Quantum Isomorphism Is Equivalent to Equality of Homomorphism Counts from Planar Graphs

Quantum isomorphism is equivalent to equality of homomorphism counts from planar graphs

Laura Manˇcinska1 David E. Roberson2

1QMATH, University of Copenhagen

2Technical University of Denmark

QIP 2021 February 4, 2021 2 / 23 Problem: Entanglement-assisted strategies for arbitrary • nonlocal games are hard to analyze Line of attack: Focus on a well-behaved class of games •

Nonlocal games provide a general framework for studying • entanglement Line of attack: Focus on a well-behaved class of games •

Nonlocal games provide a general framework for studying • entanglement Problem: Entanglement-assisted strategies for arbitrary • nonlocal games are hard to analyze Nonlocal games provide a general framework for studying • entanglement Problem: Entanglement-assisted strategies for arbitrary • nonlocal games are hard to analyze Line of attack: Focus on a well-behaved class of games • This talk

PART I

Quantum isomorphism and diﬀerent ways to think about it: Nonlocal games • Matrix formulations • Homomorphism counts •

PART II

Elements of the proof: Intertwiners of quantum groups • Bi-labeled graphs • Homomorphism matrices •

4 / 23 A map f : V(G) V(H) is an isomorphism from G to H if → f is a bijection and • g ∼ g if and only if f(g) ∼ f(g ). • 0 0

If such a map exists, we say that G and H are isomorphic and write G =∼ H.

Matrix formulation: PAGP† = AH for some permutation matrix P

Graph isomorphism

∼=

5 / 23 If such a map exists, we say that G and H are isomorphic and write G =∼ H.

Matrix formulation: PAGP† = AH for some permutation matrix P

Graph isomorphism

∼=

A map f : V(G) V(H) is an isomorphism from G to H if → f is a bijection and • g ∼ g if and only if f(g) ∼ f(g ). • 0 0

5 / 23 Matrix formulation: PAGP† = AH for some permutation matrix P

Graph isomorphism

∼=

A map f : V(G) V(H) is an isomorphism from G to H if → f is a bijection and • g ∼ g if and only if f(g) ∼ f(g ). • 0 0

If such a map exists, we say that G and H are isomorphic and write G =∼ H.

5 / 23 Graph isomorphism

∼=

A map f : V(G) V(H) is an isomorphism from G to H if → f is a bijection and • g ∼ g if and only if f(g) ∼ f(g ). • 0 0

If such a map exists, we say that G and H are isomorphic and write G =∼ H.

Matrix formulation: PAGP† = AH for some permutation matrix P

5 / 23 To win players must reply h, h • 0 such that rel(h, h0) = rel(g, g0)

No communication during game •

Fact. G =∼ H Classical players can win the game with certainty ⇔ Def. (Quantum isomorphism) ∼ 1 We say that G =qc H if quantum players can win the game with certainty.

Intuition: Alice and Bob want to convince a referee that G =∼ H.

A B

1We work in the commuting rather than the tensor-product model.

(G, H)-Isomorphism Game

6 / 23 To win players must reply h, h • 0 such that rel(h, h0) = rel(g, g0)

No communication during game •

Fact. G =∼ H Classical players can win the game with certainty ⇔ Def. (Quantum isomorphism) ∼ 1 We say that G =qc H if quantum players can win the game with certainty.

A B

1We work in the commuting rather than the tensor-product model.

(G, H)-Isomorphism Game

Intuition: Alice and Bob want to convince a referee that G =∼ H.

6 / 23 To win players must reply h, h • 0 such that rel(h, h0) = rel(g, g0)

No communication during game •

Fact. G =∼ H Classical players can win the game with certainty ⇔ Def. (Quantum isomorphism) ∼ 1 We say that G =qc H if quantum players can win the game with certainty.

1We work in the commuting rather than the tensor-product model.

