Markov Entropy Approximation Scheme · 2011-07-22 · Markov Entropy Approximation Scheme David Poulin Département de Physique Université de Sherbrooke Joint work with: Matt Leifer,

Markov Entropy Approximation Scheme

David Poulin

Département de PhysiqueUniversité de Sherbrooke

Joint work with: Matt Leifer, Ersen Bilgin, and Matt Hastings

Quantum Computation and Quantum Spin SystemsVienna, August 2009

David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 1 / 33

Faculty position @ SherbrookeCanadian Excellence Research Chair

Solid state information processingExperimentalists and Theorists

$ 10 million Research Grant≥ 300m2 of new lab or office space in construction


TaskCompute thermal properties of spin systems on lattices.

Origin of the method

Refinement of quantum generalized belief propagation.

ResultAlgorithm to produce lower bounds to free energy by solving finite-sizeconvex optimization problem with linear constraints.

Why is this interesting?

Variational methods (PEPS, MPS, MERA) provide UPPER bounds tofree energy.























Outline

1 Belief propagation

2 Quantum belief propagation

3 Markov Entropy Approximation Scheme

4 Closing


Belief propagation

Outline




4 Closing


Belief propagation

Description of the algorithm

PurposeApproximate the solution to statistical inference problems involvinga large number of random variables v (spins) located at thevertices v ∈ V of a graph G = (V ,E).The graph’s edges (u, v) ∈ E encode some king of directdependency relation of the random variables.

Algorithm architectureHighly parallelizable: one processor per vertex.Messages exchanged between processors related by an edge.Outgoing messages at v depend on local "fields" and incommingmessages.Exact when G is a tree and complexity = depth(G).Good heuristic on loopy graphs.


Belief propagation





Belief propagation





Belief propagation





Belief propagation





Belief propagation





Belief propagation





Belief propagation

Transfer matrix

Consider the 1d classical system with hamiltonianH =

∑i h(vi) +

∑〈ij〉 J(vi , vj).

Its Gibbs distribution is (µ(v) = e−βh(v) and ν(u : v) = e−βJ(u,v))

ρ(v1, v2, . . .) =1Z

e−βH(v1,v2,...)

=1Zµ(v1)ν(v1, v2)µ(v2)ν(v2, v3)µ(v3) . . .

So the reduced state of spin N can be evaluated step by step:

m1→2(v2) =∑v1

µ(v1)ν(v1 : v2)

m2→3(v3) =∑v2

mv1→v2(v2)µ(v2)ν(v2 : v3)

m3→4(v4) =∑v3

mv2→v3(v3)µ(v3)ν(v3 : v4)...

ρ(vN) = mvN−1→vN (vN)µ(vN)


Belief propagation

Transfer matrix


∑i h(vi) +



ρ(v1, v2, . . .) =1Z

e−βH(v1,v2,...)

=1Zµ(v1)ν(v1, v2)µ(v2)ν(v2, v3)µ(v3) . . .


m1→2(v2) =∑v1

µ(v1)ν(v1 : v2)

m2→3(v3) =∑v2

mv1→v2(v2)µ(v2)ν(v2 : v3)

m3→4(v4) =∑v3

mv2→v3(v3)µ(v3)ν(v3 : v4)...



Belief propagation

Transfer matrix


∑i h(vi) +



ρ(v1, v2, . . .) =1Z

e−βH(v1,v2,...)

=1Zµ(v1)ν(v1, v2)µ(v2)ν(v2, v3)µ(v3) . . .


m1→2(v2) =∑v1

µ(v1)ν(v1 : v2)

m2→3(v3) =∑v2

mv1→v2(v2)µ(v2)ν(v2 : v3)

m3→4(v4) =∑v3

mv2→v3(v3)µ(v3)ν(v3 : v4)...



Belief propagation

Transfer matrix


∑i h(vi) +



ρ(v1, v2, . . .) =1Z

e−βH(v1,v2,...)

