantum Kan Extensions — 17 July 2012 4/34 Background Right Kan Extensions Left Kan Extensions Theory Plans Kan Extensions: Examples Right Kan Extensions Left Kan Extensions products coproducts equalizers coequalizers fixed points orbits greatest lower bound least upper bound intersection union conjunction implication existential quantification∧ universal quantification⇒ left adjoints ∃ right adjoints ∀ limits colimits ends coends claws coset enumeration Right Kan extensions can be calculated from products and equalizers. Left Kan extensions can be calculated from coproducts and coequalizers. ÅÓÙÒØaiÒ BakeÖ Quantum Kan Extensions — 17 July 2012 5/34 Background Right Kan Extensions Left Kan Extensions Theory Plans The Todd-Coxeter Coset Enumeration Algorithm Dehn (1911): Find an algorithm to decide whether, in a finitely-presented group, a word in the generators represents the identity element. Todd-Coxeter (1936): Algorithm for enumerating cosets of H G. Haselgrove (1953) gave the first computer implementation. ≤ Now implemented in many computer algebra systems. Novikov, Boone and Britton (1955–1963): The word problem is unsolvable (in finite time by any Turing machine). Cannon, Dimino, Havas and Watson. Implementation and Analysis of the Todd-Coxeter Algorithm (1973). Given group G and integer m, there is a presentation of G for which Todd-Coxeter will generate at least m cosets. The number of cosets generated by Todd-Coxeter can vary with the order of the relations in the presentation. Carmody-Walters (1995): Left Kan extension algorithm for finitely-presented groups generalizes Todd-Coxeter. ÅÓÙÒØaiÒ BakeÖ Quantum Kan Extensions — 17 July 2012 6/34 Background Right Kan Extensions Left Kan Extensions Theory Plans Carmody-Walters Algorithm: Sample Calculation Setup Generators Relation Category ◦ ◦ id id q p q id q |B| B P P ◦ q q p q p ◦ P p Q P p Q q X L ◦ Sets Q P Q p p q p ◦ q p ◦ q ◦ p q ◦ p ǫ-tables X (P) L(P) X (Q) L(Q) L(P) = arrows into P L(Q) = arrows into Q 1 1 1 1 L-tables Relation-table L(P) L(Q) L(Q) L(P) L(P) L(Q) L(P) L(Q) L(P) L(P) 1 2 1 2 1 2 3 4 5 1 2 3 2 3 2 3 4 5 6 2 3 4 3 4 341233 4 5 4 5 452344 5 6 ÅÓÙÒØaiÒ BakeÖ Quantum Kan Extensions — 17 July 2012 7/34 Background Right Kan Extensions Left Kan Extensions Theory Plans Outline 1 Background Definitions Examples and Applications The Carmody-Walters Kan Extension Algorithm 2 Right Kan Extensions Products Pullbacks Equalizers 3 Left Kan Extensions Coproducts Coequalizers 4 Theory The Hidden Subgroup Problem Kan Liftings Homotopy Kan Extensions and Liftings 5 Plans ÅÓÙÒØaiÒ BakeÖ Quantum Kan Extensions — 17 July 2012 8/34 Background Right Kan Extensions Left Kan Extensions Theory Plans Products: Definition A product of sets X and Y is a triple (X Y ,π0,π1) with π π × X Y 0 X and X Y 1 Y and which enjoys the property that if × −→ × −→ (u, v) (Z, u, v) is any such triple, then there is a unique Z X Y for −→ × which the following commutes: π π X o 0 X Y 1 / Y bFF ×O x< FF xx FF (u,v) xx u FF xx v F xx Z If X = 0,..., m 1 and Y = 0,..., n 1 , then { − } { − } X Y = 0,..., (m n) 1 π (k)= m/n and π (k)= k mod n × { × − } 0 ⌊ ⌋ 1 A product of ordinals is given by multiplication. Vedral, Barenco and Ekert. Quantum Networks for Elementary Arithmetic Operations. 