(G, H)-Isomorphism Game

Intuition: Alice and Bob want to convince a referee that G =∼ H.

R g g0

A B

6 / 23 To win players must reply h, h • 0 such that rel(h, h0) = rel(g, g0)

No communication during game •

Fact. G =∼ H Classical players can win the game with certainty ⇔ Def. (Quantum isomorphism) ∼ 1 We say that G =qc H if quantum players can win the game with certainty.

1We work in the commuting rather than the tensor-product model.

(G, H)-Isomorphism Game

Intuition: Alice and Bob want to convince a referee that G =∼ H.

R g g0

A B h h0

6 / 23 No communication during game •

Fact. G =∼ H Classical players can win the game with certainty ⇔ Def. (Quantum isomorphism) ∼ 1 We say that G =qc H if quantum players can win the game with certainty.

1We work in the commuting rather than the tensor-product model.

(G, H)-Isomorphism Game

Intuition: Alice and Bob want to convince a referee that G =∼ H.

R To win players must reply h, h0 g g0 • such that rel(h, h0) = rel(g, g0)

A B h h0

6 / 23 Fact. G =∼ H Classical players can win the game with certainty ⇔ Def. (Quantum isomorphism) ∼ 1 We say that G =qc H if quantum players can win the game with certainty.

1We work in the commuting rather than the tensor-product model.

(G, H)-Isomorphism Game

Intuition: Alice and Bob want to convince a referee that G =∼ H.

R To win players must reply h, h0 g g0 • such that rel(h, h0) = rel(g, g0)

No communication during game • A B h h0

6 / 23 Def. (Quantum isomorphism) ∼ 1 We say that G =qc H if quantum players can win the game with certainty.

1We work in the commuting rather than the tensor-product model.

(G, H)-Isomorphism Game

Intuition: Alice and Bob want to convince a referee that G =∼ H.

R To win players must reply h, h0 g g0 • such that rel(h, h0) = rel(g, g0)

No communication during game • A B h h0

Fact. G =∼ H Classical players can win the game with certainty ⇔

6 / 23 1We work in the commuting rather than the tensor-product model.

(G, H)-Isomorphism Game

Intuition: Alice and Bob want to convince a referee that G =∼ H.

R To win players must reply h, h0 g g0 • such that rel(h, h0) = rel(g, g0)

No communication during game • A B h h0

Fact. G =∼ H Classical players can win the game with certainty ⇔ Def. (Quantum isomorphism) ∼ 1 We say that G =qc H if quantum players can win the game with certainty.

6 / 23 (G, H)-Isomorphism Game

Intuition: Alice and Bob want to convince a referee that G =∼ H.

R To win players must reply h, h0 g g0 • such that rel(h, h0) = rel(g, g0)

No communication during game • A B h h0

Fact. G =∼ H Classical players can win the game with certainty ⇔ Def. (Quantum isomorphism) ∼ 1 We say that G =qc H if quantum players can win the game with certainty.

1We work in the commuting rather than the tensor-product model. 6 / 23 Eg = {Egh B(H): h V(H)} where ∈ ∈ Egh 0, Egh = I. h P

ψ is a unit vector in a Hilbert space H Upon receiving g, Alice performs a local • measurement Eg to get h V(H) ∈

Bob measures with F 0 • g E and F 0 0 commute • gh g h

Alice and Bob share a quantum state ψ •

The probability that players respond with h, h0 on questions g, g0 is

p(h, h0|g, g0) = ψ, EghFg h ψ h 0 0 i

Quantum commuting strategies ∼ G =qc H := Quantum players can win the (G, H)-isomorphism game

R g g0

A h h0 B

7 / 23 ψ is a unit vector in a Hilbert space H

Eg = {Egh B(H): h V(H)} where ∈ ∈ Egh 0, Egh = I. h P

E and F 0 0 commute • gh g h

The probability that players respond with h, h0 on questions g, g0 is

p(h, h0|g, g0) = ψ, EghFg h ψ h 0 0 i

Quantum commuting strategies ∼ G =qc H := Quantum players can win the (G, H)-isomorphism game

