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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?

◮ Quantum Circuits 2.0

σ

{0, 1}

1

√

|S| 1

√

|S| 1 √ 2

f

Green et al. arXiv 1304.3390 Wecker & Svore arXiv:1402.4467

Introduction

◮ Problem: How can we best exploit the structure of quantum

mechanics?

◮ Quantum Circuits 2.0

σ

{0, 1}

1

√

|S| 1

√

|S| 1 √ 2

⇒

f

Green et al. arXiv 1304.3390 Wecker & Svore arXiv:1402.4467

Introduction

Quantum Information

FHilb

represented by

Quantum Circuits

Selinger arXiv 0908.3347

Introduction

Quantum Information

FHilb

represented by

Quantum Circuits Abstract Process Theories

†-SMC †-compact categories

represented by generalize to

Categorical Diagrams

Selinger arXiv 0908.3347

Introduction

Quantum Information

FHilb

represented by

Quantum Circuits Abstract Process Theories

†-SMC †-compact categories

represented by generalize to

Categorical Diagrams

Selinger arXiv 0908.3347

Introduction

Quantum Information

FHilb

represented by

Quantum Circuits Abstract Process Theories

†-SMC †-compact categories

represented by generalize to

Categorical Diagrams

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

Quantum circuits 1.0

A category C is a set of systems A, B ∈ Ob(C) a set of processes f : A → B ∈ Arr(C) f : A → B := B f A g ◦ f := C g B f A idA := A A

Quantum circuits 1.0

A category C is a set of systems A, B ∈ Ob(C) a set of processes f : A → B ∈ Arr(C) f : A → B := B f A g ◦ f := C g B f A idA := A A

These are sequential processes.

The framework

A monoidal category C has

• cat. tensor (− ⊗ −) : C × C → C

a unit object I ∈ Ob(C)

The framework

A monoidal category C has

• cat. tensor (− ⊗ −) : C × C → C

a unit object I ∈ Ob(C)

f ⊗ g := B f A D g C = B f A D g C idI :=

The framework

A monoidal category C has

• cat. tensor (− ⊗ −) : C × C → C

a unit object I ∈ Ob(C)

f ⊗ g := B f A D g C = B f A D g C idI :=

These are parallel processes.

• Sym. Mon. Cats. & quantum circuits

category B f A C g B f A A A monoidal category f ⊗ g := B f A D g C idI := states |ψ := A ψ symmetric monoidal categories B A B A

• Sym. Mon. Cats. & quantum circuits

category B f A C g B f A A A monoidal category f ⊗ g := B f A D g C idI := states |ψ := A ψ symmetric monoidal categories B A B A

Quantum Computation

◮ FHilb: Sym. Mon. Cat. ◮ Ob(FHilb) = f.d. Hilbert

Spaces

◮ Arr(FHilb) = linear maps ◮ ⊗ is the tensor product ◮ I = C ◮ States are |ψ : C → H

Abramsky & Coecke arXiv 0808.1023

• Sym. Mon. Cats. & quantum circuits

category B f A C g B f A A A monoidal category f ⊗ g := B f A D g C idI := states |ψ := A ψ symmetric monoidal categories B A B A

FHilb : Sym. Mon. Cat. Ob(FHilb) = f.d. Hilbert Spaces Arr(FHilb) = linear maps

• Sym. Mon. Cats. & quantum circuits

category B f A C g B f A A A monoidal category f ⊗ g := B f A D g C idI := states |ψ := A ψ symmetric monoidal categories B A B A

FHilb : Sym. Mon. Cat. Ob(FHilb) = f.d. Hilbert Spaces Arr(FHilb) = linear maps

/ U f |1 |0 H

1

H HN H2 H2 H2 f U

The dagger

A dagger functor † : C → C s.t.

• f ††

= f (1) (g ◦ f)† = f † ◦ g† (2) id†

A = idH

(3) FHilb is a dagger category with the usual adjoint.

Abramsky & Coecke arXiv 0808.1023

The dagger

A dagger functor † : C → C A f B → B f A := B f † A

Abramsky & Coecke arXiv 0808.1023

The dagger

A dagger functor † : C → C A f B → B f A := B f † A Unitarity: f f = f f =

Abramsky & Coecke arXiv 0808.1023

The dagger

A dagger functor † : C → C A f B → B f A := B f † A On states:    ψ   

†

= ψ

Abramsky & Coecke arXiv 0808.1023

The dagger

A dagger functor † : C → C A f B → B f A := B f † A On states:    ψ   

†

= ψ |φ ◦ ψ| = φ|ψ = ψ φ 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 × S S ǫ − → 1

Bases

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

• n objects in FHilb are eqv. to orthogonal (orthonormal) bases.

