Diagrammatic Methods for the Specification and Verification of Quantum Algorithms

Transcription

1 Diagrammatic Methods or the Speciication and Veriication o Quantum lgorithms William Zeng Quantum Group Department o Computer Science University o Oxord Quantum Programming and Circuits Workshop IQC, University o Waterloo June,

2 Introduction Problem: What are appropriate abstractions or describing quantum algorithms? Green et al. arxiv Wecker & Svore arxiv:

3 Introduction Problem: What are appropriate abstractions or describing quantum algorithms? 1 S {0, 1} Quantum Circuits S 1 2 σ Green et al. arxiv Wecker & Svore arxiv:

4 Introduction Problem: What are appropriate abstractions or describing quantum algorithms? 1 S {0, 1} Quantum Circuits S 1 2 σ Green et al. arxiv Wecker & Svore arxiv:

5 Introduction Quantum Inormation FHilb represented by Quantum Circuits Selinger arxiv

6 Introduction bstract Process Theories -SMC -compact categories represented by Categorical Diagrams generalize to Quantum Inormation FHilb represented by Quantum Circuits Selinger arxiv

7 Introduction bstract Process Theories -SMC -compact categories represented by Categorical Diagrams generalize to Quantum Inormation FHilb represented by Quantum Circuits Selinger arxiv

8 Introduction bstract Process Theories -SMC -compact categories represented by Categorical Diagrams generalize to Quantum Inormation FHilb represented by Quantum Circuits Selinger arxiv

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

10 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

11 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 o other results. algorithms locality oundations Outlook.

12 Quantum circuits 1.0 category C is { a set o systems, B Ob(C) a set o processes : B rr(c)

13 Quantum circuits 1.0 category C is { a set o systems, B Ob(C) a set o processes : B rr(c) B C g : B := g := B id :=

14 Quantum circuits 1.0 category C is { a set o systems, B Ob(C) a set o processes : B rr(c) B C g : B := g := B id := These are sequential processes.

15 The ramework monoidal category C has { cat. tensor ( ) : C C C a unit object I Ob(C)

16 The ramework monoidal category C has { cat. tensor ( ) : C C C a unit object I Ob(C) B D B D g := g = g id I := C C

17 The ramework monoidal category C has { cat. tensor ( ) : C C C a unit object I Ob(C) B D B D g := g = g id I := C C These are parallel processes.

18 Sym. Mon. Cats. & quantum circuits category B B C g B D monoidal category g := g id I := states ψ := symmetric monoidal categories B C ψ B

19 Sym. Mon. Cats. & quantum circuits category monoidal category B g := states ψ := symmetric monoidal categories B B C g B g D C ψ B id I := Quantum Computation FHilb: Sym. Mon. Cat. Ob(FHilb) =.d. Hilbert Spaces rr(fhilb) = linear maps is the tensor product I = C States are ψ : C H bramsky & Coecke arxiv

20 Sym. Mon. Cats. & quantum circuits category B B C g B D monoidal category g := g id I := states ψ := symmetric monoidal categories FHilb : Sym. Mon. Cat. Ob(FHilb) =.d. Hilbert Spaces rr(fhilb) = linear maps B C ψ B

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

22 The dagger dagger unctor : C C s.t. ( ) = (1) (g ) = g (2) id = id H (3) FHilb is a dagger category with the usual adjoint. bramsky & Coecke arxiv

23 The dagger dagger unctor : C C B := B B bramsky & Coecke arxiv

24 The dagger dagger unctor : C C Unitarity: B := = = B B bramsky & Coecke arxiv

25 The dagger dagger unctor : C C B := On states: ψ = ψ B B bramsky & Coecke arxiv

26 The dagger dagger unctor : C C B := On states: ψ = ψ B B φ ψ = φ ψ = φ ψ This is a scalar φ ψ : C C or I I in general and admits a generalized Born rule. bramsky & Coecke arxiv

27 Bases -special Frobenius algebra (,, ) obeys: = = = = = = =

28 Bases Given a inite set S, we use the ollowing diagrams to represent the copying and deleting unctions: S Δ S S S ɛ 1

29 Bases Given a inite set S, we use the ollowing diagrams to represent the copying and deleting unctions: S Δ S S S ɛ C s s s s 1 We treat these as linear maps acting on a ree vector space, whose basis is S.