=1Zµ(v1)ν(v1, v2)µ(v2)ν(v2, v3)µ(v3) . . .


m1→2(v2) =∑v1

µ(v1)ν(v1 : v2)

m2→3(v3) =∑v2

mv1→v2(v2)µ(v2)ν(v2 : v3)

m3→4(v4) =∑v3

mv2→v3(v3)µ(v3)ν(v3 : v4)...



Belief propagation

Transfer matrix


∑i h(vi) +



ρ(v1, v2, . . .) =1Z

e−βH(v1,v2,...)

=1Zµ(v1)ν(v1, v2)µ(v2)ν(v2, v3)µ(v3) . . .


m1→2(v2) =∑v1

µ(v1)ν(v1 : v2)

m2→3(v3) =∑v2

mv1→v2(v2)µ(v2)ν(v2 : v3)

m3→4(v4) =∑v3

mv2→v3(v3)µ(v3)ν(v3 : v4)...



Belief propagation

Transfer matrix


∑i h(vi) +



ρ(v1, v2, . . .) =1Z

e−βH(v1,v2,...)

=1Zµ(v1)ν(v1, v2)µ(v2)ν(v2, v3)µ(v3) . . .


m1→2(v2) =∑v1

µ(v1)ν(v1 : v2)

m2→3(v3) =∑v2

mv1→v2(v2)µ(v2)ν(v2 : v3)

m3→4(v4) =∑v3

mv2→v3(v3)µ(v3)ν(v3 : v4)...



Belief propagation

Transfer matrix


∑i h(vi) +



ρ(v1, v2, . . .) =1Z

e−βH(v1,v2,...)

=1Zµ(v1)ν(v1, v2)µ(v2)ν(v2, v3)µ(v3) . . .


m1→2(v2) =∑v1

µ(v1)ν(v1 : v2)

m2→3(v3) =∑v2

mv1→v2(v2)µ(v2)ν(v2 : v3)

m3→4(v4) =∑v3

mv2→v3(v3)µ(v3)ν(v3 : v4)...



Belief propagation

General setting

Graph G = (V ,E).Random variables vi for i ∈ V .Functions µi(vi) for i ∈ V and νij(vi : vj) for (i , j) ∈ E .

You can think of them as exponentiels of local fields µ(v) = e−βh(v)

and couplings ν(u, v) = e−βJ(u,v).

Task (basic case)Evaluate

ρk (vk ) =∑{vi}−vk

∏i∈V

µi(vi)∏ij∈E

νij(vi : vj)

=∑{vi}−vk

e−β(P

i∈V hi (vi )+P

i,j∈E Jij (vi ,vj ))


Belief propagation

General setting






∏i∈V

µi(vi)∏ij∈E

νij(vi : vj)

=∑{vi}−vk

e−β(P

i∈V hi (vi )+P



Belief propagation

General setting






∏i∈V

µi(vi)∏ij∈E

νij(vi : vj)

=∑{vi}−vk

e−β(P

i∈V hi (vi )+P



Belief propagation

General setting






∏i∈V

µi(vi)∏ij∈E

νij(vi : vj)

=∑{vi}−vk

e−β(P

i∈V hi (vi )+P



Belief propagation

General setting






∏i∈V

µi(vi)∏ij∈E

νij(vi : vj)

=∑{vi}−vk

e−β(P

i∈V hi (vi )+P



Belief propagation

Belief propagation algorithm

Algorithm

Initialization mu→v (v) = cte.Iterations mu→v (v) ∝∑u µ(u)ν(u : v)

∏v ′∈n(u)−v mv ′→u(u).

Beliefs b(u) ∝ µ(u)∏

v∈n(u) mv→u(u).b(u, v) ∝ µ(u)µ(v)ν(u : v)

∏w∈n(u)−v mw→u(u)

∏w∈n(v)−u mw→v (v).


Belief propagation


Algorithm


∏v ′∈n(u)−v mv ′→u(u).

... u v

a

b

g

ma!

u

mg!u

mb!u

µu!(u : v)mu!v






Belief propagation


Algorithm


∏v ′∈n(u)−v mv ′→u(u).

