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Diagrammatic Methods for the Specification and Verification of Quantum Algorithms William Zeng Quantum Group Department of Computer Science University of Oxford Quantum Programming and Circuits Workshop IQC, University of Waterloo June, 2015 http://willzeng.com/shared/qcircuitworkshop.pdf

Introduction ◮ Problem: How can we best exploit the structure of quantum mechanics? Green et al. arXiv 1304.3390 Wecker & Svore arXiv:1402.4467

Introduction ◮ Problem: How can we best exploit the structure of quantum mechanics? { 0 , 1 } 1 √ | S | ◮ Quantum Circuits 2.0 f 1 √ 1 √ σ | S | 2 Green et al. arXiv 1304.3390 Wecker & Svore arXiv:1402.4467

Introduction ◮ Problem: How can we best exploit the structure of quantum mechanics? { 0 , 1 } 1 √ | S | ◮ Quantum Circuits 2.0 ⇒ f 1 √ 1 √ σ | S | 2 Green et al. arXiv 1304.3390 Wecker & Svore arXiv:1402.4467

Introduction Quantum Information Quantum Circuits represented by FHilb Selinger arXiv 0908.3347

Introduction Abstract Process Theories Categorical Diagrams † -SMC represented by † -compact categories generalize to Quantum Information Quantum Circuits represented by FHilb Selinger arXiv 0908.3347

Introduction Abstract Process Theories Categorical Diagrams † -SMC represented by † -compact categories generalize to Quantum Information Quantum Circuits represented by FHilb Selinger arXiv 0908.3347

Introduction Abstract Process Theories Categorical Diagrams † -SMC represented by † -compact categories generalize to Quantum Information Quantum Circuits represented by FHilb Selinger arXiv 0908.3347

Overview ◮ The Framework: Circuit Diagrams 2.0 ◮ bases ∙ copying/deleting ∙ groups/representations ∙ complementarity ∙ oracles

Overview ◮ The Framework: Circuit Diagrams 2.0 ◮ bases ∙ copying/deleting ∙ groups/representations ∙ complementarity ∙ oracles ◮ Example 1. Generalized Deutsch-Jozsa algorithm ◮ Example 2. The quantum GROUPHOMID algorithm

Overview ◮ The Framework: Circuit Diagrams 2.0 ◮ bases ∙ copying/deleting ∙ groups/representations ∙ complementarity ∙ oracles ◮ Example 1. Generalized Deutsch-Jozsa algorithm ◮ Example 2. The quantum GROUPHOMID algorithm ◮ Overview of other results. ◮ algorithms ∙ locality ∙ foundations ◮ Outlook.

Quantum circuits 1.0 � a set of systems A , B ∈ Ob ( C ) A category C is a set of processes f : A → B ∈ Arr ( C )

Quantum circuits 1.0 � a set of systems A , B ∈ Ob ( C ) A category C is a set of processes f : A → B ∈ Arr ( C ) B C A g f f : A → B := g ◦ f := B id A := f A A A

Quantum circuits 1.0 � a set of systems A , B ∈ Ob ( C ) A category C is a set of processes f : A → B ∈ Arr ( C ) B C A g f f : A → B := g ◦ f := B id A := f A A A These are sequential processes.

The framework � cat. tensor ( − ⊗ − ) : C × C → C A monoidal category C has a unit object I ∈ Ob ( C )

The framework � cat. tensor ( − ⊗ − ) : C × C → C A monoidal category C has a unit object I ∈ Ob ( C ) B D B D f g f f ⊗ g := = id I := g A C A C

The framework � cat. tensor ( − ⊗ − ) : C × C → C A monoidal category C has a unit object I ∈ Ob ( C ) B D B D f g f f ⊗ g := = id I := g A C A C These are parallel processes.

Sym. Mon. Cats. & quantum circuits B C A g f category B f A A A B D monoidal g f f ⊗ g := id I := category A C A | ψ � := states ψ B A symmetric monoidal categories A B

Sym. Mon. Cats. & quantum circuits B C A Quantum Computation g f category B ◮ FHilb : Sym. Mon. Cat. f A A A ◮ Ob( FHilb ) = f.d. Hilbert B D Spaces monoidal g f f ⊗ g := id I := category A C ◮ Arr( FHilb ) = linear maps A | ψ � := states ◮ ⊗ is the tensor product ψ B A symmetric ◮ I = C monoidal categories A B ◮ States are | ψ � : C → H Abramsky & Coecke arXiv 0808.1023

Sym. Mon. Cats. & quantum circuits B C A g f category B f A A A B D monoidal g f f ⊗ g := id I := category A C A | ψ � := states ψ B A symmetric monoidal categories A B FHilb : Sym. Mon. Cat. Ob( FHilb ) = f.d. Hilbert Spaces Arr( FHilb ) = linear maps

