a quantum local lemma

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A Quantum Local Lemma Or Sattath TAU and HUJI Joint work with Andris Ambainis Julia Kempe

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A Quantum Local Lemma. Or Sattath TAU and HUJI. Joint work with Andris Ambainis Julia Kempe. Outline. The classical Lovász Local Lemma Motivation in the quantum case A quantum local lemma Application to random QSAT. The Lovász Local Lemma. - PowerPoint PPT Presentation

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Page 1: A Quantum Local Lemma

A Quantum Local Lemma

Or SattathTAU and HUJI

Joint work with Andris Ambainis

Julia Kempe

Page 2: A Quantum Local Lemma

• The classical Lovász Local Lemma

• Motivation in the quantum case

• A quantum local lemma

• Application to random QSAT

Outline

Page 3: A Quantum Local Lemma

If a large number of events are all independent, then there is a positive (small) probability that none of them occurs.

I.e.: If each of m events occurs with probability at most p<1 then

Pr[no events occur] ≥ (1-p)m >0.

The Lovász Local Lemma

But what if the events are “weakly” dependent?

Page 4: A Quantum Local Lemma

Given a k-SAT formula where each of the m clauses shares a variable with at most d other clauses.

Example: sparse k-SAT

Page 5: A Quantum Local Lemma

Given a k-SAT formula where each of the m clauses shares a variable with at most d other clause.

In a random assignment each clause is violated with probability p=2-k.

These events are independent.A random assignment satisfies with

probability (1-2-k)m >0. is satisfiable.

Example: sparse k-SATreally stupid

anyno

Page 6: A Quantum Local Lemma

Given a k-SAT formula where each of the clauses shares a variable with at most d other clauses.

In a random assignment each clause is violated with probability p=2-k.

However, these events are not independent.

Corollary of LLL [ErdősLovász75]: If then a random assignment satisfies with probability >0.

is satisfiable.

Example: sparse k-SAT

Page 7: A Quantum Local Lemma

Corollary: A k-SAT formula where each variable appears in at most 2k/(ek)

clauses is always satisfiable.

Example: sparse k-SAT

Page 8: A Quantum Local Lemma

LLL [ErdősLovász75]: Let B1,…,Bn be events with Pr[Bi] ≤ p and s.t. each event is independent of all but d of the others.

If then there is a non-zero probability that none of them occur.

The Lovász Local Lemma

Page 9: A Quantum Local Lemma

• The classical Lovász Local Lemma

• Motivation in the quantum case

• A quantum local lemma

• Application to random QSAT

Outline

Page 10: A Quantum Local Lemma

A classical bit can be either “0” or “1”.A quantum bit (qubit), can be in either |0 or |

1, or a linear combination:

A qubit is a vector in a 2 dimensional vector space. |0 and |1 form an orthonormal basis for this vector space.

Quantum bit

ba

1b0av

Page 11: A Quantum Local Lemma

n classicals bits can be either “00…0”, “00…1”,… or “11…1”.

n qubit state can be in either |00…0, |00…1,…, |11…1, or some linear combination:

|v=a00…0|00…0+a00…1|00…1+…+a11…1|11…1.

n qubit state is a vector in a 2n dimensional vector space. |0…0,…,|1…1 form an orthonormal basis for this vector space.

n qubits

Page 12: A Quantum Local Lemma

k-SAT: each clause excludes 1 configuration out of the 2k possible configurations.

k-QSAT[Bravyi06]: each quantum clause excludes one dimensional subspace out of 2k dimensions of the involved qubits.

Clauses Rank-1 Projectors

Quantum SAT

0, vvv vv

Satisfying State

Excluded 2n-k dimensional subspace

Page 13: A Quantum Local Lemma

QSAT generalizes SAT:

is satisfiable iff is satisfiable.

The state |0011 is a satisfying state:

|0011=0, |0011=0

QSAT example

)()( 432321 xxxxxx

000123 ,Π 111

234{ }

123000 234

111

Page 14: A Quantum Local Lemma

Formal Def: (k-QSAT)Given a collection of k-local rank-1 projectors

on n qubits,

Is there a state |s inside the allowed subspace of 0 for i=1..m.

Importance: known to be QMA1-Complete (quantum analogue of NP) , for k≥4[Bravyi06].

Quantum SAT

v1,...,Π vm

v i

Page 15: A Quantum Local Lemma

If each projector acts on a set of mutually disjoint qubits,

then |s= |s1…|sm is a satisfying state.

But what if each qubit appears in a few projections?

