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

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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

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

ABǫ =⇒ 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

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, Dürr, 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. Quantum Amplitude Amplification and Estimation. 2000. Thm. 18. Approx Count with 1 <ε≤1 outputs E with |E − E|<εE 3 X e e with probability 2/3 and uses an expected number of evaluations of f in the order of pX /φ + pE (X − E)/φ where φ = ⌊εE⌋ + 1. Childs and Eisenberg. Quantum Algorithms for Subset Finding. 2003. e e Thm. 1. The query complexity of E-subset finding is O(X E/(E+1)). e

What if Approx Count miscounts? ÅÓÙÒØaiÒ BakeÖ Quantum Kan Extensions — 17 July 2012 14/34 Background Right Kan Extensions Left Kan Extensions Theory Plans Equalizers as Right Kan Extensions

Equalizers are a particular kind of right Kan extension:

! 0 1 / ) {∗} H ◦ e }} H , f } HH (e = } HH }E (f , g) HH }} H$ ⇐ ~}} Sets

Quantum search ﬁnds the members of the equalizer. Quantum counting computes the size of a right Kan extension. Conversely, all right Kan extensions can be computed from products and equalizers. Theorem: Quantum computers can not improve upon the classical

complexity of exact right Kan extension calculations. ÅÓÙÒØaiÒ BakeÖ Quantum Kan Extensions — 17 July 2012 15/34 Background Right Kan Extensions Left Kan Extensions Theory Plans Outline

5 Plans ÅÓÙÒØaiÒ BakeÖ Quantum Kan Extensions — 17 July 2012 16/34 Background Right Kan Extensions Left Kan Extensions Theory Plans Coproducts: Deﬁnition

A coproduct of sets X and Y is a triple (X Y , i0, i1) with i : X X Y and i : Y X Y and which enjoys the property 0 → 1 → ` that if (Z, u, v) is any such triple, then there is a unique function ` ` X Y Z for which the following commutes: → i i 0 / X Y o 1 ` X F Y FF xx FF u v xx u FF ` xxv FF xx F# ` {xx Z If X = 0,..., m 1 and Y = 0,..., n 1 , then { − } { − } X Y = 0,..., m + n 1 i (x)= x and i (y)= y + m { − } 0 1 a

That is, a coproduct of ordinals is a sum (with two inclusion maps). ÅÓÙÒØaiÒ BakeÖ Quantum Kan Extensions — 17 July 2012 17/34 Background Right Kan Extensions Left Kan Extensions Theory Plans Quantum Algorithms for Coproducts

Adaptation of classical algorithms Vedral, Barenco and Ekert. Quantum Networks for Elementary Arithmetic Operations. 1996 O(n) depth and O(n) ancillary qubits Draper, Kutin, Rains and Svore. A Logarithmic-Depth Quantum Carry-Lookahead Adder. 2008 O(log(n)) depth and O(n) ancillary qubits Cuccaro, Draper, Kutin and Moulton. A New Quantum Ripple-Carry Addition Circuit. 2008 O(n) depth and 1 ancillary qubit Approximate Fourier transform Draper. Addition on a Quantum Computer. 2000 Barenco, Ekert, Suominen and T¨orm¨a. Approximate Quantum Fourier

Transform and Decoherence. 2008 ÅÓÙÒØaiÒ BakeÖ Quantum Kan Extensions — 17 July 2012 18/34 Background Right Kan Extensions Left Kan Extensions Theory Plans Sample Coproduct Calculation

Input first set of the coproduct X = 31 Input second set of the coproduct Y = 25 Compute coproduct of X = 31 and Y = 25 Here is the input to the CDKM ripple-carry addition circuit: |011010111110> (1.0, 0.0) Here are the calculation stages: Stage 0 |011010111110> (1.0, 0.0) Stage 1 |101010111110> (1.0, 0.0) Stage 2 |100110111110> (1.0, 0.0) Stage 3 |100101111110> (1.0, 0.0) Stage 4 |100101001110> (1.0, 0.0) Stage 5 |100101000010> (1.0, 0.0) Stage 6 |100101000011> (1.0, 0.0) Stage 7 |100101001111> (1.0, 0.0) Stage 8 |100101111111> (1.0, 0.0) Stage 9 |100110111111> (1.0, 0.0) Stage 10 |101010111111> (1.0, 0.0) Stage 11 |001010111111> (1.0, 0.0) Here is the output from the algorithm: |001010111111> (1.0, 0.0)