... u v

a

b

g

ma!

u

mg!u

mb!u

µu!(u : v)mu!v






Belief propagation


Algorithm


∏v ′∈n(u)−v mv ′→u(u).

... u v

a

b

g

ma!

u

mg!u

mb!u

µu!(u : v)mu!v






Belief propagation

Bethe free energy

Let H =∑

v hv +∑〈u,v〉 Juv be a local Hamiltonian on G.

Given a probability P(V ), the Gibbs free energy G = E − TSwhere

E =∑

V

P(V )H(V ) and S = −∑

V

P(V ) log P(V )

The Gibbs distribution P(V ) = 1Z e−βH is the stationary point of G.

E only depends on the two-body probabilities

E =∑〈u,v〉

P(u, v)Juv +∑

u

P(u)hu

So does S if P is the product of local functions on a tree (whichincludes Gibbs distributions)

S = −∑〈u,v〉

P(u, v) log P(u, v)−∑

u

(1− dv )P(u) log P(u)


Belief propagation

Bethe free energy

Let H =∑



E =∑

V

P(V )H(V ) and S = −∑

V

P(V ) log P(V )



E =∑〈u,v〉

P(u, v)Juv +∑

u

P(u)hu


S = −∑〈u,v〉


u



Belief propagation

Bethe free energy

Let H =∑



E =∑

V

P(V )H(V ) and S = −∑

V

P(V ) log P(V )



E =∑〈u,v〉

P(u, v)Juv +∑

u

P(u)hu


S = −∑〈u,v〉


u



Belief propagation

Bethe free energy

Let H =∑



E =∑

V

P(V )H(V ) and S = −∑

V

P(V ) log P(V )



E =∑〈u,v〉

P(u, v)Juv +∑

u

P(u)hu


S = −∑〈u,v〉


u



Belief propagation

Bethe free energy

Let H =∑



E =∑

V

P(V )H(V ) and S = −∑

V

P(V ) log P(V )



E =∑〈u,v〉

P(u, v)Juv +∑

u

P(u)hu


S = −∑〈u,v〉


u



Belief propagation

Bethe free energy

The Bethe free energy is extending this expression to arbitrarygraphs

GBethe =∑〈u,v〉

P(u, v)(T log P(u, v)+Juv )+∑

u

(1−du)P(u)(T log P(u)+hu).

This is seemingly easier to handle because it involves onlytwo-body distributions.


Belief propagation

Bethe free energy

The Bethe free energy is extending this expression to arbitrarygraphs

GBethe =∑〈u,v〉

P(u, v)(T log P(u, v)+Juv )+∑

u

(1−du)P(u)(T log P(u)+hu).

This is seemingly easier to handle because it involves onlytwo-body distributions.


Belief propagation

Bethe free energy and BP

Theorem (Yedidia, Freeman, and Weiss)

The fixed point of the belief propagation algorithms b(u, v) and b(u)are stationary points of the Bethe free energy.

Use BP to solve Bethe free energy minimization.Exact on tree, complexity = depth(G).Most successful on trees with only large loops:

LDPC and Turbo codes.Spin glasses on Bethe lattices.Random k -SAT assignments.


Belief propagation







Belief propagation







Belief propagation







Belief propagation







Belief propagation

Generalized belief propagation

When the graph contains small loops, “collapse" them into asingle super-site.Messages exchanged between (possibly overlapping) regions.

If clustered graph is a tree, method is exact.


Belief propagation





Belief propagation





Belief propagation





Belief propagation





Belief propagation

Kikuchi free energy

Fix a set of regions on the graph.Approximate the entropy by the sum of the entropy of the regions.Correct for the over counting of the sub-regions.Correct for the sub-sub-regions ...

1 2 3

4 5 6

7 8 9

Corresponds to Bethe’s free energy when regions are nearestneighbor pairs.