Quantum SAT

∀i Π visi = 0

Page 16: A Quantum Local Lemma

If each projector excludes a p-fraction of the space and shares a qubit with at most d other projectors, then the k-QSAT instance is satisfiable as long as

Or:Let I be an instance of k-QSAT. If each qubit appears in at most 2k/(ek)

projectors, then I is satisfiable.

A statement we would like

Page 17: A Quantum Local Lemma

Correspondences:Probability space: Vector space VEvents: Subspaces X VProbabilities Pr: relative dimension R

Conditional Probability Pr(X|Y):

Independence: X, Y are R-independent if R(X|Y)=R(X) (equivalently R(XY)=R(X)R(Y) )

Events and independence

Page 18: A Quantum Local Lemma

Properties or R:1) 0 ≤R(X)≤12) XY R(X)≤R(Y)3) Chain rule:

4) “Inclusion/Exclusion”: Let X+Y={x+y|xX,yY}

So far complete analogy to classical probability.

Properties of relative dim

Page 19: A Quantum Local Lemma

Properties of classical Pr:Let Xc be the complement of X.Then Pr(X)+Pr(Xc)=1 and Pr(X|Y)+Pr(Xc|Y)=1

(needed in proof of LLL).This is not true for R. We can define a

“complement”: Xc=X=subspace orthogonal to X

R(X)+R(X)=1 but we only have R(X|Y)+R(X|Y)≤1 !

The complement

Page 20: A Quantum Local Lemma

Example: R(X|Y)+R(X|Y)<1

The complement

XY Xc

R(X|Y)=R(X|Y)=0

Page 21: A Quantum Local Lemma

Classically, if X and Y are independent, then Xc and Y are also independent.

For relative dimension this is wrong!

Care needed in the formulation of the Local Lemma.

The complement

Page 22: A Quantum Local Lemma

QLLL: Let X1,…,Xm be subspaces of V with R(Xi)≥1-p and such that Xi is R-independent of all but at most d of the others.

If then

In particular

Proof: Use properties of R, especially chain-rule and inclusion-exclusion. Induction.

A quantum local lemma

Page 23: A Quantum Local Lemma

Corollary of QLLL: Let 1,…,m be k-local projectors on n qubits s.t. each qubit appears in at most 2k/(ek) projectors. Then there is a state satisfying all i.

We show: If i and j do not share a qubit, then their satisfying subspaces Xi and Xj are R-independent.

Sparse QSAT

Page 24: A Quantum Local Lemma

Corollary of QLLL: Let 1,…,m be k-local projectors on n qubits s.t. each qubit appears in at most 2k/(ek) projectors. Then there is a state satisfying all i.

Proof: Xi=satisfying subspace for i. Then R(Xi)=1-2-k, i.e. p=2-k. Each Xi is R-dependent only on d=2k/e-1 others (d+1)pe1.

Sparse QSAT

Page 25: A Quantum Local Lemma

• The classical Lovász Local Lemma

• Motivation in the quantum case

• A quantum local lemma

• Application to random QSAT

Outline

Page 26: A Quantum Local Lemma

Classically: Properties of random k-SAT formulas have been studied in order to understand easy and hard instances as a function of clause density ( = #clauses/#variables).

Generating random k-SAT on n variables:For i=1,…,m=n• Pick a random set of k variables (random

hyperedge – Gk(n,m) model )• Negate each variable with probability ½.

Random SAT

Page 27: A Quantum Local Lemma

Threshold phenomenon[Friedgut99]: For every k, there exists c(k) such that

Random-k-SAT Threshold

)(if0

)(if1)esatisfiablisPr(

kk

c

c

n

Page 28: A Quantum Local Lemma

Results:• Various properties [KS94,MPZ02,MMZ05].• c(2)=1 [CR92,Goe92]• 3.52 ≤ c(3)≤ 4.49 [KKL03,HS03]• 2kln2-O(k) ≤ c(k)≤ 2kln2 [AP04]

What about k-QSAT?

Random SAT and QSAT

Page 29: A Quantum Local Lemma

A random k-QSAT on n qubits is constructed as follows:

For i=1,…,m =n :• Pick a random set of k qubits (random

hyperedge – Gk(n,m) model)• Pick a uniformly random k-qubit state |vi

on those k qubits and exclude it.

Random k-QSAT

Page 30: A Quantum Local Lemma

[LaumannMSS09,Bravyi07]:Threshold at density ½

The satisfying states in the satisfiable phase are tensor product states.

2-QSAT is fully understood.

