Expansion in SLd and SU(d)

Applications

1

EXPANDER GRAPHS

Sparse graphs with high connectivity

Many applications

• efficient communication networks

• error-correcting codes

• derandomization of random algorithms

• quantum computation

• group theory

• geometry

• number theory

2

3

EXPANSION PROPERTY AND RATIO

Every subset has a large number of

neighbors

h(X) = min|S|≤n2

|∂S||S|

∂S = edges from S to its complement

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h = 2

4

DO EXPANDER GRAPHS EXIST?

M. Pinsker (73)

‘Given k ≥ 3, a random (= typical)

k-regular graph on n vertices

(n→ ∞) is an expander graph’

5

HOW TO PRODUCE

EXPLICITLY EXPANDER GRAPHS?

USE OF ALGEBRAIC METHODS

First construction: G. Margulis (73)

Expansion of CAYLEY GRAPHS on

groups

6

SELBERG’S THEOREM

〈S〉 of finite index in SL2(Z)

πp : SL2(Z) → SL2(Fp) p→ ∞

(mod p reduction)

G(

SL2(Fp), πp(S))

is expander family

7

Example

S1 =

1 10 1

,

1 01 1

S2 =

1 20 1

,

1 02 1

S3 =

1 30 1

,

1 03 1

(?)

8

Lubotzky-Weiss Conjecture

Let S be a finite subset of SLd(Z)

generating a Zariski dense

subgroup of SLd. Then there is

q0 ∈ Z such that the family of

Cayley Graphs

G(

SLd(Z/qZ), πq(S))

(q, q0) = 1

forms a family of expanders

9

CONNECTEDNESS OF THE GRAPH

ROLE OF STRONG APPROXIMATION

PROPERTY

Theorem. Let G be a Zariski dense

subgroup of SLd(Z). There is

q0 ∈ Z such that πq(G) = SLd(Z/qZ)

if (q, q0) = 1

πq: reduction mod q

Matthews-Vaserstein-Weisfeiler-Pink

10

Theorem

Lubotzky-Weiss conjecture is true

J. B.

E. BREUILLARD

A. GAMBURD

B. GREEN

H. HELFGOTT

L. PYBER

P. SARNAK

E. SZABO

T. TAO

P. VARJU

11

EXTENSION TO NUMBER FIELDS

[K : Q] = r σ1, . . . , σr : K → C embeddings

OK integers of K

S ⊂ SLd(OK) finite symmetric

Γ = 〈S〉 subgroup of SLd

12

Theorem (P. Varju)

Let σ = σ1 ⊕ · · · ⊕ σr : K −→ Cr and

assume σ(Γ) Zariski dense in

SLd(C)r

Then there is an ideal J ⊂ OK such

that

G(

SLd(O/I), πI(S))

is family of expanders if I ⊂ Oranges over square free ideals prime

to J13

SL2 – APPLICATIONS

• Generalization of Selberg’s theorem

(B-G-S)

• Hyperbolic lattice point counting

(B-G-S)

• Prime sieving in orbits of linear groups

(B-G-S, K, B-K, K-O)

• Topology

Covers of hyperbolic 3-manifolds

with positive Heegaard gradient

(work of Long-Lubotzky-Reid)14

SU(2)

g1, . . . , gk ∈ SU(2) algebraic and free

T : L2(G) → L2(G) Hecke operator

Tf(x) =∑(

f(gjx) + f(g−1j x)

)

Theorem. There is spectral gap

λ1(T ) < 2K − γ

γ = γ(g1, . . . , gk) controlled by non

commutative diophantine

property

15

Non CommutativeDiophantine Property

G = {g1, . . . , gk}

Wℓ(G) = words of length ℓ

DC : g ∈Wℓ(G)\{1} ⇒ ‖1−g‖ > A−ℓ

Satisfied for G ⊂ Mat2×2(Q) where A

depends on the height

16

Applications

• Banach-Ruziewicz Problem

(finitely additive invariant measures on S2)