Belief propagation

Kikuchi free energy


1 2 3

4 5 6

7 8 9

S = S1245 + S2356 + S478 + S5689



Belief propagation

Kikuchi free energy


1 2 3

4 5 6

7 8 9

S = S1245 + S2356 + S478 + S5689

!S4 ! S8 ! S25 ! S56



Belief propagation

Kikuchi free energy


1 2 3

4 5 6

7 8 9

S = S1245 + S2356 + S478 + S5689

!S4 ! S8 ! S25 ! S56



Belief propagation

Kikuchi free energy


1 2 3

4 5 6

7 8 9

S = S1245 + S2356 + S478 + S5689

!S4 ! S8 ! S25 ! S56



Belief propagation

Kikuchi free energy and generalized BP

Theorem (Yedidia, Freeman, and Weiss)The fixed point of the generalized belief propagation algorithms arestationary points of the Kikushi free energy computed using the sameregions.a

aTechnical detail: the counting number of each sub-region must be non-positive.

Greatly improves numerical precision for graphs containing smallloops.Can use GBP instead of minimizing Kikuchi free energy.


Belief propagation






Belief propagation






Quantum belief propagation

Outline




4 Closing



The algorithm

Defining the log-exp product A� B = elog A+log B, the quantum andclassical setting become very similar.

Let G = (V ,E) be a graph with Hamiltonian H =∑

hi +∑

Jij .Defining µi = e−βhi , νij = e−βJij , the Gibbs state is

ρV =1Z

(⊗i∈V

µi

)�( ⊙

(i,j)∈E

νij

)We can execute BP by substituting ordinary products for �.

TheoremIf G is a tree and (G, ρV ) is a quantum Markov random field, then thebeliefs bu converge to the correct marginals ρu = TrV−u{ρV} in a timeproportional to depth(G).

This doesn’t happen very often, but similar conditions forgeneralized BP are more natural.



The algorithm



hi +∑


ρV =1Z

(⊗i∈V

µi

)�( ⊙

(i,j)∈E

νij






The algorithm



hi +∑


ρV =1Z

(⊗i∈V

µi

)�( ⊙

(i,j)∈E

νij






The algorithm



hi +∑


ρV =1Z

(⊗i∈V

µi

)�( ⊙

(i,j)∈E

νij






The algorithm



hi +∑


ρV =1Z

(⊗i∈V

µi

)�( ⊙

(i,j)∈E

νij






The algorithm



hi +∑


ρV =1Z

(⊗i∈V

µi

)�( ⊙

(i,j)∈E

νij






Effective thermal hamiltonian

-5 -4 -3 -2 -1 0 1 2 3 4 5... ...

! =1Z

exp{!"H}H =!!

i="!hi + Ji,i+1

Effective Thermal Hamiltonian =∑∞

i=1 hi + Ji,i+1 +V1 +V2+V3+V4 . . .




-5 -4 -3 -2 -1 0 1 2 3 4 5... ...

! =1Z

exp{!"H}


i=1 hi + Ji,i+1 +V1 +V2+V3+V4 . . .




-5 -4 -3 -2 -1 0 1 2 3 4 5... ...

!! = Tr"#..."1{!} ! =1Z

exp{!"H}


i=1 hi + Ji,i+1 +V1 +V2+V3+V4 . . .




-5 -4 -3 -2 -1 0 1 2 3 4 5... ...

!! = Tr"#..."1{!} !! =1Z ! exp{!"He!}


i=1 hi + Ji,i+1 +V1 +V2+V3+V4 . . .




-5 -4 -3 -2 -1 0 1 2 3 4 5... ...

!! = Tr"#..."1{!} !! =1Z ! exp{!"He!}


i=1 hi + Ji,i+1 +V1 +V2+V3+V4 . . .

10 20 30 40 50 60 70 80 90 10010!12

10!10

10!8

10!6

10!4

10!2

100

!!z 0!

z j"

j

! = 20

! = 1




-5 -4 -3 -2 -1 0 1 2 3 4 5... ...

!! = Tr"#..."1{!} !! =1Z ! exp{!"He!}


i=1 hi + Ji,i+1 +V1 +V2+V3+V4 . . .