Alice and Bob share a quantum state ψ R • g g0 Upon receiving g, Alice performs a local • measurement Eg to get h V(H) ∈ A h h0 B

Bob measures with F 0 • g

7 / 23 Eg = {Egh B(H): h V(H)} where ∈ ∈ Egh 0, Egh = I. h P

E and F 0 0 commute • gh g h

The probability that players respond with h, h0 on questions g, g0 is

p(h, h0|g, g0) = ψ, EghFg h ψ h 0 0 i

Quantum commuting strategies ∼ G =qc H := Quantum players can win the (G, H)-isomorphism game

Alice and Bob share a quantum state ψ • R ψ is a unit vector in a Hilbert space H g g0 Upon receiving g, Alice performs a local • measurement Eg to get h V(H) ∈ A h h0 B

Bob measures with F 0 • g

7 / 23 E and F 0 0 commute • gh g h

The probability that players respond with h, h0 on questions g, g0 is

p(h, h0|g, g0) = ψ, EghFg h ψ h 0 0 i

Quantum commuting strategies ∼ G =qc H := Quantum players can win the (G, H)-isomorphism game

Alice and Bob share a quantum state ψ • R ψ is a unit vector in a Hilbert space H g g0 Upon receiving g, Alice performs a local • measurement Eg to get h V(H) ∈ A B Eg = {Egh B(H): h V(H)} where h h0 ∈ ∈ Egh 0, h Egh = I. ψ

Bob measures with FP0 • g

7 / 23 The probability that players respond with h, h0 on questions g, g0 is

p(h, h0|g, g0) = ψ, EghFg h ψ h 0 0 i

Quantum commuting strategies ∼ G =qc H := Quantum players can win the (G, H)-isomorphism game

Bob measures with FP0 • g E and F 0 0 commute • gh g h

7 / 23 The probability that players respond with h, h0 on questions g, g0 is

p(h, h0|g, g0) = ψ, EghFg h ψ h 0 0 i

Quantum commuting strategies ∼ G =qc H := Quantum players can win the (G, H)-isomorphism game

Bob measures with FP0 • g E and F 0 0 commute • gh g h

7 / 23 The probability that players respond with h, h0 on questions g, g0 is

p(h, h0|g, g0) = ψ, EghFg h ψ h 0 0 i

Quantum commuting strategies ∼ G =qc H := Quantum players can win the (G, H)-isomorphism game

Bob measures with FP0 • g E and F 0 0 commute • gh g h

7 / 23 Quantum commuting strategies ∼ G =qc H := Quantum players can win the (G, H)-isomorphism game

Bob measures with FP0 • g E and F 0 0 commute • gh g h

The probability that players respond with h, h0 on questions g, g0 is

p(h, h0|g, g0) = ψ, EghFg h ψ h 0 0 i

7 / 23 Quantum commuting strategies ∼ G =qc H := Quantum players can win the (G, H)-isomorphism game

Bob measures with FP0 • g E and F 0 0 commute • gh g h

The probability that players respond with h, h0 on questions g, g0 is

p(h, h0|g, g0) = ψ, EghFg h ψ h 0 0 i

7 / 23 Example: G =∼ H but G =∼ H 6 qc

x1 + x2 + x3 = 0 x4 + x5 + x6 = 0 x7 + x8 + x9 = 0

000 011 101 110 000 011 101 110 000 011 101 110

000 011 101 110 000 011 101 110 111 100 010 001

x1 + x4 + x7 = 0 x2 + x5 + x8 = 0 x3 + x6 + x9 = 1

Construction based on reduction from linear system games.