◮ [Evans et al. 0909.4453] †-(special) commutative Frobenius algebras

• n 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

G × G

m

− → 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

G ⊗ G

m

− → 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

G ⊗ G

m

− → G We linearize this to obtain the group algebra multiplication.

◮ A one-dimensional representation G

ρ

− → C is:

ρ m

=

ρ ρ

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

G ⊗ G

m

− → G We linearize this to obtain the group algebra multiplication.

◮ A one-dimensional representation G

ρ

− → C is:

ρ m

=

ρ ρ

The adjoint C

ρ

− → 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 |x |y |f(x) ⊕ y Oracle f

Unitary Oracles

◮ From these can construct the internal structure of oracles:

|x |x |y |f(x) ⊕ y Oracle f

◮ [WZ & Vicary 1406.1278]: For f to map between bases is a

self-conjugate comonoid homomorphism. Oracles with this abstract structure are unitary in general.

Ex 1. The Deutsch-Jozsa Algorithm

◮ Blackbox function f : {0, 1}N → {0, 1} is balanced when it

takes each possible value the same number of times 00 01 10 11 1

Ex 1. The Deutsch-Jozsa Algorithm

◮ Blackbox function f : {0, 1}N → {0, 1} is balanced when it

takes each possible value the same number of times 00 01 10 11 1

Definition (The Deutsch-Jozsa problem)

Given a blackbox function f promised to be either constant or balanced, identify which.

◮ Classically we require at most 2N−1 + 1 queries of f ◮ The quantum algorithm only requires a single query.

Ex 1. The Deutsch-Jozsa Algorithm

◮ Blackbox function f : {0, 1}N → {0, 1} is balanced when it

takes each possible value the same number of times 00 01 10 11 1

◮ Let σ be non-trivial irrep. of Z2 i.e. σ(0) = 1, σ(1) = −1.

balanced:

f σ

= 0 constant:

f

=

x

Vicary arXiv 1209.3917

Ex 1. The Deutsch-Jozsa Algorithm

σ {0, 1} Oracle

1

√

|S| 1

√

|S| 1 √ 2

f

Vicary arXiv 1209.3917

Ex 1. The Deutsch-Jozsa Algorithm

We can use our higher level description to decompose the algorithm: σ† {0, 1}

1

√

|S| 1

√

|S| 1 √ 2

f

m

Vicary arXiv 1209.3917

Ex 1. The Deutsch-Jozsa Algorithm

We can use our higher level description to decompose the algorithm: σ† {0, 1}

1

√

|S| 1

√

|S| 1 √ 2

Representation (1, −1) of Z2 f

m

Vicary arXiv 1209.3917

Ex 1. The Deutsch-Jozsa Algorithm

We can use our higher level description to decompose the algorithm: σ† {0, 1}

1

√

|S| 1

√

|S| 1 √ 2

Representation (1, −1) of Z2 Linear maps from set basis f

m

Vicary arXiv 1209.3917

Ex 1. The Deutsch-Jozsa Algorithm

We can use our higher level description to decompose the algorithm: σ† {0, 1}

1

√

|S| 1

√

|S| 1 √ 2

Representation (1, −1) of Z2 Linear maps from set basis Function f : S → {0, 1} f

m

Vicary arXiv 1209.3917

Ex 1. The Deutsch-Jozsa Algorithm

We can use our higher level description to decompose the algorithm: σ† {0, 1}

1

√

|S| 1

√

|S| 1 √ 2

Representation (1, −1) of Z2 Linear maps from set basis Function f : S → {0, 1} Multiplication operation on Z2 f

m

Vicary arXiv 1209.3917

Ex 1. The Deutsch-Jozsa Algorithm

We can use our higher level description to decompose the algorithm: σ† {0, 1}

1

√

|S| 1

√

|S| 1 √ 2

Representation (1, −1) of Z2 Linear maps from set basis Function f : S → {0, 1} Multiplication operation on Z2 Projection onto

s |s (measure X to be 0)