10 20 30 40 50 60 70 80 90 10010!12

10!10

10!8

10!6

10!4

10!2

100

!!z 0!

z j"

j

! = 20

! = 1




-5 -4 -3 -2 -1 0 1 2 3 4 5... ...

!! = Tr"#..."1{!} !! =1Z ! exp{!"He!}


i=1 hi + Ji,i+1 +V1 +V2+V3+V4 . . .

10 20 30 40 50 60 70 80 90 10010!12

10!10

10!8

10!6

10!4

10!2

100

!!z 0!

z j"

j

! = 20

! = 1




-5 -4 -3 -2 -1 0 1 2 3 4 5... ...

!! = Tr"#..."1{!} !! =1Z ! exp{!"He!}


i=1 hi + Ji,i+1 +V1 +V2+V3+V4 . . .

10 20 30 40 50 60 70 80 90 10010!12

10!10

10!8

10!6

10!4

10!2

100

!!z 0!

z j"

j

! = 20

! = 1




-5 -4 -3 -2 -1 0 1 2 3 4 5... ...

!! = Tr"#..."1{!} !! =1Z ! exp{!"He!}


i=1 hi + Ji,i+1 +V1 +V2+V3+V4 . . .

10 20 30 40 50 60 70 80 90 10010!12

10!10

10!8

10!6

10!4

10!2

100

!!z 0!

z j"

j

! = 20

! = 1




-5 -4 -3 -2 -1 0 1 2 3 4 5... ...

!! = Tr"#..."1{!} !! =1Z ! exp{!"He!}


i=1 hi + Ji,i+1 +V1 +V2+V3+V4 . . .

10 20 30 40 50 60 70 80 90 10010!12

10!10

10!8

10!6

10!4

10!2

100

!!z 0!

z j"

j

! = 20

! = 1

1 2 3 4 5 610!12

10!10

10!8

10!6

10!4

10!2

100

!Vj! 2

j

! = 1

! = 20



Generalized quantum belief propagation

σ1−4 = e−β(h1+h2+h3+h4+J12+J23+J34)

σ′2−4 = Tr1{σ1−4} h′2−4 = −1β

logσ′2−4

σ2−5 = e−β(h′2−4+h5+J45)

σ′3−5 = Tr2{σ2−5} h′3−5 = −1β

logσ′3−5

σ3−6 = e−β(h′3−5+h6+J56)

...ρN ≈ bN = TrN−3,N−2,N−1{σN−3,N−2,N−1,N}




σ1−4 = e−β(h1+h2+h3+h4+J12+J23+J34)

σ′2−4 = Tr1{σ1−4} h′2−4 = −1β

logσ′2−4

σ2−5 = e−β(h′2−4+h5+J45)

σ′3−5 = Tr2{σ2−5} h′3−5 = −1β

logσ′3−5

σ3−6 = e−β(h′3−5+h6+J56)

...ρN ≈ bN = TrN−3,N−2,N−1{σN−3,N−2,N−1,N}




σ1−4 = e−β(h1+h2+h3+h4+J12+J23+J34)

σ′2−4 = Tr1{σ1−4} h′2−4 = −1β

logσ′2−4

σ2−5 = e−β(h′2−4+h5+J45)

σ′3−5 = Tr2{σ2−5} h′3−5 = −1β

logσ′3−5

σ3−6 = e−β(h′3−5+h6+J56)

...ρN ≈ bN = TrN−3,N−2,N−1{σN−3,N−2,N−1,N}




{!1!4

σ1−4 = e−β(h1+h2+h3+h4+J12+J23+J34)

σ′2−4 = Tr1{σ1−4} h′2−4 = −1β

logσ′2−4

σ2−5 = e−β(h′2−4+h5+J45)

σ′3−5 = Tr2{σ2−5} h′3−5 = −1β

logσ′3−5

σ3−6 = e−β(h′3−5+h6+J56)

...ρN ≈ bN = TrN−3,N−2,N−1{σN−3,N−2,N−1,N}




{

h!2"4

σ1−4 = e−β(h1+h2+h3+h4+J12+J23+J34)