• Quantum-Computation

(Solovay-Kitaev algorithm)

• Conway-Radin Quaqua-versal Tiling of R3

• Dimension Expansion Problem

17

SU(d)

g1, . . . , gk ∈ SU(d) ∩ Matd×d(Q)

Γ = 〈g1, . . . , gk〉Assume Γ topologically dense

Tf(x) =∑

(

f(gjx)+f(g−1j x)

)

Theorem

T has spectral gap

18

Comments

• May assume k = 2 and {g1, g2} free

generators of the free group F2

(Breuillard-Gelander)

• Define

ν =1

4(δg1 + δg2 + δ

g−11

+ δg−12

)

on G = SU(d)

Escape Property. There is κ > 0

such that if H is a non-trivial

closed subgroup of G, then

ν(ℓ)(H) < e−κℓ for ℓ→ ∞

19

Denote for δ > 0

Pδ =1B(1, δ)

|B(1, δ)|an approximate identity on G

Proposition. Given τ > 0, there is a

positive integer ℓ < C(τ) log 1δ such that

‖ν(ℓ) ∗ Pδ‖∞ < δ−τ

Main Lemma. Given γ > 0, there is κ > 0

such that for δ > 0 small enough,

ℓ ∼ log 1δ , if

‖ν(ℓ) ∗ Pδ‖2 > δ−γ

Then

‖ν(2ℓ) ∗ Pδ‖2 < δκ‖ν(ℓ) ∗ Pδ‖2

20

Proof of Spectral Gap

Alternative Approach to

Sarnak-Xue Argument

First treat the case SO(3) ≃ SU(2)/Z2

G = SO(3)

ρ : G −→ L2(S2) ρgf = f ◦ g

T = 14(ρg1 + ρg2 + ρ

g−11

+ ρg−12

)

Assume f ∈ L20(S), ‖f‖2 = 1 and

‖Tf‖2 > 1 − ε

〈g1, g2〉 dense ⇒ f → 0 weakly if ε→ 0

21

Introduce Littlewood-Paley decomposition

f ∈ L20(S)

∆1f = f ∗ P2−1

∆kf = (f ∗ P2−k) − (f ∗ P2−k+1) for k ≥ 2

f =∑

∆kf

‖f‖22 ∼∑

‖∆kf‖22

Fix ℓ0 ∈ Z+. For ε > 0 small enough

‖f ∗ ν(ℓ0)‖2 > (1 − ε)ℓ0 >1

2

There is k ∈ Z+, k large, such that

‖F ∗ ν(ℓ0)‖2 > c where F =∆kf

‖∆kf‖2

22

δ = 4−k ⇒ F ≈ F ∗ Pδ

Fix τ > 0 and take

ℓ0ℓ > C(τ) log1

δ

such that

‖µ‖2 < δ−τ µ = ν(2ℓ0ℓ) ∗ Pδ

Then

cℓ < ‖F ∗ ν(ℓℓ0)‖2

2<∫

|〈ρgF, F 〉|µ(dg)

< δ−τ( ∫

G|〈ρgF, F 〉|2dg

)12

Hence

δ2τc2ℓ <∫

G

∫

S×S F(gx)F(x)F(gy)F(y)dxdydg

=∫

S×S×SF(x)F(y)F(z)Aθ(x,y)F(z)dxdydz

23

AθF(z) = average on [ζ ∈ S2 : θ(ζ, z) = θ] ≃ S1

Since F = ∆kF , smoothing bounds imply

‖AθF‖2 < C(1 + 2k|θ|)−12

Therefore

δ2τc2ℓ < C2−12k∫

S×S

|F(x)| |F(y)||x− y|

12

< C2−k/2 < Cδ1/4

where

ℓ ∼ 1

ℓ0log

1

δ

⇒ contradiction24

G = SU(d)

ρ= regular representation on L2(G)

f =∑

k≥1

∆kf for f ∈ L20(G)