30 Bases Given a inite set S, we use the ollowing diagrams to represent the copying and deleting unctions: S Δ S S S ɛ C s s s s 1 We treat these as linear maps acting on a ree vector space, whose basis is S. s t δ s,t s 1 s s

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

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

33 Complementarity [Coecke & Duncan ]: Two -SCF s on the same object are complementary when: d() =

34 Complementarity [Coecke & Duncan ]: Two -SCF s on the same object are complementary when: d() = This is the Hop law. Two complementary -SCF s that also orm a bialgebra are called strongly complementary.

35 Strongly Complementary Bases [Kissinger et al ]: Strongly complementary observables in FHilb are characterized by belian groups. Given a inite group G, its multiplication is: m G G m G

36 Strongly Complementary Bases [Kissinger et al ]: Strongly complementary observables in FHilb are characterized by belian groups. Given a inite group G, its multiplication is: m G G m G We linearize this to obtain the group algebra multiplication.

37 Strongly Complementary Bases [Kissinger et al ]: Strongly complementary observables in FHilb are characterized by belian groups. Given a inite group G, its multiplication is: m G G m G We linearize this to obtain the group algebra multiplication. one-dimensional representation G ρ C is: ρ m = ρ ρ Vicary arxiv It is copied by the multiplication vertex.

38 Strongly Complementary Bases [Kissinger et al ]: Strongly complementary observables in FHilb are characterized by belian groups. Given a inite group G, its multiplication is: m G G m G We linearize this to obtain the group algebra multiplication. one-dimensional representation G ρ C is: m = ρ ρ ρ Vicary arxiv The adjoint C ρ G is also copied on the lower legs.

39 Strongly Complementary Bases [Kissinger et al ]: Strongly complementary observables in FHilb are characterized by belian groups. [Gogioso & WZ]: Pairs o strongly complementary observables correspond to Fourier transorms between their bases.*

40 Unitary Oracles From these can construct the internal structure o oracles: x (x) y Oracle x y

41 Unitary Oracles From these can construct the internal structure o oracles: x (x) y Oracle x y [WZ & Vicary ]: For to map between bases is a sel-conjugate comonoid homomorphism. Oracles with this abstract structure are unitary in general.

42 Ex 1. The Deutsch-Jozsa lgorithm Blackbox unction : {0, 1} N {0, 1} is balanced when it takes each possible value the same number o times

43 Ex 1. The Deutsch-Jozsa lgorithm Blackbox unction : {0, 1} N {0, 1} is balanced when it takes each possible value the same number o times Deinition (The Deutsch-Jozsa problem) Given a blackbox unction promised to be either constant or balanced, identiy which. Classically we require at most 2 N queries o The quantum algorithm only requires a single query.

44 Ex 1. The Deutsch-Jozsa lgorithm Blackbox unction : {0, 1} N {0, 1} is balanced when it takes each possible value the same number o times Let σ be non-trivial irrep. o Z 2 i.e. σ(0) = 1, σ(1) = 1. balanced: σ = 0 constant: = x Vicary arxiv

45 Ex 1. The Deutsch-Jozsa lgorithm 1 S {0, 1} Oracle 1 S 1 2 σ Vicary arxiv

46 Ex 1. The Deutsch-Jozsa lgorithm We can use our higher level description to decompose the algorithm: {0, 1} 1 S m 1 S 1 2 σ Vicary arxiv

47 Ex 1. The Deutsch-Jozsa lgorithm We can use our higher level description to decompose the algorithm: 1 S {0, 1} m 1 S 1 σ 2 Representation (1, 1) o Z 2 Vicary arxiv

48 Ex 1. The Deutsch-Jozsa lgorithm We can use our higher level description to decompose the algorithm: 1 S {0, 1} m Linear maps rom set basis 1 S 1 σ 2 Representation (1, 1) o Z 2 Vicary arxiv

49 Ex 1. The Deutsch-Jozsa lgorithm We can use our higher level description to decompose the algorithm: 1 S {0, 1} m Function : S {0, 1} Linear maps rom set basis 1 S 1 σ 2 Representation (1, 1) o Z 2 Vicary arxiv

50 Ex 1. The Deutsch-Jozsa lgorithm We can use our higher level description to decompose the algorithm: 1 S {0, 1} Multiplication operation on Z 2 m Function : S {0, 1} Linear maps rom set basis 1 S 1 σ 2 Representation (1, 1) o Z 2 Vicary arxiv

