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 Department o Computer Science, University o Oxord Quantum Physics and Logic, 2014

Unitary Oracles Oracles are common structures in algorithms. They are blackboxes with unknown internal structure.

Unitary Oracles Oracles are common structures in algorithms. They are blackboxes with unknown internal structure. Most known quantum algorithms are constructed using quantum oracles, the Deutsch-Josza algorithm, Shor s algorithm, Grover s algorithm... Physical realizations o oracles place conditions on their unknown structure. (Unitarity in the quantum case)

Unitary Oracles The traditional Deutsch-Joza circuit is: s {0, 1} F {0,1} N m Oracle U F {0,1} N 0 F {0,1} σ 1

Unitary Oracles Here is its abstract structure: {0, 1} {0, 1} s 1 S F {0,1} N m Oracle U Oracle F {0,1} N F {0,1} σ 0 1 1 S 1 2 σ

Unitary Oracles This is the oracle s internal structure: x (x) y Oracle x y

Unitary Oracles This is the oracle s internal structure: x (x) y Oracle x y Theorem Oracles with this abstract structure are unitary in general.

Categorical Quantum Inormation Deinition: A special -Frobenius algebra obeys: = = = = = = =

Categorical Quantum Inormation Deinition: A special -Frobenius algebra obeys: = = = = = = = This represents the abstract structure o an observable.

Complementary observables Deinition [Coecke & Duncan]: Two -Frobenius algebras on the same object are complementary when: d(a) =

Complementary observables Complementary observables in FHilb come rom inite abelian groups Copying Group multiplication :: g g g :: g 1 :: g 1 g 2 1 D g 1 g 2 :: 1 D 0

Classical Maps Deinition: A classical map : (A,, ) (B,, ) obeys: = =

Classical Maps Deinition: A classical map : (A,, ) (B,, ) obeys: = = These are sel-conjugate comonoid homomorphisms.

Unitarity Theorem Three -Frobenius algebras, (,, )

Unitarity Theorem Three -Frobenius algebras, (,, ) A pair are complementary ( and )

Unitarity Theorem Three -Frobenius algebras, (,, ) A pair are complementary ( and ) A classical map : (A,, ) (B,, ) Produce the unitary morphism: d(a)

Abstract proo o unitarity d(a)

Abstract proo o unitarity Frob. Hom. d(a)

Abstract proo o unitarity d(a)

Abstract proo o unitarity

Unitary Oracles We have deined (diagrammatically) an abstract structure required to make oracles physical.

Unitary Oracles We have deined (diagrammatically) an abstract structure required to make oracles physical. This lits the property o unitarity or quantum oracles to the more abstract setting o dagger monoidal categories. Can we take advantage o this abstract setting to gain new insights?

The group homomorphism identiication problem Deinition. (Group homomorphism identiication problem) Given inite groups G and A where A is abelian, and a blackbox unction : G A promised to be a group homomorphism, identiy. Group representations are ρ : G Mat(n) Abelian C ρ Non-abelian Mat(n) ρ G G

The group homomorphism identiication problem Deinition. (Group homomorphism identiication problem) Given inite groups G and A where A is abelian, and a blackbox unction : G A promised to be a group homomorphism, identiy. Graphical rules or group representations: Mat(n) Mat(n) Mat(n) Mat(n) ρ = ρ ρ ρ = G G G G G G

The group homomorphism identiication algorithm Case: Let A be a cyclic group Z n. σ Measure the let system G A G Apply a unitary map 1 G ρ Prepare initial states

The group homomorphism identiication algorithm σ G A G 1 G ρ

The group homomorphism identiication algorithm G σ ρ A G ρ 1 G

The group homomorphism identiication algorithm σ ρ A G ρ

The group homomorphism identiication algorithm σ G A ρ ρ

The group homomorphism identiication algorithm σ G A ρ ρ ρ is an irreducible representation o A.

The group homomorphism identiication algorithm σ G A ρ ρ ρ is an irreducible representation o A. Choose ρ to be a aithul representation o A.

The group homomorphism identiication algorithm σ ρ G A ρ ρ is an irreducible representation o A. Choose ρ to be a aithul representation o A. Then measuring ρ identiies (up to isomorphism)

The group homomorphism identiication algorithm σ ρ G A ρ ρ is an irreducible representation o A. Choose ρ to be a aithul representation o A. Then measuring ρ identiies (up to isomorphism) One-dimensional representations are isomorphic only i they are equal.

The group homomorphism identiication algorithm Homomorphism : G A We generalize with proo by induction via the Structure Theorem. A = Z p1... Z pk Can identiy the group homomorphism in k oracle queries. The naive classical solution requires a number o queries equal to the number o actors o G rather than A.

Comparison to the hidden subgroup algorithm Group ID Hidden Subgroup σ σ G A G X G G 1 G ρ 1 G

Comparison to the hidden subgroup algorithm Group ID Hidden Subgroup σ σ G A G X G G 1 G ρ 1 G = i ρ i

FinRel k [Baez, Erlebe, Fong 2014] Deinition The category FinRel k o linear relations is deined in the ollowing way, or any ield k: Objects are inite dimensional k-vector spaces

FinRel k [Baez, Erlebe, Fong 2014] Deinition The category FinRel k o linear relations is deined in the ollowing way, or any ield k: Objects are inite dimensional k-vector spaces A morphism : V W is a linear relation, deined as a subspace S V W Composition o linear relations : U V and g : V W is deined as the ollowing subspace o U W : {(u, w) v V with (u, v) S and (v, w) S g } This deines a linear subspace o U W.

The signal-low calculus FinRel k r Addition Zero Copying Deletion Multiplier : k k k : {0} k : k k k : k {0} r : k k (a, b, a + b) (0, 0) (a, a, a) (a, 0) (a, ra) r

The signal-low calculus FinRel k r Addition Zero Copying Deletion Multiplier : k k k : {0} k : k k k : k {0} r : k k (a, b, a + b) (0, 0) (a, a, a) (a, 0) (a, ra) r Resistor i v + ir r i v

Conclusions (arxiv: 1406.1278) Theorem A pair o complementary dagger-frobenius algebras, equipped with a classical map onto one o the algebras, produce a unitary morphism: d(a)

Conclusions (arxiv: 1406.1278) Theorem A pair o complementary dagger-frobenius algebras, equipped with a classical map onto one o the algebras, produce a unitary morphism: d(a) Abstract understanding o oracle in quantum computation Apply this to develop a new algorithm or the deterministic identiication o group homomorphisms into abelian groups. Find the same structure in the theory o signal-low networks.

The Non-Abelian Case σ σ ρ G = 1 G ρ ψ ρ ψ