f

m

Vicary arXiv 1209.3917

Ex 1. The Deutsch-Jozsa Algorithm

Diagrammatic moves allow us to verify the algorithm in generality: σ† {0, 1}

1

√

|S| 1

√

|S| 1 √ 2

f

Vicary arXiv 1209.3917

Ex 1. The Deutsch-Jozsa Algorithm

Diagrammatic moves allow us to verify the algorithm in generality: σ† {0, 1}

1

√

|S| 1

√

|S| 1 √ 2

f

◮ Slide up σ†

Vicary arXiv 1209.3917

Ex 1. The Deutsch-Jozsa Algorithm

Diagrammatic moves allow us to verify the algorithm in generality: σ σ {0, 1}

1

√

|S| 1

√

|S| 1 √ 2

f

◮ Slide up σ† ◮ Pull σ† through the whitedot

Vicary arXiv 1209.3917

Ex 1. The Deutsch-Jozsa Algorithm

Diagrammatic moves allow us to verify the algorithm in generality: σ

1

√

|S| 1

√

|S|

f

◮ Slide up σ† ◮ Pull σ† through the whitedot ◮ Neglect the right-side system

Vicary arXiv 1209.3917

Ex 1. The Deutsch-Jozsa Algorithm

Diagrammatic moves allow us to verify the algorithm in generality: σ

1 |S|

f

◮ Slide up σ† ◮ Pull σ† through the whitedot ◮ Neglect the right-side system ◮ Topological contraction of blackdot

Vicary arXiv 1209.3917

Ex 1. The Deutsch-Jozsa Algorithm

σ

1 |S|

f Gives the amplitude for the input state

1

√

|S|

• s |s to be in the σ state

at measurement.

Vicary arXiv 1209.3917

Ex 1. The Deutsch-Jozsa Algorithm

σ

1 |S|

f Gives the amplitude for the input state

1

√

|S|

• s |s to be in the σ state

at measurement. What if f is balanced? σ f = 0 so the system is never measured in σ. What if f is constant? Then f = x ⇒ σ

1 |S|

f = x σ

1 |S|

= ±1 So the system is always measured in σ.

Vicary arXiv 1209.3917

Ex 1. Summary for Deutsch-Josza

◮ Verify: Abstractly verify the algorithm

Ex 1. Summary for Deutsch-Josza

◮ Verify: Abstractly verify the algorithm ◮ Generalize:

◮ Abstract definition for balanced generalizes [Høyer Phys. Rev. A

59, 3280 1999] and [Batty, Braunstein, Duncan 0412067]. See [Vicary 1209.3917].

Ex 1. Summary for Deutsch-Josza

◮ Verify: Abstractly verify the algorithm ◮ Generalize:

◮ Abstract definition for balanced generalizes [Høyer Phys. Rev. A

59, 3280 1999] and [Batty, Braunstein, Duncan 0412067]. See [Vicary 1209.3917].

◮ The algorithm can be executed with complementary rather than

strongly complementary observables

Ex 2. The GROUPHOMID Algorithm

◮ Given finite groups G and A where A is abelian, and a blackbox

function f : G → A promised to be a group homomorphism, identify f.

◮ Case: Let A be a cyclic group Zn.

f

G A

σ

1

√

|G|

ρ Prepare initial states Apply a unitary map Measure the left system

• |G|

WZ & Vicary arXiv 1406.1278

Ex 2. The GROUPHOMID Algorithm

◮ Given finite groups G and A where A is abelian, and a blackbox

function f : G → A promised to be a group homomorphism, identify f.

◮ Case: Let A be a cyclic group Zn.

G A

f σ

1

√

|G|

ρ

• |G|

WZ & Vicary arXiv 1406.1278

Ex 2. The GROUPHOMID Algorithm

◮ Given finite groups G and A where A is abelian, and a blackbox

function f : G → A promised to be a group homomorphism, identify f.

◮ Case: Let A be a cyclic group Zn.

G A

f σ ρ

1

√

|G|

ρ

• |G|

◮ Pull ρ through whitedot

WZ & Vicary arXiv 1406.1278

Ex 2. The GROUPHOMID Algorithm

◮ Given finite groups G and A where A is abelian, and a blackbox

function f : G → A promised to be a group homomorphism, identify f.

◮ Case: Let A be a cyclic group Zn.

f

G A

σ ρ ρ

◮ Pull ρ through whitedot ◮ Contract set scalars

WZ & Vicary arXiv 1406.1278

Ex 2. The GROUPHOMID Algorithm

◮ Given finite groups G and A where A is abelian, and a blackbox

function f : G → A promised to be a group homomorphism, identify f.

◮ Case: Let A be a cyclic group Zn.

f

A G

σ ρ ρ

◮ Pull ρ through whitedot ◮ Contract set scalars ◮ Topological equivalence

WZ & Vicary arXiv 1406.1278

Ex 2. The GROUPHOMID Algorithm

◮ Given finite groups G and A where A is abelian, and a blackbox

function f : G → A promised to be a group homomorphism, identify f.

◮ Case: Let A be a cyclic group Zn.

f

A G

σ ρ ρ

◮ ρ ◦ f is an irrep. of G.

WZ & Vicary arXiv 1406.1278

Ex 2. The GROUPHOMID Algorithm

◮ Given finite groups G and A where A is abelian, and a blackbox

function f : G → A promised to be a group homomorphism, identify f.