8 / 23 Example: G =∼ H but G =∼ H 6 qc

x1 + x2 + x3 = 0 x4 + x5 + x6 = 0 x7 + x8 + x9 = 0

000 011 101 110 000 011 101 110 000 011 101 110

x1 + x4 + x7 = 0 x2 + x5 + x8 = 0 x3 + x6 + x9 = 0

Construction based on reduction from linear system games.

9 / 23 p is a projection, i.e., p2 = p = p for all i, j • ij ij ij ij∗ p = 1 = p for all i, j • k ik ` `j P P Remark. A QPM with entries from C is a permutation matrix.

Thm. (Lupini, M., Roberson) ∼ G =qc H PAGP† = AH for some quantum ⇔ permutation matrix P

Quantum isomorphism and quantum groups

Def. A matrix P = (pij) whose entries are elements of a C∗-algebra is a quantum permutation matrix (QPM), if

10 / 23 p = 1 = p for all i, j • k ik ` `j P P Remark. A QPM with entries from C is a permutation matrix.

Thm. (Lupini, M., Roberson) ∼ G =qc H PAGP† = AH for some quantum ⇔ permutation matrix P

Quantum isomorphism and quantum groups

Def. A matrix P = (pij) whose entries are elements of a C∗-algebra is a quantum permutation matrix (QPM), if p is a projection, i.e., p2 = p = p for all i, j • ij ij ij ij∗

10 / 23 Remark. A QPM with entries from C is a permutation matrix.

Thm. (Lupini, M., Roberson) ∼ G =qc H PAGP† = AH for some quantum ⇔ permutation matrix P

Quantum isomorphism and quantum groups

Def. A matrix P = (pij) whose entries are elements of a C∗-algebra is a quantum permutation matrix (QPM), if p is a projection, i.e., p2 = p = p for all i, j • ij ij ij ij∗ p = 1 = p for all i, j • k ik ` `j P P

10 / 23 Thm. (Lupini, M., Roberson) ∼ G =qc H PAGP† = AH for some quantum ⇔ permutation matrix P

Quantum isomorphism and quantum groups

10 / 23 Quantum isomorphism and quantum groups

Thm. (Lupini, M., Roberson) ∼ G =qc H PAGP† = AH for some quantum ⇔ permutation matrix P

10 / 23 Can we describe quantum isomorphism in combinatorial terms?

11 / 23 Example

C7 C5 →

hom(F, G) := # of homomorphisms from F to G.

Graph homomorphisms

Def. A map ϕ : V(F) V(G) is a homomorphism from F to G if → ϕ(u) ∼ ϕ(v) whenever u ∼ v.

12 / 23 hom(F, G) := # of homomorphisms from F to G.

Graph homomorphisms

Def. A map ϕ : V(F) V(G) is a homomorphism from F to G if → ϕ(u) ∼ ϕ(v) whenever u ∼ v.

Example

C7 C5 →

12 / 23 Graph homomorphisms

Def. A map ϕ : V(F) V(G) is a homomorphism from F to G if → ϕ(u) ∼ ϕ(v) whenever u ∼ v.

Example

C7 C5 →

hom(F, G) := # of homomorphisms from F to G.

12 / 23 Theorem. (Lov´asz,1967) Homomorphism counts determine a graph up to isomorphism, i.e.

G =∼ H hom(F, G) = hom(F, H) for all graphs F. ⇔

Theorem. (M., Roberson) ∼ G =qc H hom(F, G) = hom(F, H) for all planar graphs F. ⇔

Counting homomorphisms

13 / 23 Theorem. (M., Roberson) ∼ G =qc H hom(F, G) = hom(F, H) for all planar graphs F. ⇔

Counting homomorphisms

Theorem. (Lov´asz,1967) Homomorphism counts determine a graph up to isomorphism, i.e.

G =∼ H hom(F, G) = hom(F, H) for all graphs F. ⇔

13 / 23 Counting homomorphisms

Theorem. (Lov´asz,1967) Homomorphism counts determine a graph up to isomorphism, i.e.