Follow same argument

⇒ cℓδτ <( ∫

G

∣∣∣〈ρgF, F 〉

∣∣∣2dg)1/2

= ‖F∗F‖2

Contradicts

Convolution Lemma

Assume F1, F2 ∈ L∞(G), |F1|, |F2| ≤ 1 and

‖F1 ∗ Pδ‖2 < δc

Then

‖F1 ∗ F2‖2 < δc′

25

Sketch of Convolution Lemma

• G = SU(2) Follows from preceding

• Take subgroup H ≃ SU(2) in G

Write∫

G

∣∣∣∣

∫

GF1(xg

−1)F2(g)dg∣∣∣∣dx ≤

∫

G

∫

G

[ ∫

H

∣∣∣∣

∫

HF1(xyh

−1g−1)F2(gh)dh∣∣∣∣dy

︸ ︷︷ ︸

‖ϕ1∗ϕ2‖L1(H)

]

dxdg

ϕ1(y) = F1(xyg−1) ϕ2(y) = F2(gy)

with x, g fixed

26

L2 - Flattening Lemma

Reduction to ‘Approximate Groups’

(BSG - Tao)

δ > 0 µ = ν(ℓ) ℓ ∼ log1

δ

‖µ ∗ µ ∗ Pδ‖2 ∼ ‖µ ∗ Pδ‖2

∃H ⊂ G,H union of δ-balls

∃X ⊂ G finite set

Satisfying

• H = H−1

• H.H ⊂ H.X ∩X.H• |X| < δ−ε

• µ(aH) > δε for some a ∈ G

• |H| < δγ

27

Main Steps

• Construction of diagonal sets in H ′

• Construction of 1-dim structures

(use of discretized ring theorem)

• Amplification

(using Lie-algebra and escape property)

⇒ |H ′| > δε (contradiction)

28

Discretized Ring Theorem in R

Theorem. Assume µ a probability measure

on [0,1] s.t.

µ(I) < Cρκ if I is a ρ-interval, δ < ρ < 1

Then for some s < s(κ, C) ∈ Z+ the

sum-product set sA(s)−sA(s) is δ-dense

in [0,1]

Definition. N(A, δ) = maximum number of

δ-separated points in A

Corollary. Given σ > 0, there is γ(σ) > 0

and s(σ) ∈ Z+ s.t. for δ > 0 small

enough, if A ⊂ [0,1] and

N(A, δ) > δ−σ

then[0, δγ] ⊂ sA(s) − sA(s) + [0, δγ+1]

29

Discretized Ring Theorem in C and Cd

Theorem. Given σ > 0, there is γ > 0 and

s ∈ Z+ s.t. if δ > 0 is small enough

and A ⊂ C ∩B(0,1),

N(A, δ) > δ−σ

Then there is ξ ∈ C, |ξ| = 1 s.t.

[0, δγ]ξ ⊂ sA(s) − sA(s) +B(0, δγ+1)

Theorem. If A ⊂ Cd ∩B(0,1) and

N(A, δ) > δ−σ

then there is ξ ∈ Cd, |ξ| = 1 s.t.

[0, δγ]ξ ⊂ sA(s) − sA(s) +B(0, δγ+1)

Cd equipped with product ring structure30

Diagonal Sets

Proposition. {g1, g2} in U(d)∩ Matd×d(Q)

generating free group

H ⊂Wℓ(g1, g2) such that log |H| > cℓ

Then there is A ⊂ H(s), s < C and δ > 0,

log 1δ ∼ ℓ s.t.