The coproduct of X = 31 and Y = 25 is X+Y = 56. ÅÓÙÒØaiÒ BakeÖ Quantum Kan Extensions — 17 July 2012 19/34 Background Right Kan Extensions Left Kan Extensions Theory Plans Implementation Design

Java implementation using the Apache Commons library for complex number and matrix calculations.

XML Instance XML Instance Documents: Algorithms Documents:

, , F and X Parser Parser L and ǫ A B Inputs Outputs

JAXB Compiler

XML Schema: categories, functors, natural

transformations, Kan extensions ÅÓÙÒØaiÒ BakeÖ Quantum Kan Extensions — 17 July 2012 20/34 Background Right Kan Extensions Left Kan Extensions Theory Plans Coproducts as Kan Extensions

Coproducts are a particular kind of left Kan extension:

! 0, 1 / { }H ) {∗} H , i 1 HH (i 0 HH (X ,Y ) HH = X Y H# ⇒| Sets `

Quantum addition computes a left Kan extension. Conversely, all left Kan extensions can be computed from coproducts

and coequalizers. ÅÓÙÒØaiÒ BakeÖ Quantum Kan Extensions — 17 July 2012 21/34 Background Right Kan Extensions Left Kan Extensions Theory Plans Coequalizers: Deﬁnition

f g A coequalizer of functions X Y and X Y is a pair (Q, q) for q −→ −→ which Y Q and q f = q g and which enjoys the property that, −→ ◦ ◦ ϕ if (Q′, q′) is any such pair, then there is a unique Q Q′ for which −→ ϕ q = q′. ◦ Q′ q′ x; O xx f xx ϕ ) xx X 5 Y q / Q g Coequalizer formulas: Q = Y / and q(y) = [y]. ∼ where is the equivalence relation generated by all f (x) g(x).

∼ ∼ ÅÓÙÒØaiÒ BakeÖ Quantum Kan Extensions — 17 July 2012 22/34 Background Right Kan Extensions Left Kan Extensions Theory Plans Quantum Algorithm Concept for Coequalizers

Coequalizer example: f q 0 0 1 2 0, 2 1 { }{ } g Classical algorithms: Sedgewick: Quick-Union and Union-Find We seek a coequalizer quantum circuit that can compute the following and generalize to larger problems. 0= 00 | i 00 01 00 + 10 01 10 1= 01 | i| i| i | i| i| i | i √2 2= 10 | i f = 00 | i g = 10 | i

ancilae Coequalizer Circuit ÅÓÙÒØaiÒ BakeÖ Quantum Kan Extensions — 17 July 2012 23/34 Background Right Kan Extensions Left Kan Extensions Theory Plans Outline

5 Plans ÅÓÙÒØaiÒ BakeÖ Quantum Kan Extensions — 17 July 2012 24/34 Background Right Kan Extensions Left Kan Extensions Theory Plans Kan Extensions and the Hidden Subgroup Problem

Let G be a group, S be a set, and f : G S be a function. → If H G is a subgroup, then f is constant on the cosets of H if: for ≤ any x, y G, if x−1 y H, then ∈ ∗ ∈ f (x)= f (y).

The Hidden Subgroup Problem is: given a set map f : G S, → such that we are assured there exists an H for which f is constant on the cosets, ﬁnd H. f G / S C = CC {{ CC {{ CC {{ C! ! {{ G/H The challenge is to calculate and present H in an eﬀective manner

(e.g., by ﬁnding a set of generators). ÅÓÙÒØaiÒ BakeÖ Quantum Kan Extensions — 17 July 2012 25/34 Background Right Kan Extensions Left Kan Extensions Theory Plans The Hidden Subgroup Problem as a Kan Extension: I

Kan extensions can be used to solve the Hidden Subgroup Problem and, hence, to implement most quantum algorithms. In fact, we can do it as a left adjoint of a forgetful functor. Fix a group G and a set S.

