Compositionality in Categorical Quantum Mechanics Ross Duncan Simon Perdrix Bob Coecke Kevin Dunne Niel de Beaudrap Compositionality x 3 • Plain old monoidal category theory: —— quantum computing in string diagrams • Rewriting and substitution: —— taking the syntax seriously • “Quantum theory” as a composite theory —— Lack’s composing PROPS An application: compiling for quantum architecture 1. Quantum theory as string diagrams How much quantum theory can be expressed as an internal language in some monoidal category? PAST / HEAVEN FUTURE / HELL F.D. Pure state QM • States : Hilbert spaces • Compound systems : Tensor product • Dynamics : Unitary maps • Non-degenerate measurements : O.N. Bases F.D. Pure state QM Ambient mathematical framework: †-symmetric monoidal categories • States : Hilbert spaces • Compound systems : Tensor product • Dynamics : Unitary maps • Non-degenerate measurements : O.N. Bases F.D. Pure state QM Ambient mathematical framework: †-symmetric monoidal categories • States : Hilbert spaces • Compound systems : Tensor product • Dynamics : Unitary maps • Non-degenerate measurements : O.N. Bases Choose some good generators and relations to capture this stuff Frobenius Algebras Theorem: in fdHilb orthonormal bases are in bijection with †-special commutative Frobenius algebras. ! !δ : A A Aµ: A A A ! ! ⌦ ⌦ ! ✏ : A I ⌘ : I A ! ! ! Via: ! δ :: a a a µ = δ† | ii!| ii⌦| ii ✏ :: a 1 ⌘ = ⌘† | ii! Coecke, Pavlovic, and Vicary, “A new description of orthogonal bases”, MSCS 23(3), 2013. arxiv:0810.0812 Frobenius Algebras Definition A -special commutativeFrobenius Frobenius Algebras algebra ( -SCFA) in ( , , I ) † Represent observables by †-special commutative† C ⌦ consists of: An objectFrobenius Aalgebras:, ! 2 C ! ! µ = , ⌘ = µ† = , ⌘† = satisfying... of directed equations (f2,f1) (f Õ ,fÕ ) commuting morphisms of T2 past those æ 1 2 f-1 f-2 of T1. The composite PROP T1; T2 has morphisms of the form n z m where f1 is an arrow of T1 and f2 of T2; its syntactic presentation is that of T1 + T2 with the additional equations of ⁄. Example 1.4. As a simple example we can view P as a PRO with a single generator c :2 2 quotiented by c2 = id and the usual hexagon diagrams. Let Frobenius Algebrasæ G be some any; we define G◊ to be the PRO with hom-sets G◊(n, n)= n G, and G◊(n, m)= if n = m. Composition is done component-wise in G.The ÿ ” r generators of G◊ are just the elements g :1 1 for each g G quotiented by æ œ the equations of G. We can define the composite P; G◊ via the distributive law: g1 g2 Definition ⁄ : = g2 g1 for each g andFrobeniusg in G. This yields the PROAlgebras – actually a -PROP – whose A -special commutative1 2 Frobenius algebra ( -SCFA)† in ( , , I ) morphisms n n are a permutation on n followed by an n-vector of elements of † Representæ observables by †-special commutative† C ⌦ G. It’s easy to see that this construction yields a functor P : Grp -PROP. Frobenius algebras: æ † consistsNotice of: An that objectPG is againA a groupoid,, and every morphism is unitary. ! 2 C Example 1.5.! A second cluster of examples, stolen shamelessly from [17], pro- vides the main! structures of interest of this paper. Let M denote the PROP of commutative monoids; it has two generators, µ :2 1 and ÷ :0: 1,whichwe µ = , ⌘æ = æ write graphically as and , subject to the equations: = µ† = , = = ⌘† = = (M) We can define the PROP of cocommutative comonoids as C = Mop. The gener- ators are ” :1 2 and ‘ :1 0; the equations are those of (M) but flipped æ æ satisfying...upside down. Bialgebras and Frobenius algebras combine a monoid and comonoid in different ways; both can be built using distributive laws between M and C. The PROP B of commutative bialgebras is constructed via a distributive law ⁄B : M; C C; M generated by the equations æ = = = = (B) where the dashed box represents the empty diagram. The PROP F of Frobenius algebras is also defined by distributive law, ⁄F : C; M M; C, given by the equations: æ = = = (F) = = = (F) Frobenius Algebrasæ C M M C ; ; , given by the equations: F F The PROP of Frobenius algebras is also defined by distributive law, : ⁄ where the dashed box represents the empty diagram. = = = = (B) æ M C C M B : ; ; generated by the equations Definition⁄ Frobenius Algebras B The PROP of commutative bialgebras is constructed via a distributive law M C fferent ways; both can be built using distributive laws between and . in di Represent observables by †-special commutative A -specialupside commutative down. Bialgebras and Frobenius Frobenius algebras algebra combine a monoid( -SCFA) and comonoid in ( , , I ) æ Frobeniusæ algebras: ators are :1 2 and :1 0; the equations are those of (M) but flipped ‘ † ”! † C ⌦ C M We can define the PROP of cocommutative comonoids as = . The gener- consists of: An object! A , op ! 