51 Ex 1. The Deutsch-Jozsa lgorithm We can use our higher level description to decompose the algorithm: {0, 1} 1 S Projection onto s s (measure X to be 0) m Multiplication operation on Z 2 Function : S {0, 1} Linear maps rom set basis 1 S 1 σ 2 Representation (1, 1) o Z 2 Vicary arxiv

52 Ex 1. The Deutsch-Jozsa lgorithm Diagrammatic moves allow us to veriy the algorithm in generality: {0, 1} 1 S 1 S 1 2 σ Vicary arxiv

53 Ex 1. The Deutsch-Jozsa lgorithm Diagrammatic moves allow us to veriy the algorithm in generality: 1 S {0, 1} Slide up σ 1 2 σ 1 S Vicary arxiv

54 Ex 1. The Deutsch-Jozsa lgorithm Diagrammatic moves allow us to veriy the algorithm in generality: 1 S {0, 1} 1 2 σ Slide up σ Pull σ through the whitedot σ 1 S Vicary arxiv

55 Ex 1. The Deutsch-Jozsa lgorithm Diagrammatic moves allow us to veriy the algorithm in generality: 1 S Slide up σ σ Pull σ through the whitedot Neglect the right-side system 1 S Vicary arxiv

56 Ex 1. The Deutsch-Jozsa lgorithm Diagrammatic moves allow us to veriy the algorithm in generality: Slide up σ 1 S σ Pull σ through the whitedot Neglect the right-side system Topological contraction o blackdot Vicary arxiv

57 Ex 1. The Deutsch-Jozsa lgorithm Gives the amplitude or the input state 1 s to be in the σ state S s at measurement. σ 1 S Vicary arxiv

58 Ex 1. The Deutsch-Jozsa lgorithm Gives the amplitude or the input state 1 s to be in the σ state S s at measurement. What i is balanced? 1 S σ σ = 0 so the system is never measured in σ. What i is constant? Then = x σ σ 1 S = x = ±1 Vicary arxiv So the system is always measured in σ. 1 S

59 Ex 1. Summary or Deutsch-Josza Veriy: bstractly veriy the algorithm

60 Ex 1. Summary or Deutsch-Josza Veriy: bstractly veriy the algorithm Generalize: bstract deinition or balanced generalizes [Høyer Phys. Rev. 59, ] and [Batty, Braunstein, Duncan ]. See [Vicary ].

61 Ex 1. Summary or Deutsch-Josza Veriy: bstractly veriy the algorithm Generalize: bstract deinition or balanced generalizes [Høyer Phys. Rev. 59, ] and [Batty, Braunstein, Duncan ]. See [Vicary ]. The algorithm can be executed with complementary rather than strongly complementary observables

62 Ex 2. The GROUPHOMID lgorithm Given inite groups G and where is abelian, and a blackbox unction : G promised to be a group homomorphism, identiy. Case: Let be a cyclic group Z n. σ Measure the let system G G pply a unitary map 1 G ρ Prepare initial states WZ & Vicary arxiv

63 Ex 2. The GROUPHOMID lgorithm Given inite groups G and where is abelian, and a blackbox unction : G promised to be a group homomorphism, identiy. Case: Let be a cyclic group Z n. σ G G 1 G ρ WZ & Vicary arxiv

64 Ex 2. The GROUPHOMID lgorithm Given inite groups G and where is abelian, and a blackbox unction : G promised to be a group homomorphism, identiy. Case: Let be a cyclic group Z n. Pull ρ through whitedot σ ρ ρ G G 1 G WZ & Vicary arxiv

65 Ex 2. The GROUPHOMID lgorithm Given inite groups G and where is abelian, and a blackbox unction : G promised to be a group homomorphism, identiy. Case: Let be a cyclic group Z n. σ ρ G ρ Pull ρ through whitedot Contract set scalars WZ & Vicary arxiv

66 Ex 2. The GROUPHOMID lgorithm Given inite groups G and where is abelian, and a blackbox unction : G promised to be a group homomorphism, identiy. Case: Let be a cyclic group Z n. σ G ρ Pull ρ through whitedot Contract set scalars Topological equivalence ρ WZ & Vicary arxiv

67 Ex 2. The GROUPHOMID lgorithm Given inite groups G and where is abelian, and a blackbox unction : G promised to be a group homomorphism, identiy. Case: Let be a cyclic group Z n. ρ is an irrep. o G. σ G ρ ρ WZ & Vicary arxiv