• |A| > δ−c

• Elements of A are δ-separated

• In an appropriate orthonormal basis

dist (x,∆) < δ for x ∈ A

∆ = {diagonal matrices}

(uses Helfgott’s method)

31

Use of Lie-algebra

H : approximate group

H ′ = (H ∪H−1)(s) s ∈ Z+ bounded

V = su(d) ⊗ C = {traceless matrices}

Escape Property. There is κ > 0 such that

if L is a proper subspace of V , then

(for k large enough)

ν(k)[g ∈ G; g−1Lg = L] < e−κk

Corollary. If a, b ∈ V \{0} and k > ε log 1δ ,

there is g ∈ H ′ ∩Wk s.t.

|Tr g−1agb| > C−k‖a‖ ‖b‖

32

Verification of Escape Property

• Tits method Proximal (but reducible)

representation on wedge product of

adjoint representation

(in appropriate local field)

• Random matrix product theory

Theorem. Assume ρ : Γ → GL(V ) acts

strongly irreducibly and has proximal

element. Let Γ = 〈supp ν〉 with ν a

symmetric, finitely supported,

probability measure. Then

ν(k)[g ∈ Γ; ρg(L) = L] < e−σk (k → ∞)

if L ⊂ V is nontrivial subspace33

Quantum computation

Aperiodic tilings

34

QUANTUM COMPUTATION

PURE QUBIT STATES

|ψ >= α|0 > +β|1 >

|α|2 + |β|2 = 1

35

OPERATIONS ON QUBITS

performed by unitary matrices

Hadamard gate

H =1√2

1 1

1 −1

Phase shifter gates

R(θ) =

1 0

0 e2πiθ

(θ = phase shift)

36

Computationally universal gates

Definition: Finite set A of

‘base-gates’ such that any gate

can be arbitrary well approximated

by a product of elements of A

and their inverses

Important in context of

FAULT-TOLERANT QC: direct

fault-tolerant constructions

typically only available for

special gates

How good can an arbitrary gate be

approximated by a short word in A ?37

The Solovay-Kitaev algorithm

SU(2) = special unitary group

a b−b a

|a|2 + |b|2 = 1

‘If A ⊂ SU(2) is computationally

universal, then A fills SU(2) quickly’

38

Approximation of any

SU(2)-gate within ε > 0 by

a product of sequence of

base-gates of length

0(

log3,97(

1ε

))

Sequence produced by

algorithm with running time

0(

log2,71(

1ε

))

39

How short can the string be made?

ε-approximation requires sequences

of length at least

0(

log1

ε

)

(at least ∼ ε−3 elements need to

be produced)

Definition: A ⊂ SU(2) is efficiently

universal if any SU(2)-gate can

be approximated with ε-precision

by a string of length 0(log 1ε)

40

Example: (A. Lubotzky, R. Phillips, P. Sarnak)

V1 =1√5

1 2i2i 1

V2 =1√5

1 2−2 1

V3 =1√5

1 + 2i 0

0 1 − 2i

Theorem: Any system of

computationally universal gates

is efficiently universal

41

Open Problem

Find polynomial time algorithm to

generate an ε-approximating string

of length 0(

log 1ε

)

42

Aperiodic Tilings

Models for Quasi-Cristals

43

‘Penrose Tilings’

Penrose “kite and dart” tiling

Penrose tilings are aperiodic tilings

of the plane using copies of 2

polytopes which appear in 10

different orientations

Number of orientations is slowly

growing in the volume44

The Quaquaversal Tiling(J.Conway, C.Radin)

Number of orientations of tiles

(=congruent triangular prisms)

rapidly growing in the volume

45

Construction Process

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1

1

√3

AQQ tile

2

Decomposition in 8 congruent

daughter tiles

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Rescale by factor 2 and iterate46

Theorem. (C&R): QQ tilings

statistically invariant under all

rotations

At what rate do the orientations

approach uniform distribution?

Theorem. (B&G) Exponentially fast

in the number n of subdivisions

Conjectured rate based on

numerics: (0,9938...)n

47

B.Draco, L.Sadun, D.Van Wierden

Tests on special harmonics

ℓ Eigenvalue

1 0.500002 0.787854 0.926936 0.949128 0.98454

14 0.9879718 0.9904832 0.9924345 0.9932456 0.9933572 0.99362

248 0.99367258 0.99381

Records Eigenvalues

48

Dimension Expanders

F field

Fn n-dim linear space over F

Find a small (constant) set of linear

transformations (Ai)i∈I so that for some α > 0

dim(∑

i∈IAi(V )

)