G =∼ H hom(F, G) = hom(F, H) for all graphs F. ⇔

Theorem. (M., Roberson) ∼ G =qc H hom(F, G) = hom(F, H) for all planar graphs F. ⇔

13 / 23 Folklore. G and H cospectral hom(F, G) = hom(F, H) for all cycles F ⇔ Thm. (Dvoˇr´ak,2010; Dell, Grohe, Rattan, 2018) ∼ G =f H hom(F, G) = hom(F, H) for all trees F ∼ ⇔ G =k H hom(F, G) = hom(F, H) for all F of treewidth k ⇔ 6

Complexity: Except for the class of planar graphs, equality of homomorphism counts from all of the above graph classes can be tested in at worst quasi-polynomial time.

Context: Homomorphism counting Thm. (Lov´asz,1967) G =∼ H hom(F, G) = hom(F, H) for all graphs F ⇔ Thm. (M., Roberson, 2019) ∼ G =qc H hom(F, G) = hom(F, H) for all planar graphs F ⇔

14 / 23 Thm. (Dvoˇr´ak,2010; Dell, Grohe, Rattan, 2018) ∼ G =f H hom(F, G) = hom(F, H) for all trees F ∼ ⇔ G =k H hom(F, G) = hom(F, H) for all F of treewidth k ⇔ 6

Complexity: Except for the class of planar graphs, equality of homomorphism counts from all of the above graph classes can be tested in at worst quasi-polynomial time.

14 / 23 ∼ G =k H hom(F, G) = hom(F, H) for all F of treewidth k ⇔ 6

Complexity: Except for the class of planar graphs, equality of homomorphism counts from all of the above graph classes can be tested in at worst quasi-polynomial time.

Context: Homomorphism counting Thm. (Lov´asz,1967) G =∼ H hom(F, G) = hom(F, H) for all graphs F ⇔ Thm. (M., Roberson, 2019) ∼ G =qc H hom(F, G) = hom(F, H) for all planar graphs F ⇔ Folklore. G and H cospectral hom(F, G) = hom(F, H) for all cycles F ⇔ Thm. (Dvoˇr´ak,2010; Dell, Grohe, Rattan, 2018) ∼ G =f H hom(F, G) = hom(F, H) for all trees F ⇔

14 / 23 Complexity: Except for the class of planar graphs, equality of homomorphism counts from all of the above graph classes can be tested in at worst quasi-polynomial time.

14 / 23 : Except for the class of planar graphs, equality of homomorphism counts from all of the above graph classes can be tested in at worst quasi-polynomial time.

Complexity

14 / 23 Context: Homomorphism counting Thm. (Lov´asz,1967) G =∼ H hom(F, G) = hom(F, H) for all graphs F ⇔ Thm. (M., Roberson, 2019) ∼ G =qc H hom(F, G) = hom(F, H) for all planar graphs F ⇔ Folklore. G and H cospectral hom(F, G) = hom(F, H) for all cycles F ⇔ Thm. (Dvoˇr´ak,2010; Dell, Grohe, Rattan, 2018) ∼ G =f H hom(F, G) = hom(F, H) for all trees F ∼ ⇔ G =k H hom(F, G) = hom(F, H) for all F of treewidth k ⇔ 6

Complexity: Except for the class of planar graphs, equality of homomorphism counts from all of the above graph classes can be tested in at worst quasi-polynomial time.

14 / 23 Before: Diﬃcult to prove that they are not quantum isomorphic.

Now: Only one (the Rook graph) contains K4.

Application: Certiﬁcate for G =∼ H 6 qc Are these two graphs quantum isomorphic? Rook graph Shrikhande graph

15 / 23 Now: Only one (the Rook graph) contains K4.

Application: Certiﬁcate for G =∼ H 6 qc Are these two graphs quantum isomorphic? Rook graph Shrikhande graph

Before: Diﬃcult to prove that they are not quantum isomorphic.