The case k=2

Page 31: A Quantum Local Lemma

Lower bound [LaumannLMSS09] :“Matching condition”: if there is a

matching between clauses and qubits contained in a clause, there is a satisfying product state

Random k-QSAT at k≥3

-clauses (projectors)-qubits

Page 32: A Quantum Local Lemma

Random k-QSAT at k≥31

2 34

5

1

23 4

5 6

Random left-k-regular

Matching condition there is a left matching in G.

True w.h.p for random graphs if m≤rkn , i.e. density ≤ rk 1

1

234

5

1

2

3

4

5

6Clauses Projectors

Page 33: A Quantum Local Lemma

Lower bound: [LaumannLMSS09]As long as density <1 there is a satisfiable

product state. Nothing was known about non-

product states or above density 1.Upper bound: [BravyiMooreRussell09]For k=3: critical density <3.549…For k≥4: critical density <2k0.573... Large gap between lower bound 1 and

upper bounds.

Random k-QSAT at k≥3

Page 34: A Quantum Local Lemma

Remark: Let Gk(n,d) be a random k-uniform hypergraph of fixed degree d.

Matching dk. By [LaumannLMSS09], d≤k satisfiable product state. Nothing was known for d>k.

Random k-QSAT at k≥3

deg ddeg =k

Page 35: A Quantum Local Lemma

Remark: Let Gk(n,d) be a random k-uniform hypergraph of fixed degree d.

Matching dk. By [LaumannLMSS09], d≤k satisfiable product state. Corollary of QLLL: If d ≤2k/(ek) there is a satisfying state.

[LaumannLMSS09] conjecture that there is no satisfying product state above degree d=k.

Would show that QLLL can deal with entangled satisfying states.

Random k-QSAT at k≥3

Page 36: A Quantum Local Lemma

What about Gk(n,m) (random hypergraph with n vertices and m hyperedges)?

Problem: QLLL can deal with degree up to 2k/(ek). But Gk(n,m) of average degree 2k/(ek) will have some vertices with much higher degree.

Degrees are Poisson distributed.

Random k-QSAT and QLLL

Page 37: A Quantum Local Lemma

Theorem [using QLLL]: Gk(n,m) of density c2k/k2 has a satisfying groundstate with high probability.

Random k-QSAT and QLLL

Page 38: A Quantum Local Lemma

satisfiable unsatisfiable

c2k/k2

QLLL

ln2 2k

classical thresholdfor large k

1

[LaumannLMSS09]

Product states 0.573 2k

[BravyiMooreRussell09]

Entangled states suspected

clause density

Random k-QSAT and QLLL

Page 39: A Quantum Local Lemma

Theorem [using QLLL]: Gk(n,m) of density c2k/k2 has a satisfying groundstate with high probability.

Idea: hybrid approach – split the graph into two parts: a high degree part H and a low degree part L.

Random k-QSAT and QLLL

L

H

Page 40: A Quantum Local Lemma

Gluing Lemma: If the vertices of the hypergraph G can be partitioned into H and L s.t.:1) All vertices in L have a degree somewhat below the QLLL threshold.2) All edges that involve only H can be satisfied.3) All edges that involve both H and L have the form: .

Then there is a satisfying assignment for G.

Gluing Lemma

H

L

Page 41: A Quantum Local Lemma

Proof sketch: We know there is a satisfying state for all edges that involve H.

If edge e involves both L and H:

Define two new projectors on qubits 2,3,4.

Any state on qubits 2,3,4 orthogonal to |0 and |1 will be orthogonal to | effectively decoupled H and L

Apply QLLL to qubits 2,3,4 with 2 new constraints.

Random QSAT and QLLL

constraint ||

12 34

10,

Page 42: A Quantum Local Lemma

L

HH

Constructing the partition H and L:Random QSAT and QLLL

H

We show w.h.p. |H| is small.This is enoguh.Intuition: Smaller sets have smaller density H density becomes much smaller than 1.Þ w.h.p. it has a matchingÞ H is satisfiable.

Page 43: A Quantum Local Lemma

• Lovász Local Lemma generalizes to the geometric/quantum setting.

• Allows making statements about satisfiability of (sparse) QSAT. We avoid the “tensor product structure”, by using the probabilistic method!

• Allows to improve lower bounds on threshold for random k-QSAT and to deal with entangled satisfying states.

Notable points

Page 44: A Quantum Local Lemma

QLLL is a geometric statement about subspaces: are there any other applications?

Finding the satisfying state? Recent breakthrough by [Moser09] gives efficient algorithm to find it classically. Essentially Walk-SAT.

Is there a generalization of Moser’s algorithm to the quantum case?

Open Questions

Page 45: A Quantum Local Lemma

Thank you!

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