Consider the following category CG,S : Objects of CG,S are triples (f , H, f ′) where H G and f : G S and f ′ : G/H S are ≤ ′ → → functions satisfying f = f πH (with πH : G G/H the natural ◦ → map g gH induced by H) as shown in the diagram below. 7→ f G / S NNN pp7 NNN ppp NNN ppp πH NN pp f ′ NN' ppp G/H Commutativity of this diagram is equivalent to the condition that f is

constant on the cosets of H. ÅÓÙÒØaiÒ BakeÖ Quantum Kan Extensions — 17 July 2012 26/34 Background Right Kan Extensions Left Kan Extensions Theory Plans The Hidden Subgroup Problem as a Kan Extension: II

′ In CG,S , the set of morphisms from an object (f1, H1, f1 ) to an object ′ (f2, H2, f2) is deﬁned by (H , H ) if f = f and H H ′ ′ { 2 1 } 1 2 2 ≤ 1 CG,S (f1, H1, f1 ), (f2, H2, f2 ) = ( φ otherwise.

In words, there is a unique morphism from H2 to H1. In pictures: f1 = f2 G O /7 S >O ′ o@ >OOO f oooÐ >> OOOπ 1 ooo ÐÐ >> OO oo Ð > OOO ooo ÐÐ >> ' o ÐÐ >> G/H1 ÐÐ π > Ð f ′ >> O ÐÐ 2 >> proj ÐÐ >> ÐÐ > ÐÐ

G/H2 ÅÓÙÒØaiÒ BakeÖ Quantum Kan Extensions — 17 July 2012 27/34 Background Right Kan Extensions Left Kan Extensions Theory Plans The Hidden Subgroup Problem as a Kan Extension: III

Consider DG,S , the discrete category whose objects are set maps f : G S. There are no non-trivial morphisms. → There is a forgetful functor

U : CG,S DG,S → which takes (f , H, f ′) to f . The left adjoint to U solves the Hidden Subgroup Problem. U + CG,S DG,S k ⊤ L Let f : G S. Then L(f ) = (f , H , f ′) where H is the largest → ∗ ∗ subgroup of G that satisﬁes the property. In other words, there may be many subgroups H of G such that f is

constant on the cosets G/H but L ﬁnds a maximal one. ÅÓÙÒØaiÒ BakeÖ Quantum Kan Extensions — 17 July 2012 28/34 Background Right Kan Extensions Left Kan Extensions Theory Plans The Hidden Subgroup Problem as a Kan Extension: IV

There is a need for a reduction of one problem to another. A surjective group homomorphism G G ′ induces inclusion functors → CG ′ S CG S and DG ′ S DG S . , → , , → , Similarly a function S S ′ induces functors → CG S CG S′ and DG S DG S′ . , → , , → , We should be able to use this to describe the notion of a “reduction” of one hidden subgroup problem to another and perhaps even the notion of a complete problem in the sense of NP-Complete (i.e., the

hardest Hidden Subgroup Problems or hardest quantum algorithm). ÅÓÙÒØaiÒ BakeÖ Quantum Kan Extensions — 17 July 2012 29/34 Background Right Kan Extensions Left Kan Extensions Theory Plans Variations on the Kan Extension Theme: Kan Liftings I

There are other types of constructions in the Kan extension family that seems worthy of more study. One of them is a Kan lifting. Witold Hurewicz is supposed to have said that all questions in algebraic topology can be formulated as either an extension or a lifting problem.

Extend Lift / / XX1 FY X1 Y 11 11 11 11 11 11 inc 1 f f 1 proj 11 11 11 11 + 1K 1 Ö Ö

Z Z ÅÓÙÒØaiÒ BakeÖ Quantum Kan Extensions — 17 July 2012 30/34 Background Right Kan Extensions Left Kan Extensions Theory Plans Variations on the Kan Extension Theme: Kan Liftings II

Let’s do the same for categories.

