Quantum computing with relations. Outline. Dusko Pavlovic. Quantum. programs. Quantum. categories. Classical interfaces.

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1 Outline Wat do quantum prorammers do? Cateories or quantum prorammin Kestrel Institute and Oxord University or cateorical quantum QI 009 Saarbrücken, Marc 009 ll tat in te cateory o Outline Wat do quantum prorammers do? Wat do quantum prorammers do? Cateories or quantum prorammin x Z m or cateorical quantum (x) Z n ll tat in te cateory o Wat do quantum prorammers do? Wat do quantum prorammers do? x x x x m C Zm x Z m (x) Z n U y (x) y y (x) y n C Zn

2 Wat do quantum prorammers do? Simon s aloritm z H m x z P x ( 1)z x x x H m z x P z ( 1)x z z : Z m Zn : x (x) : Z m+n Z m+n : x, y x, (x) y U : C Zm+n C Zm+n : x, y x, (x) y U Simon (H m id)u (H m id) 0, 0 ( 1) x z z, (x) y (x) y x,z Z m Simon s aloritm Sor s aloritm : Z m Zn : x (x) : Z m+n Z m+n : x, y x, (x) y U : C Zm+n C Zm+n : x, y x, (x) y : Z m+n q U : C Zm+n q : Z m q Zn q : x ax mod q Z m+n q : x, y x, a x + y mod q C Zm+n q : x, y x, a x + y mod q Simon (H m id)u (H m id) 0, 0 ( 1) x z z, (x) x,z Z m Sor (FT m id)u (FT m id) 0, 0 ( 1) x z z, (x) x,z Z m q...toindaiddensubroup measurement ind c suc tat (x + c) (x)...toindaiddensubroup measurement ind c suc tat a x+c a x mod q Hallren s aloritm sotware enineerin ;) : Z m Z n : x I x (raction ideal) : Z m+n Z m+n : x, y x, y (x) U : C Zm+n C Zm+n : x, y x, y (x) Hallren (FT m id)u (FT m id) d, d x,z Z m ( 1) x z z, (x) QUN T MES T...toindaiddensubroup measurement ind R suc tat (x + R) (x)

3 resources resources QUN T superposition entanlement QUN T superposition entanlement MES T MES T quantum prorammin unctional prorammin + superposition + entanlement Standard universes Outline FSet, FSet op, FFMod R... Wat do quantum prorammers do? QUN T FHilb, CPM(FHilb)... Cateories or quantum prorammin MES T FHilb, CPM(FHilb)... or cateorical quantum ll tat in te cateory o ormalisms ormalisms standard universe: Hilbert spaces standard universe: Hilbert spaces von Neumann ( 37): "I don t believe in Hilbert spaces"

4 ormalisms ormalisms standard universe: Hilbert spaces von Neumann ( 37): "I don t believe in Hilbert spaces" loic o Hilbert spaces: ortonormal lattices standard universe: Hilbert spaces von Neumann ( 37): "I don t believe in Hilbert spaces" loic o Hilbert spaces: ortonormal lattices no compound systems ormalisms Cateories in pictures: Objects standard universe: Hilbert spaces von Neumann ( 37): "I don t believe in Hilbert spaces" loic o Hilbert spaces: ortonormal lattices no compound systems structure o Hilbert spaces: -monoidal Cateories in pictures: Identities Cateories in pictures: Operators id

5 Cateories in pictures: Tensors Cateories in pictures: Composition C C C C Cateories in pictures: vectors and covectors Cateories in pictures: Symmetry b C b C b C b C c c a a I a a I Cateories in pictures: djoints Claim a c I a c ll details o te HSP aloritms can be speciied usin tis structure. C b C b