> (1 + α) dimV

for any linear subspace V ⊂ Fn, dim V < n2

Barak-Impagliazzo-Shpilka-Wigderson (04)

W (0.4)

char F = 0

Solved by Lubotzky-Zelmanov (08) using

property (T ) and an adjoint representation

acting unitarily on Hilbert-Schmidt operators49

Role of Monotone Expanders (Dvir-Shpilka)

[n] = {1, . . . , n}

Mon [n] = set of monotone partial functions

ϕ : [n] → [n] ∪ {⊥}

ϕ(x) = ⊥ if ϕ undefined at x

ϕ(x) < ϕ(y) if x < y and ϕ defined at x, y

ϕ1, . . . , ϕd ∈ Mon [n] ⇒ graph G = ([n];ϕ1, . . . , ϕd)

For S ⊂ [n], let

ΓG(S) = {y ∈ [n]; ∃x ∈ S, ∃i = 1, . . . , d with y = ϕi(x)}

vertex expansion coefficient

µ(G) = minS⊂[n]|S|≤n

2

|ΓG(S)\S||S|

50

ϕ ∈ Mon [n] ⇒ linear map Tϕ on Fn

Tϕ(∑

xiei

)

=∑

xieϕ(i)

Theorem. (D-S)

If ([n];ϕ1, . . . , ϕd) is an expander graph, then

(Tϕ1, . . . , Tϕd) is dimensional expander

Construction for d ∼ logn with monotone

maps obtained from translation

operators on Z/nZ

Problem. Construct monotone expanders

of bounded degree51

Use of Moebius transformations

Theorem 1. There is c0 > 0 and an explicit

finite family Ψ of smooth increasing

maps ψ : [0,1] → [0,1] s.t. if A ⊂ [0,1]

is a measurable set, 0 < |A| < 12, then

maxψ∈Ψ

|ψ(A)\A| ≥ c0|A|

Maps ψ obtained from Moebius transformations

g =

a bc d

∈ SL2(R) acts on R∪{∞}

g(x) =ax+ b

cx+ dand

g′(x) =1

(cx+ d)2> 0

52

Theorem 2. ∀ε > 0,∃ finite G ⊂ SL2(R), s.t.

‖1 − g‖ < ε for g ∈ Gand

maxg∈G

‖f − f ◦ g‖2 >1

2

if f ∈ L2(R), supp f ⊂ [0,1], ‖f‖2 = 1 and f⊥E

where E is some finite dimensional subspace of L2(R)

ρ = unitary representation of SL2(R) on L2(R)

ρg−1f = (g′)12(f ◦ g).

Since |1− (g)′| < 0(ε) on bounded sets, suffices

to prove⟨∑

gν(g)ρgf, f

⟩

<1

2‖f‖2

where

ν =1

2|G|∑

g∈G(δg + δg−1)

53

Construction of the set G

Lemma. Given ε > 0, there is Q ∈ Z+ and

G ⊂ SL2(R) ∩(1

QMat2(Z)

)

(i) 1ε < Q <

(1ε

)C

(ii) |G| > Qc

(iii) G generates freely free group on |G|generators

(iv) ‖1 − g‖ < ε for g ∈ G

Wk(G) = reduced words of length k

⊂ Q−kMat2(Z)

Non-commutative diophantine property

g ∈Wk(G)\{1} ⇒ ‖1−g‖ > Q−k > |G|−Ck

54

Main Theorem

G = {g1, . . . , gR} ⊂ SL2(R), R large

(1) G generates free group on R elements

(2) g ∈Wk(G)\{1} ⇒ ‖1 − g‖ > R−Ck

(3) ‖g‖ < 2 for g ∈ G

ρ = representation of SL2(R) by Moebius

transformation

Then∥∥∥∥

∑

g∈G(ρgf+ρg−1f)

∥∥∥∥2< R−κ κ = κ(C)

if f ∈ L2(R), ‖f‖2 = 1, supp f ⊂ [0,1] and

f⊥E

(for some finite dimensional space E)

55