σ′2−4 = Tr1{σ1−4} h′2−4 = −1β

logσ′2−4

σ2−5 = e−β(h′2−4+h5+J45)

σ′3−5 = Tr2{σ2−5} h′3−5 = −1β

logσ′3−5

σ3−6 = e−β(h′3−5+h6+J56)

...ρN ≈ bN = TrN−3,N−2,N−1{σN−3,N−2,N−1,N}




{!2!5

σ1−4 = e−β(h1+h2+h3+h4+J12+J23+J34)

σ′2−4 = Tr1{σ1−4} h′2−4 = −1β

logσ′2−4

σ2−5 = e−β(h′2−4+h5+J45)

σ′3−5 = Tr2{σ2−5} h′3−5 = −1β

logσ′3−5

σ3−6 = e−β(h′3−5+h6+J56)

...ρN ≈ bN = TrN−3,N−2,N−1{σN−3,N−2,N−1,N}




{

h!3"5

σ1−4 = e−β(h1+h2+h3+h4+J12+J23+J34)

σ′2−4 = Tr1{σ1−4} h′2−4 = −1β

logσ′2−4

σ2−5 = e−β(h′2−4+h5+J45)

σ′3−5 = Tr2{σ2−5} h′3−5 = −1β

logσ′3−5

σ3−6 = e−β(h′3−5+h6+J56)

...ρN ≈ bN = TrN−3,N−2,N−1{σN−3,N−2,N−1,N}




{

!3!6

σ1−4 = e−β(h1+h2+h3+h4+J12+J23+J34)

σ′2−4 = Tr1{σ1−4} h′2−4 = −1β

logσ′2−4

σ2−5 = e−β(h′2−4+h5+J45)

σ′3−5 = Tr2{σ2−5} h′3−5 = −1β

logσ′3−5

σ3−6 = e−β(h′3−5+h6+J56)

...ρN ≈ bN = TrN−3,N−2,N−1{σN−3,N−2,N−1,N}




{

!N!3,N!2,N!1,N

σ1−4 = e−β(h1+h2+h3+h4+J12+J23+J34)

σ′2−4 = Tr1{σ1−4} h′2−4 = −1β

logσ′2−4

σ2−5 = e−β(h′2−4+h5+J45)

σ′3−5 = Tr2{σ2−5} h′3−5 = −1β

logσ′3−5

σ3−6 = e−β(h′3−5+h6+J56)

...ρN ≈ bN = TrN−3,N−2,N−1{σN−3,N−2,N−1,N}



When does this work?

... ...A B C

This quantum generalized belief propagation is exact when

I(A : C|B) = S(AB) + S(BC)− S(B)− S(ABC) = 0

This is probably never exactly the case, but as B gets larger it seemsto be a good approximation.



When does this work?

... ...A B C

This quantum generalized belief propagation is exact when

I(A : C|B) = S(AB) + S(BC)− S(B)− S(ABC) = 0

This is probably never exactly the case, but as B gets larger it seemsto be a good approximation.



Results on trees

0 0.5 1 1.5 20

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Bethe ansatzQBP

0.46 0.47 0.48 0.49 0.50.349

0.3492

0.3494

0.3496

0.3498

Spec

ific h

eat

Temperature

Sliding window

Bethe Ansatz

Temperature

Tra

nsvers

e F

ield

Str

ength

Edwards!Anderson Order Parameter on a Cayley Tree with Transverse Ising Hamiltonian

0.2 0.4 0.6 0.8 1 1.2 1.4

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Ising spin glass on Cayley tree

Tran

sver

se fi

eld

Bx

Temperature 1.13

17

Finally, we note that much of the phase diagram is surprisingly stable to variation in Nt.