R R / / A _@@ ? CA @@ C @@ @@ F @@ X X @@ G @ @ B B A functor F : induces a functor B→A F ∗ : A B. C →C Left and right adjoints to F ∗ are left and right Kan extensions. A functor G : induces a functor C→B G : A A. ∗ C →B

Left and right adjoints to G∗ are left and right Kan liftings. ÅÓÙÒØaiÒ BakeÖ Quantum Kan Extensions — 17 July 2012 31/34 Background Right Kan Extensions Left Kan Extensions Theory Plans Variations on the Kan Extension Theme: Homotopy

Other variations that we are interested in are homotopy Kan extensions and liftings. Here, rather than just dealing with a category, we are dealing with a category and a type of relation where we are telling when certain morphisms are “essentially” the same. The ideas come from homotopy theory of topological spaces or simplical sets. We want to make certain constructions based on “homotopy types” of objects rather than objects. The ﬁnal result will also be a presentation of a “homotopy type”.

R R _4* _4* G7 @ A gX@@@@ C A@@@@@@ C @@@@@@ @@@@ @@@@ @@@@@@ F @@@@@ X X @@@@ G @@ @' ×w B B Special cases of these are homotopy limits and colimits. These are popular now in

string theory, motivic theory and other areas. ÅÓÙÒØaiÒ BakeÖ Quantum Kan Extensions — 17 July 2012 32/34 Background Right Kan Extensions Left Kan Extensions Theory Plans Outline

5 Plans ÅÓÙÒØaiÒ BakeÖ Quantum Kan Extensions — 17 July 2012 33/34 Background Right Kan Extensions Left Kan Extensions Theory Plans Plans

Conduct further analysis of the use of approximate counting to estimate right Kan extensions Develop, analyze and implement quantum circuits for computing coequalizers Integrate quantum coproducts and coequalizers into a left Kan extension algorithm Integrate quantum algorithms into our XML data ﬂow design Complete our report on Kan liftings and Grover’s algorithm

Write a report on homotopy Kan extensions and liftings ÅÓÙÒØaiÒ BakeÖ Quantum Kan Extensions — 17 July 2012 34/34 Background Right Kan Extensions Left Kan Extensions Theory Plans

Backup slides ÅÓÙÒØaiÒ BakeÖ Quantum Kan Extensions — 17 July 2012 35/34 Background Right Kan Extensions Left Kan Extensions Theory Plans Background

Kan extensions: powerful, abstract constructions Applications in algebra, topology, logic and computer science Hidden Subgroup Problems may be formulated as Kan extensions. Particular cases are implemented in widely-used software packages. (Mathematica, Magma, GAP, . . . ) Classical Kan extension algorithm generalized coset enumeration. Left Kan extensions: constructed from coproducts and coequalizers Coproducts: quantum addition Coequalizers: quantum random walk Right Kan extensions: constructed from products and equalizers Products: quantum multiplication

Equalizers: quantum amplitude estimation and L-subset ﬁnding ÅÓÙÒØaiÒ BakeÖ Quantum Kan Extensions — 17 July 2012 36/34 Background Right Kan Extensions Left Kan Extensions Theory Plans Quantum Algorithms for Products

Adaptation of classical algorithms Vedral, Barenco and Ekert. Quantum Networks for Elementary Arithmetic Operations. 1996 Multiplication mod N via repeated conditional addition Circuits for modular addition and exponentiation Quantum multiplication computes the product size.