2 C = = = (M) = µ = , ⌘ = write graphically as and , subject to the equations: æ æ commutative monoids; it has two generators, :2 1 and :0: 1,whichwe µ ÷ M vides the main structures of interest of this paper. Let denote the PROP of A second cluster of examples, stolen shamelessly from [17], pro- Example 1.5. µ† = , ⌘† = P Notice that is again a groupoid, and every morphism is unitary. G æ † P Grp PROP . It’s easy to see that this construction yields a functor : - . G æ morphisms are a permutation on followed by an -vector of elements of n n n n † 1 2 for each and in . This yields the PRO – actually a -PROP – whose g G satisfying... g : = ⁄ 2 1 g g 1 2 g g P ◊ the equations of . We can define the composite ; via the distributive law: G G æ œ ◊ generators of are just the elements :1 1 for each quotiented by G g g G r ÿ ” ◊ and ( )= if = . Composition is done component-wise in .The G n, m n m G n ◊ ◊ be some any; we define to be the PRO with hom-sets ( )= , G G G n, n G æ generator :2 2 quotiented by = id and the usual hexagon diagrams. Let c c 2 P Example 1.4. As a simple example we can view as a PRO with a single T T 1 2 + with the additional equations of . ⁄ T T 1 1 2 2 where is an arrow of and of ; its syntactic presentation is that of f f T T T 1 1 2 of . The composite PROP ; has morphisms of the form n z m - - 1 2 f f æ 1 2 T 2 1 2 Õ Õ of directed equations ( ) ( ) commuting morphisms of past those f ,f f ,f of directed equations (f2,f1) (f Õ ,fÕ ) commuting morphisms of T2 past those æ 1 2 f-1 f-2 of T1. The composite PROP T1; T2 has morphisms of the form n z m where f1 is an arrow of T1 and f2 of T2; its syntactic presentation is that of T1 + T2 with the additional equations of ⁄. Example 1.4. As a simple example we can view P as a PRO with a single generator c :2 2 quotiented by c2 = id and the usual hexagon diagrams. Let æ G be some any; we define G◊ to be the PRO with hom-sets G◊(n, n)= n G, and G◊(n, m)= if n = m. Composition is done component-wise in G.The ÿ ” r generators of G◊ are just the elements g :1 1 for each g G quotiented by æ œ the equations of G. We can define the composite P; G◊ via the distributive law: g1 g2 ⁄ : = g2 g1 for each g and g in G. This yields the PRO – actually a -PROP – whose 1 2 † morphisms n n are a permutation on n followed by an n-vector of elements of æ G. It’s easy to see that this construction yields a functor P : Grp -PROP. æ † Notice that PG is again a groupoid, and every morphism is unitary. Example 1.5. A second cluster of examples, stolen shamelessly from [17], pro- vides the main structures of interest of this paper. Let M denote the PROP of commutative monoids; it has two generators, µ :2 1 and ÷ :0: 1,whichwe æ æ write graphically as and , subject to the equations: = = = = (M) We can define the PROP of cocommutative comonoids as C = Mop. The gener- ators are ” :1 2 and ‘ :1 0; the equations are those of (M) but flipped æ æ upside down.Frobenius Bialgebras Algebras and Frobenius algebras combine a monoid and comonoid in different ways; both can be built using distributive laws between M and C. The PROP B of commutative bialgebras is constructed via a distributive law ⁄B : M; C C; M generated by the equations æ DefinitionFrobenius Algebras A -special commutative Frobenius algebra ( -SCFA) in ( , , I ) † Represent observables by †-special† commutativeC ⌦ = consists of: An object= A , = = (B) Frobenius algebras:2 C ! ! µ = , ⌘ = where the dashed! box represents the empty diagram. The PROP of Frobenius algebras is also defined by distributive law, ⁄ : F µ† = , ⌘† = F C; M M; C, given by the equations: æ satisfying... = = = (F) Call Z↵ the phases for this Frobenius algebra or, the -phases Call Z↵ the phases for this Frobenius algebra or, the -phases Observables and Phases Observables and Phases Phases , Abasis 0 1 A -SCFA Abasis 0 , 1 A -SCFA {| i | i } † { • Defn: a phase}{| isi unitary| i } map that commutes† with{ } the Frobenius algebra like this: Z↵ 0 = 0 Z↵ Z↵ 0 Z=↵ 0 Z Z | i | i = , = | i | i ↵ = , = ↵ Z 1 = 1 Z↵ Z ↵ Z↵ 1 = 1 Z↵ Z↵ ↵| i | i | i | i • Thm: the phases form an abelian group Example: Z-spin • The following define a Frobenius algebra on the qubit: 0 00 0 1 δ : ✏ : |1i 7! |11i |1i 7! 1 | i 7! | i | i 7! • Its group of phases is: 0 0 Z : ↵ |1i 7! e| i↵i 1 | i 7! | i Example: Z-spin 0 0 00 0 1 δ : ✏ : | |1i 7! |11i |1i 7! 1 | i 7! | i | i 7! 10 1 Z = ↵ 0 ei↵ | ✓ ◆ = = = = = = = – Frob.