6 Formal concepts erived structure Over ormal vectors we deine: inner product Universe S spaces: S {,,...} : C() C() I ( ) (ψ, ϕ : I ) ϕ I ψ I operators: S(, ) {,,...} vectors: S() S(I, ) scalars: I S(I, I) erived structure Over ormal vectors we deine: erived structure Over ormal vectors we deine: inner product inner product : C() C() I ( ) (ψ, ϕ : I ) ϕ I ψ I : C() C() I ( ) (ψ, ϕ : I ) ϕ I ψ I partial inner product partial inner product : C() C( ) C() ( ) ϕ (ψ : I, ϕ : I ) I ψ : C() C( ) C() ( ) ϕ (ψ : I, ϕ : I ) I ψ entanled vectors η C( ), suctat ϕ C() η ϕ ϕ erived structure Usin entanled vectors η : I and η : I teir adjoints η : I and η : I erived structure Usin entanled vectors η : I and η : I teir adjoints η : I and η : I deine or every : te dual : η η

7 erived structure Usin entanled vectors η : I and η : I teir adjoints η : I and η : I deine or every : Outline Wat do quantum prorammers do? Cateories or quantum prorammin te dual : η η or cateorical quantum te conjuate : ll tat in te cateory o data data Question How do we reconize classical data in a quantum world? Idea data can be copied and deleted. data cannot be copied or deleted. data data Idea data can be copied and deleted. data cannot be copied or deleted. : Z m Zn : x (x) : Z m+n Z m+n : x, y x, (x) y Question ut ow do we really tell tem apart in a proram? U : C Zm+n C Zm+n : x, y x, (x) y Simon (H m id)u (H m id) 0, 0 ( 1) x z z, (x) x,z Z m

8 data djoinin variables to alebras nswer data are wat is denoted by te variables. Z ad x Z[x] a x S a djoinin variables to djoinin variables to S S[x] 1 x ad x F C F a 1 a F b C c a b C id x I c r x a x I I I Variable abstraction in structure S(, ) S[x](, ) κx. κx. κx. id I

9 structure: comonoid structure: Frobenius alebra Sel-dual structure: Frobenius alebra structure: Frobenius alebra...orequivalently...orstillequivalently structures are bases structures are bases Teorem (Coecke, P & Vicary) structures over Hilbert spaces and linear maps are in a bijective correspondence wit te bases. Teorem structures over sets and are disjoint unions o abelian roups.

10 Outline Wat quantum prorammers do now? Wat do quantum prorammers do? (x) x FSet [x :m](n) Cateories or quantum prorammin (x, y) x, y (x) FSet [x, y :m + n](m + n) U x, y (x,y) FHilb [ x, y : (m+n)]( (m+n)) or cateorical quantum were C and ( ) : FSet [x, y :m + n] FHilb [ x, y : (m+n)] ll tat in te cateory o Wat can tey do in Rel? Wat can tey do in Rel? Te role o can be played by Ξ Z Z,were Te role o can be played by Ξ Z Z,were Ξ {00, 01, 10, 11} (i0) { i0, i0, i1, i1 } (i1) { i0, i1, i1, i0 } {00, 10} (Ξ) {β 0 {00, 01}, β 1 {10, 11}} Ξ n Z n {ij 0 i, j n 1} n (ij) { ik, il j k + l} {i0 0 i n 1} (Ξ n ) {β i {ij} 0 i, j n 1} Qubits in Rel Qubits in Rel Te point is tat Ξ n supports a simple Fourrier transorm into te complementary basis Te point is tat Ξ n supports a simple Fourrier transorm into te complementary basis FT n : Ξ n Ξ n ij ji FT n : Ξ n Ξ n ij ji Use H FT to transorm m-bitstrins by H m : Ξ m Ξ m or Simon s aloritm.

Abstract structure of unitary oracles for quantum algorithms

Abstract structure of unitary oracles for quantum algorithms Abstract structure o unitary oracles or quantum algorithms William Zeng 1 Jamie Vicary 2 1 Department o Computer Science University o Oxord 2 Centre or Quantum Technologies, University o Singapore and