15 / 23 Application: Certiﬁcate for G =∼ H 6 qc Are these two graphs quantum isomorphic? Rook graph Shrikhande graph

Before: Diﬃcult to prove that they are not quantum isomorphic.

Now: Only one (the Rook graph) contains K4. 15 / 23 Part II Elements of the proof

∼ Thm. G =qc H hom(F, G) = hom(F, H) for all planar graphs F ⇔ Main component of our proof: Provide a combinatorial description of the intertwiners of Qut(G).

Intertwiners of Qut(G) = U, M, AG , , ,lin, where h i◦ ⊗ ∗

U = ei, M(ei ej) = δijei i, j V(G). ⊗ ∀ ∈ i V(G) ∈X

The starting point

Thm. (Lupini, M., Roberson) Quantum isomorphism can be characterized in terms of the quantum automorphism group of a graph, denoted Qut(G).

17 / 23 Intertwiners of Qut(G) = U, M, AG , , ,lin, where h i◦ ⊗ ∗

U = ei, M(ei ej) = δijei i, j V(G). ⊗ ∀ ∈ i V(G) ∈X

The starting point

Thm. (Lupini, M., Roberson) Quantum isomorphism can be characterized in terms of the quantum automorphism group of a graph, denoted Qut(G).

Main component of our proof: Provide a combinatorial description of the intertwiners of Qut(G).

17 / 23 The starting point

Thm. (Lupini, M., Roberson) Quantum isomorphism can be characterized in terms of the quantum automorphism group of a graph, denoted Qut(G).

Main component of our proof: Provide a combinatorial description of the intertwiners of Qut(G).

Intertwiners of Qut(G) = U, M, AG , , ,lin, where h i◦ ⊗ ∗

U = ei, M(ei ej) = δijei i, j V(G). ⊗ ∀ ∈ i V(G) ∈X

17 / 23 Example. ~F = K4, (2, 1), (2, 2)

2 3 4

Bi-labeled graphs

Def. (Lov´asz,Large Networks and Graph Limits) An (`, k)-bi-labeled graph is a triple ~F = (F, a~, ~b) where F is a graph • a~ = (a ,..., a ) and ~b = (b ,..., b ) are tuples of vertices • 1 ` 1 k of F.

18 / 23 1

2 3 4

Bi-labeled graphs

Example. ~F = K4, (2, 1), (2, 2)

18 / 23 Bi-labeled graphs

Example. ~F = K4, (2, 1), (2, 2)

2 3 4

18 / 23 1

2 3 4

~F = K4, (2, 1), (2, 2)

~ ~ ~ U = K1, (1), ∅ M = K1, (1), (1, 1) A = K2, (1), (2)

How to draw bi-labeled graphs

19 / 23 ~ ~ ~ U = K1, (1), ∅ M = K1, (1), (1, 1) A = K2, (1), (2)

How to draw bi-labeled graphs

2 3 4

~F = K4, (2, 1), (2, 2)

19 / 23 How to draw bi-labeled graphs

2 3 4

~F = K4, (2, 1), (2, 2)

~ ~ ~ U = K1, (1), ∅ M = K1, (1), (1, 1) A = K2, (1), (2)

19 / 23 Def. (G-homomorphism matrix of ~F) For u, v V(G), the uv-entry of the homomorphism matrix T~F is ∈ |{homs ϕ : F G | ϕ(a) = u, ϕ(b) = v}| . →

Example. A~ = (K2, (1), (2)) 1 2

~ 1 if u ∼ v T A = u,v 0 otherwise

A~ U~ M~ So T = AG. Similarly, T = U, T = M.

Homomorphism matrices

Let G be a graph and ~F = (F, (a), (b)) an (1, 1)-bi-labeled graph.