Other arithmetic operations give the projections. ÅÓÙÒØaiÒ BakeÖ Quantum Kan Extensions — 17 July 2012 37/34 Background Right Kan Extensions Left Kan Extensions Theory Plans Pullbacks as Kan Extensions

Pullbacks are a particular kind of right Kan extension: 1 ! / {∗} ) | 0 2 π 1 | H ◦ | H , f | H 2 | H , π = | HH (π 1 | (f , g) HH || X ×Z Y HH ⇐ || H# }|| Sets

Quantum algorithms for ﬁnding claws, therefore, compute a particular kind of right Kan extension. Quantum computers can not, in general, improve upon classical

complexity of (exact) right Kan extension calculations. ÅÓÙÒØaiÒ BakeÖ Quantum Kan Extensions — 17 July 2012 38/34 Background Right Kan Extensions Left Kan Extensions Theory Plans Right Kan Extensions from Products and Equalizers

All right Kan extensions can be computed via products and equalizers. F

ABǫ = X ⇐ R C For each object B of , RB is the equalizer constructed as follows: B XA′ 3 O π(F (a)◦b,A′) π(b,a)

XA f XA′ RB / . 0 b Fa b∈BY(B,FA) g B→FAY→FA′

π(b,A) π(b,a) XA / ′

Xa XA ÅÓÙÒØaiÒ BakeÖ Quantum Kan Extensions — 17 July 2012 39/34 Background Right Kan Extensions Left Kan Extensions Theory Plans Coequalizers as Kan Extensions

Coequalizers are a particular kind of left Kan extension:

! 0 1 / q) {∗} H f , } H ◦ }} HH (q } HH = }Q (f , g) HH ⇒ }} H$ ~}} Sets

Conversely, all left Kan extensions can be computed from coproducts

and coequalizers. ÅÓÙÒØaiÒ BakeÖ Quantum Kan Extensions — 17 July 2012 40/34 Background Right Kan Extensions Left Kan Extensions Theory Plans Left Kan Extensions from Coproducts and Coequalizers

All left Kan extensions can be computed using coproducts and coequalizers. F ABǫ =⇒ X L C For each object B of , LB is the coequalizer constructed as follows: B XA i(A,b′◦F (a)) i(a,b′) ' XA f XA - / LB Fa b′ 1 FA→aFA′→B g b∈B(FA,B) O aO i(a,b′) i(A′,b′) XA / ′

Xa XA ÅÓÙÒØaiÒ BakeÖ Quantum Kan Extensions — 17 July 2012 41/34 Background Right Kan Extensions Left Kan Extensions Theory Plans Coset Enumeration as a Kan Extension

Let H G be groups construed as one-object categories. ≤ Let 1 : H Sets be the functor that maps the one object of H to a → one-point set. H inclusion / G D DD DD 1 DD L D" | Sets

A left Kan extension L of 1 along the inclusion functor maps: the one object of G to the set G/H of cosets g G to the function g ′H gg ′H.

∈ 7→ ÅÓÙÒØaiÒ BakeÖ Quantum Kan Extensions — 17 July 2012 42/34 Background Right Kan Extensions Left Kan Extensions Theory Plans The Carmody-Walters Kan Extension Algorithm

Inputs: Finite presentations of the categories and Deﬁnitions of X and F on objects andA generatorsB of Outputs: A Deﬁnition of L on objects and generating arrows of Natural transformation ǫ B Pseudo code: F initialize tables A B ǫ while there are undefined elements{ =⇒ define new elements X L fill in consequences Sets while there are coincidences { deal with coincidences fill in consequences } }

Quantum algorithms need not resemble their classical counterparts. ÅÓÙÒØaiÒ BakeÖ Quantum Kan Extensions — 17 July 2012 43/34 Background Right Kan Extensions Left Kan Extensions Theory Plans Calculation of Left Kan Extensions when = Sets C

L(B) = B(F (A), B) × X (A) ∼ L(β)[b, x] = [β ◦ b, x]  ∈A , AX   a ′ ′ A A′ (b ◦ F (a), x) ∼ (b , X (a)(x)) ) ) ), B ′ ), B (A (A F (a) B(F B(F F A F A′ ( ) ( ) ′ X (A ) x • ′ )(x) b b X (A) • X (a B L(B) [b, x] a) ′ ′◦F ( b b b b A A′ β L(β) ′ ) , B (A) B(F ′ ′ B L(B ) x [β ◦ b, x]

X (A) X (a) • ′ ) X (A) X (A ) ′ F (a b ◦ b β◦

A ÅÓÙÒØaiÒ BakeÖ Quantum Kan Extensions — 17 July 2012 44/34