◮ Case: Let A be a cyclic group Zn.

f

A G

σ ρ ρ

◮ ρ ◦ f is an irrep. of G. ◮ Choose ρ to be a faithful

representation of A.

WZ & Vicary arXiv 1406.1278

Ex 2. The GROUPHOMID Algorithm

◮ Given finite groups G and A where A is abelian, and a blackbox

function f : G → A promised to be a group homomorphism, identify f.

◮ Case: Let A be a cyclic group Zn.

f

A G

σ ρ ρ

◮ ρ ◦ f is an irrep. of G. ◮ Choose ρ to be a faithful

representation of A.

◮ Then measuring ρ ◦ f identifies f

(up to isomorphism)

WZ & Vicary arXiv 1406.1278

Ex 2. The GROUPHOMID Algorithm

◮ Given finite groups G and A where A is abelian, and a blackbox

function f : G → A promised to be a group homomorphism, identify f.

◮ Case: Let A be a cyclic group Zn.

f

A G

σ ρ ρ

◮ ρ ◦ f is an irrep. of G. ◮ Choose ρ to be a faithful

representation of A.

◮ Then measuring ρ ◦ f identifies f

(up to isomorphism)

◮ One-dimensional representations

are isomorphic only if they are equal.

WZ & Vicary arXiv 1406.1278

Ex 2. The GROUPHOMID Algorithm

The General Case: Homomorphism f : G → A

◮ We generalize with proof by induction via the Structure

• Theorem. A = Zp1 ⊕ ... ⊕ Zpk

◮ [WZ & Vicary 1406.1278] Given types, the quantum algorithm

can identify a group homomorphism in k oracle queries.

Ex 2. The GROUPHOMID Algorithm

The General Case: Homomorphism f : G → A

◮ We generalize with proof by induction via the Structure

• Theorem. A = Zp1 ⊕ ... ⊕ Zpk

◮ [WZ & Vicary 1406.1278] Given types, the quantum algorithm

can identify a group homomorphism in k oracle queries.

◮ Note that the quantum algorithm depends on the structure of A

while a classical algorithm will depend on the structure of G.

◮ Theorem [WZ] For large G this algorithm makes a quantum

• ptimal number of queries, while classical algorithms are lower

bounded by log |G|.

Quantum algorithms: old, generalized and new

σ† {0, 1} 1

• |S|

1

• |S|

1 √ 2 f s† σ† {0, 1} 1

• |S|

1 √ 2 D f ρ (n) S 1

• |S|

n |G| f Deutsch-Jozsa Single-shot Grover Hidden subgroup

Vicary arXiv 1209.3917 WZ & Vicary arXiv 1406.1278

Other results

◮ Automated graphical reasoning:

quantomatic.github.io

Kissinger arXiv:1203.0202 Dixon et al. arXiv 1007.3794

Other results

◮ Automated graphical reasoning:

quantomatic.github.io

◮ [Coecke & Abramsky 0808.1023]

Teleportation

Other results

◮ Automated graphical reasoning:

quantomatic.github.io

◮ [Coecke & Abramsky 0808.1023]

Teleportation

◮ [Zamdzhiev 2012, WZ & Gogioso

arXiv tmrw] Quantum Secret Sharing

◮ [Cohn-Gordon 2012] Quantum Bit

Commitment

a secret + − +

1

−

1

+

N

−

N

window of attack

Other results

◮ Automated graphical reasoning:

quantomatic.github.io

◮ [Coecke & Abramsky 0808.1023]

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 := N Mary f : N → N := M N f − − → likes ∈ N ⊗ S ⊗ N := N S N likes g ◦ f = N g N M f − − − → Mary ⊗ − − → likes := N Mary likes N S N

i

ii| := N N

Coecke et al. 1003.4394 Grefenstette & Sadrzadeh arXiv 1106.4058

Other results

◮ Automated graphical reasoning:

quantomatic.github.io

◮ [Coecke & Abramsky 0808.1023]

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. 1203.4988, WZ &

Gogioso arXiv tmrw] Foundations: Mermin Non-locality.

◮ [WZ 1503.05857] Models of quantum

algorithms in sets and relations. Outlook: Use this knowledge of quantum structure to better advantage in quantum programming.

Other results

◮ Automated graphical reasoning:

quantomatic.github.io

◮ [Coecke & Abramsky 0808.1023]

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. 1203.4988, WZ &

Gogioso arXiv tmrw] Foundations: Mermin Non-locality.

◮ [WZ 1503.05857] Models of quantum

algorithms in sets and relations. Outlook: Use this knowledge of quantum structure to better advantage in quantum programming.