20 / 23 Example. A~ = (K2, (1), (2)) 1 2

~ 1 if u ∼ v T A = u,v 0 otherwise

A~ U~ M~ So T = AG. Similarly, T = U, T = M.

Homomorphism matrices

Let G be a graph and ~F = (F, (a), (b)) an (1, 1)-bi-labeled graph.

Def. (G-homomorphism matrix of ~F) For u, v V(G), the uv-entry of the homomorphism matrix T~F is ∈ |{homs ϕ : F G | ϕ(a) = u, ϕ(b) = v}| . →

20 / 23 ~ 1 if u ∼ v T A = u,v 0 otherwise

A~ U~ M~ So T = AG. Similarly, T = U, T = M.

Homomorphism matrices

Let G be a graph and ~F = (F, (a), (b)) an (1, 1)-bi-labeled graph.

Def. (G-homomorphism matrix of ~F) For u, v V(G), the uv-entry of the homomorphism matrix T~F is ∈ |{homs ϕ : F G | ϕ(a) = u, ϕ(b) = v}| . →

Example. A~ = (K2, (1), (2)) 1 2

20 / 23 A~ U~ M~ So T = AG. Similarly, T = U, T = M.

Homomorphism matrices

Let G be a graph and ~F = (F, (a), (b)) an (1, 1)-bi-labeled graph.

Def. (G-homomorphism matrix of ~F) For u, v V(G), the uv-entry of the homomorphism matrix T~F is ∈ |{homs ϕ : F G | ϕ(a) = u, ϕ(b) = v}| . →

Example. A~ = (K2, (1), (2)) 1 2

~ 1 if u ∼ v T A = u,v 0 otherwise

20 / 23 Similarly, T U~ = U, T M~ = M.

Homomorphism matrices

Let G be a graph and ~F = (F, (a), (b)) an (1, 1)-bi-labeled graph.

Def. (G-homomorphism matrix of ~F) For u, v V(G), the uv-entry of the homomorphism matrix T~F is ∈ |{homs ϕ : F G | ϕ(a) = u, ϕ(b) = v}| . →

Example. A~ = (K2, (1), (2)) 1 2

~ 1 if u ∼ v T A = u,v 0 otherwise

A~ So T = AG.

20 / 23 Homomorphism matrices

Let G be a graph and ~F = (F, (a), (b)) an (1, 1)-bi-labeled graph.

Def. (G-homomorphism matrix of ~F) For u, v V(G), the uv-entry of the homomorphism matrix T~F is ∈ |{homs ϕ : F G | ϕ(a) = u, ϕ(b) = v}| . →

Example. A~ = (K2, (1), (2)) 1 2

~ 1 if u ∼ v T A = u,v 0 otherwise

A~ U~ M~ So T = AG. Similarly, T = U, T = M.

20 / 23 Thm. For a graph G and bi-labeled graphs ~F1,~F2,

~F1 ~F2 ~F1 ~F2 T T = T ◦ ,

where ~F1 ~F2 is deﬁned as ◦ F~ F~2 F~ F~ 1 1 ◦ 2 1 a 1 4, a, c 4 { } b = 2 ◦ 2 c 5, b, d 5 { } e 3 e 3 d

Operations on bi-labeled graphs: Products

21 / 23 Operations on bi-labeled graphs: Products

Thm. For a graph G and bi-labeled graphs ~F1,~F2,

~F1 ~F2 ~F1 ~F2 T T = T ◦ ,

where ~F1 ~F2 is deﬁned as ◦ F~ F~2 F~ F~ 1 1 ◦ 2 1 a 1 4, a, c 4 { } b = 2 ◦ 2 c 5, b, d 5 { } e 3 e 3 d

21 / 23 So we want to know what bi-labeled graphs are in U~ , M~ , A~ , , . h i◦ ⊗ ∗ Def.