68 Ex 2. The GROUPHOMID lgorithm Given inite groups G and where is abelian, and a blackbox unction : G promised to be a group homomorphism, identiy. Case: Let be a cyclic group Z n. σ G ρ ρ is an irrep. o G. Choose ρ to be a aithul representation o. ρ WZ & Vicary arxiv

69 Ex 2. The GROUPHOMID lgorithm Given inite groups G and where is abelian, and a blackbox unction : G promised to be a group homomorphism, identiy. Case: Let be a cyclic group Z n. σ G ρ ρ is an irrep. o G. Choose ρ to be a aithul representation o. Then measuring ρ identiies (up to isomorphism) ρ WZ & Vicary arxiv

70 Ex 2. The GROUPHOMID lgorithm Given inite groups G and where is abelian, and a blackbox unction : G promised to be a group homomorphism, identiy. Case: Let be a cyclic group Z n. σ ρ G ρ ρ is an irrep. o G. Choose ρ to be a aithul representation o. Then measuring ρ identiies (up to isomorphism) One-dimensional representations are isomorphic only i they are equal. WZ & Vicary arxiv

71 Ex 2. The GROUPHOMID lgorithm The General Case: Homomorphism : G We generalize with proo by induction via the Structure Theorem. = Z p1... Z pk [WZ & Vicary ] Given types, the quantum algorithm can identiy a group homomorphism in k oracle queries.

72 Ex 2. The GROUPHOMID lgorithm The General Case: Homomorphism : G We generalize with proo by induction via the Structure Theorem. = Z p1... Z pk [WZ & Vicary ] Given types, the quantum algorithm can identiy a group homomorphism in k oracle queries. Note that the quantum algorithm depends on the structure o while a classical algorithm will depend on the structure o G. Theorem [WZ] For large G this algorithm makes a quantum optimal number o queries, while classical algorithms are lower bounded by log G.

73 Quantum algorithms: old, generalized and new {0, 1} {0, 1} (n) S 1 S s n G ρ D 1 S 1 2 σ 1 S 1 2 σ 1 S Deutsch-Jozsa Single-shot Grover Hidden subgroup Vicary arxiv WZ & Vicary arxiv

74 Other results utomated graphical reasoning: quantomatic.github.io Kissinger arxiv: Dixon et al. arxiv

75 Other results utomated graphical reasoning: quantomatic.github.io [Coecke & bramsky ] Teleportation

76 Other results utomated graphical reasoning: quantomatic.github.io [Coecke & bramsky ] Teleportation [Zamdzhiev 2012, WZ & Gogioso arxiv tmrw] Quantum Secret Sharing [Cohn-Gordon 2012] Quantum Bit Commitment a N + N secret w i n d o w o a tta ck

77 Other results utomated graphical reasoning: quantomatic.github.io [Coecke & bramsky ] Teleportation [Zamdzhiev 2012, WZ & Gogioso arxiv tmrw] Quantum Secret Sharing [Cohn-Gordon 2012] Quantum Bit Commitment Connections to other theories in -SMC s: [WZ & Coecke] DisCo NLP. Mary N := l i k e s N S N := Mary N N l i k e s S N : N N := M N g = M g N N Mary l i k e s := Mary N N l i k e s S N ii := i N N Coecke et al Greenstette & Sadrzadeh arxiv

78 Other results utomated graphical reasoning: quantomatic.github.io [Coecke & bramsky ] Teleportation [Zamdzhiev 2012, WZ & Gogioso arxiv tmrw] Quantum Secret Sharing [Cohn-Gordon 2012] Quantum Bit Commitment Connections to other theories in -SMC s: [WZ & Coecke] DisCo NLP. [Kissinger et al , WZ & Gogioso arxiv tmrw] Foundations: Mermin Non-locality. [WZ ] Models o quantum algorithms in sets and relations. Outlook: Use this knowledge o quantum structure to better advantage in quantum programming.

79 Other results utomated graphical reasoning: quantomatic.github.io [Coecke & bramsky ] Teleportation [Zamdzhiev 2012, WZ & Gogioso arxiv tmrw] Quantum Secret Sharing [Cohn-Gordon 2012] Quantum Bit Commitment Connections to other theories in -SMC s: [WZ & Coecke] DisCo NLP. [Kissinger et al , WZ & Gogioso arxiv tmrw] Foundations: Mermin Non-locality. [WZ ] Models o quantum algorithms in sets and relations. Outlook: Use this knowledge o quantum structure to better advantage in quantum programming.