1996 Multiplication mod N via repeated conditional addition ÅÓÙÒØaiÒ BakeÖ Quantum Kan Extensions — 17 July 2012 9/34 Background Right Kan Extensions Left Kan Extensions Theory Plans Products as Kan Extensions Products are a particular kind of right Kan extension: ! 0, 1 / ) { }H 1 {∗} H , π HH π 0 HH ( = (X ,Y ) HH X ×Y H# ⇐ | Sets Quantum multiplication computes a right Kan extension. Conversely, all right Kan extensions can be computed from products and equalizers. ÅÓÙÒØaiÒ BakeÖ Quantum Kan Extensions — 17 July 2012 10/34 Background Right Kan Extensions Left Kan Extensions Theory Plans Pullbacks: Definition f g A pullback of functions X Z and Y Z consists of a set X Z Y π0 → → π1 × and functions X z Y X and X z Y Y for which × −→ × −→ f π = g π and enjoys the property that if f u = g v, then ◦ 0 ◦ 1 ◦ ◦ there is a unique ϕ for which π ϕ = u and π ϕ = v. 0 ◦ 1 ◦ T v ϕ % π1 ! X Z Y / Y u × π0 g ( X / Z f A pullback of functions is given by: X Z Y = (x, y) X Y f (x)= g(y) × ∈ × | The pullback is the set of all claws of f and g. ÅÓÙÒØaiÒ BakeÖ Quantum Kan Extensions — 17 July 2012 11/34 Background Right Kan Extensions Left Kan Extensions Theory Plans Quantum Algorithms for Pullbacks Classical complexity: O(N log X ) (N = max X , Y ) to find all claws { } O(X log X ) comparisons to sort the values f (x) For each y, O(log X ) comparisons to search for x with f (x)= g(y) Buhrman, D¨urr, Heiligman, Høyer, Magniez, Santha and de Wolf. Quantum Algorithms for Element Distinctness. 2005. O X 1/2Y 1/4 log X comparisons to (with high probability) find a claw (if X Y X 2) and O(Y log X ) if Y > X 2 ≤ ≤ Theorem: Quantum computers cannot improve upon classical (probabilistic) complexity of exact pullback calculations. Assume we have a quantum algorithm that calculates pullbacks. Given f : X → Y , form the pullback P of f with itself. P − X counts the number of collisions (i.e., remove the diagonal). Consequently, an efficient pullback algorithm gives an an efficient algorithm for exactly counting the number of collisions. This contradicts Theorem 6.1 of the reference cited above. What about approximate quantum counting? ÅÓÙÒØaiÒ BakeÖ Quantum Kan Extensions — 17 July 2012 12/34 Background Right Kan Extensions Left Kan Extensions Theory Plans Equalizers: Definition f g An equalizer of functions X Y and X Y is a pair (E, e) for e −→ −→ which E X and f e = g e and which enjoys the property that, −→ ◦ ◦ ϕ if (E ′, e′) is any such pair, then there is a unique E ′ E for which −→ e ϕ = e′. E ′ ◦ G ′ ϕ GGe GG f G# ) E e / X 5 Y g Equalizer formulas: E = x X f (x)= g(x) and e(x)= x. { ∈ | } An equalizer is the collection of all “needles” in X where 1 if f (x)= g(x) χ(x)= ( 0 otherwise. ÅÓÙÒØaiÒ BakeÖ Quantum Kan Extensions — 17 July 2012 13/34 Background Right Kan Extensions Left Kan Extensions Theory Plans Quantum Algorithms for Equalizers Classical complexity of finding (E, e) is O(X ). Theorem: Quantum computers can not improve upon the classical complexity of exact equalizer calculations. Algorithms for equalizers and products give an algorithm for pullbacks. π X 1 jjj4 QQQf e jjj QQQ E / X Y ( Z TT mm6 × π TTT mmgm 2 T* Y m Approximate quantum algorithm: Brassard, Høyer, Mosca and Tapp.
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