α1 β1

a1 b1 a1 b1

a2 F b2 α2 a2 F b2 β2

a3 b3 a3 b3

α3 β3

F ◦

P = {~F : F◦ has planar embedding w/ enveloping cycle bounding outer face}

Thm. (informal) Intertwiners of Qut(G) are given by the span of homomorphism matrices of planar bi-labeled graphs.

Planar bi-labeled graphs

Recall: Intertwiners of Qut(G) = U, M, AG , , ,lin h i◦ ⊗ ∗

22 / 23 Def.

α1 β1

a1 b1 a1 b1

a2 F b2 α2 a2 F b2 β2

a3 b3 a3 b3

α3 β3

F ◦

P = {~F : F◦ has planar embedding w/ enveloping cycle bounding outer face}

Thm. (informal) Intertwiners of Qut(G) are given by the span of homomorphism matrices of planar bi-labeled graphs.

Planar bi-labeled graphs

Recall: Intertwiners of Qut(G) = U, M, AG , , ,lin h i◦ ⊗ ∗ So we want to know what bi-labeled graphs are in U~ , M~ , A~ , , . h i◦ ⊗ ∗

22 / 23 P = {~F : F◦ has planar embedding w/ enveloping cycle bounding outer face}

Thm. (informal) Intertwiners of Qut(G) are given by the span of homomorphism matrices of planar bi-labeled graphs.

Planar bi-labeled graphs

Recall: Intertwiners of Qut(G) = U, M, AG , , ,lin h i◦ ⊗ ∗ So we want to know what bi-labeled graphs are in U~ , M~ , A~ , , . h i◦ ⊗ ∗ Def.

α1 β1

a1 b1 a1 b1

a2 F b2 α2 a2 F b2 β2

a3 b3 a3 b3

α3 β3

F ◦

22 / 23 Thm. (informal) Intertwiners of Qut(G) are given by the span of homomorphism matrices of planar bi-labeled graphs.

Planar bi-labeled graphs

Recall: Intertwiners of Qut(G) = U, M, AG , , ,lin h i◦ ⊗ ∗ So we want to know what bi-labeled graphs are in U~ , M~ , A~ , , . h i◦ ⊗ ∗ Def.

α1 β1

a1 b1 a1 b1

a2 F b2 α2 a2 F b2 β2

a3 b3 a3 b3

α3 β3

F ◦

P = {~F : F◦ has planar embedding w/ enveloping cycle bounding outer face}

22 / 23 Planar bi-labeled graphs

Recall: Intertwiners of Qut(G) = U, M, AG , , ,lin h i◦ ⊗ ∗ So we want to know what bi-labeled graphs are in U~ , M~ , A~ , , . h i◦ ⊗ ∗ Def.

α1 β1

a1 b1 a1 b1

a2 F b2 α2 a2 F b2 β2

a3 b3 a3 b3

α3 β3

F ◦

P = {~F : F◦ has planar embedding w/ enveloping cycle bounding outer face}

Thm. (informal) Intertwiners of Qut(G) are given by the span of homomorphism matrices of planar bi-labeled graphs.

22 / 23 Thank you!

Summary Graph isomorphism can be formulated in terms of a nonlocal game.

R g g0

A h h0 B

G =∼ H := Quantum players can win the isomorphism game • qc Thm. G =∼ H PA P = A for some quantum • qc G † H ⇔ permutation matrix P ∼ Thm. G =qc H hom(F, G) = hom(F, H) for all planar F • ⇔

23 / 23 Thank you!

Summary Graph isomorphism can be formulated in terms of a nonlocal game.

R g g0

A h h0 B

23 / 23 Summary Graph isomorphism can be formulated in terms of a nonlocal game.

R g g0

A h h0 B

G =∼ H := Quantum players can win the isomorphism game • qc Thm. G =∼ H PA P = A for some quantum • qc G † H ⇔ permutation matrix P ∼ Thm. G =qc H hom(F, G) = hom(F, H) for all planar F • ⇔ Thank you!

23 / 23