Sym. Mon. Cats. & quantum circuits B C A g / f category B U f | 1 � f A A A B D | 0 � H monoidal g f f ⊗ g := id I := category A C H 2 H 2 H 2 A | ψ � := states ψ f B A symmetric monoidal categories A B U FHilb : Sym. Mon. Cat. H Ob( FHilb ) = f.d. Hilbert Spaces Arr( FHilb ) = linear maps 1 0 H N

The dagger A dagger functor † : C → C s.t. f † � † � = f (1) ( g ◦ f ) † = f † ◦ g † (2) id † A = id H (3) FHilb is a dagger category with the usual adjoint. Abramsky & Coecke arXiv 0808.1023

The dagger A dagger functor † : C → C B A A f † f f �→ := A B B Abramsky & Coecke arXiv 0808.1023

The dagger A dagger functor † : C → C Unitarity: B A A f f = = f † f f �→ := f f A B B Abramsky & Coecke arXiv 0808.1023

The dagger A dagger functor † : C → C On states: †   ψ B A A =    ψ  f † f f �→ := A B B Abramsky & Coecke arXiv 0808.1023

The dagger A dagger functor † : C → C On states: †   ψ B A A =    ψ  f † f f �→ := φ A B B | φ � ◦ � ψ | = � φ | ψ � = ψ This is a scalar � φ | ψ � : C → C or I → I in general and admits a generalized Born rule. Abramsky & Coecke arXiv 0808.1023

Bases A † -special Frobenius algebra ( A, , ) obeys: = = = = = = =

Bases Given a finite set S , we use the following diagrams to represent the ‘copying’ and ‘deleting’ functions: S ∆ S ǫ − → S × S − → 1

Bases Given a finite set S , we use the following diagrams to represent the ‘copying’ and ‘deleting’ functions: S ∆ S ǫ − → S ⊗ S − → C | s � �→ | s � ⊗ | s � | s � �→ 1 We treat these as linear maps acting on a free vector space, whose basis is S .

Bases Given a finite set S , we use the following diagrams to represent the ‘copying’ and ‘deleting’ functions: S ∆ S ǫ − → S ⊗ S − → C | s � �→ | s � ⊗ | s � | s � �→ 1 We treat these as linear maps acting on a free vector space, whose basis is S . | s � ⊗ | t � �→ δ s , t | s � 1 �→ � s | s �

Bases and Topology These linear maps form a † -special commutative Frobenius algebra. Their composites are determined entirely by their connectivity, e.g.: =

Bases and Topology These linear maps form a † -special commutative Frobenius algebra. Their composites are determined entirely by their connectivity, e.g.: = ◮ [Coecke et al. 0810.0812] † -(special) commutative Frobenius algebras on objects in FHilb are eqv. to orthogonal (orthonormal) bases. ◮ [Evans et al. 0909.4453] † -(special) commutative Frobenius algebras on objects in Rel are eqv. to groupoids.

Complementarity ◮ [Coecke & Duncan 0906.4725]: Two † -SCFA’s on the same object are complementary when: d(A) =

Complementarity ◮ [Coecke & Duncan 0906.4725]: Two † -SCFA’s on the same object are complementary when: d(A) = ◮ This is the Hopf law. Two complementary † -SCFA’s that also form a bialgebra are called strongly complementary.

Strongly Complementary Bases ◮ [Kissinger et al. 1203.4988]: Strongly complementary observables in FHilb are characterized by Abelian groups. ◮ Given a finite group G , its multiplication is: m m G × G − → G

Strongly Complementary Bases ◮ [Kissinger et al. 1203.4988]: Strongly complementary observables in FHilb are characterized by Abelian groups. ◮ Given a finite group G , its multiplication is: m m G ⊗ G − → G We linearize this to obtain the group algebra multiplication.

Strongly Complementary Bases ◮ [Kissinger et al. 1203.4988]: Strongly complementary observables in FHilb are characterized by Abelian groups. ◮ Given a finite group G , its multiplication is: m m G ⊗ G − → G We linearize this to obtain the group algebra multiplication. ρ ◮ A one-dimensional representation G − → C is: ρ = ρ ρ m It is copied by the multiplication vertex. Vicary arXiv 1209.3917

Strongly Complementary Bases ◮ [Kissinger et al. 1203.4988]: Strongly complementary observables in FHilb are characterized by Abelian groups. ◮ Given a finite group G , its multiplication is: m m G ⊗ G − → G We linearize this to obtain the group algebra multiplication. ρ ◮ A one-dimensional representation G − → C is: ρ = m ρ ρ ρ The adjoint C − → G is also copied on the lower legs. Vicary arXiv 1209.3917

Strongly Complementary Bases ◮ [Kissinger et al. 1203.4988]: Strongly complementary observables in FHilb are characterized by Abelian groups. ◮ [Gogioso & WZ]: Pairs of strongly complementary observables correspond to Fourier transforms between their bases.*

Unitary Oracles ◮ From these can construct the internal structure of oracles: | x � | f ( x ) ⊕ y � Oracle f | x � | y �