We have explored various regions of the phase space at Nt = 6, 7, 8, 9, 10, 11. The classical

line (Bt = 0) at all temperatures is completely stable down to Nt = 1 as expected. Perhaps

more surprisingly, moving between Nt = 8 and Nt = 10, qEA is essentially stable below

Bt = 1 down to temperatures ! ! 0.15. Of course, the high field, low temperature part

of the phase transition curve moves downward as the finite discretization asymptote goes

towards the ! = 0 axis. See Figure 6 for the low temperature critical curves estimated using

vertical stripes run at five di!erent temperatures (corresponding to " = 3.5, 4, 4.5, 5, 5.5) at

various Nt.

FIG. 4: (a) Phase diagram at q = 3. The solid phase transition curve has been calculated at

Nt = 10, Nrods = 2500, Niter = 1000Nrods on a fine mesh in the (!, Bt) plane. The vertical dotted

line is the asymptotic critical line for large Bt at Nt = 10 (ie ! = !c/Nt). The points marked x

with error bars indicate Nt " # fits based on Figure 6. The dashed transition curve is a weighted

quadratic fit through the estimated low temperature points and the Nt = 10 points in the range

0.5 < ! < 1. This leads to an estimated Bct = 1.775 ± 0.03. As this fit is clearly heuristic, we

have suggested a much larger range for our estimate of Bct in the Figure. Our phase diagram

clearly disagrees with that of [18], who treat the identical model using a spherical approximation.

The stars and stripes indicate points in the phase space which we have investigated in more detail

below. (b) The average von Neumann entropy of a central spin as a function of (!, Bt).Laumann, Scardicchio, and Sondhi ’07, Bilgin and Poulin.



2D anti-ferromagnetic Heisenberg model

Doesn’t work so well in 2D...

Quantum Monte Calrlo: M.S. Makivic and H.-Q. Ding PRB’91.10th-order J/T expansion.Generalized Quantum Belief propagation, window size 7.



2D anti-ferromagnetic Heisenberg model

Doesn’t work so well in 2D...

!"" #"" $"" %""

!&$

!&'

!&(

)!*

Quantum Monte Calrlo: M.S. Makivic and H.-Q. Ding PRB’91.10th-order J/T expansion.Generalized Quantum Belief propagation, window size 7.



Outline




4 Closing



Entropy is a mess...

F (T ) = minρ{E(ρ)− TS(ρ)} = min

ρ{Tr(ρH) + T · Tr(ρ log ρ)}

Energy can be computed from pieces of ρ:

E(ρ) =∑

i

Tr(ρihi) +∑

ij

Tr(ρijJij)

Hence it is easy to compute, involves small matrices.The same cannot be achieved with entropy.







E(ρ) =∑

i

Tr(ρihi) +∑

ij

Tr(ρijJij)








E(ρ) =∑

i

Tr(ρihi) +∑

ij

Tr(ρijJij)








E(ρ) =∑

i

Tr(ρihi) +∑

ij

Tr(ρijJij)




Breaking entropy apart




1 2 3

N

...




S = SN




S = SN !SN!1 + SN!1




S = SN !SN!1 + SN!1 !SN!2 + SN!2




S = SN !SN!1 + SN!1 !SN!2 + SN!2

=!

j

Sj ! Sj!1

=!

j

S(j|1, 2, ..., j ! 1)




S(j|1, 2, ..., j ! 1)




S(j|1, 2, ..., j ! 1)

! S(j|{1, 2, ..., j " 1} #N (j))

Strong sub-additivity



Bounding the free energy

S∗ =∑

j

S(j |{1,2, ..., j − 1} ∩ N (j)) ≥ S

can be computed from pieces ρP(j) of ρ (P(j) = {1,2, ..., j} ∩ N (j)).Provides approximation for free energy

F (T ) ≥ F ∗(T ) = minρP(j):Consistent

{E({ρP(j)})− TS∗({ρP(j)})

}Consistent means ∃ρ : ρP(j) = TrP(j)(ρ). QMA-complete.Relax constraint to local consistency, i.e.

TrP(j)∩P(k)(ρP(j) − ρP(k)) = 0

F ∗(T ) ≥ F ∗∗(T ) = minρP(j):Local consistent

{E({ρP(j)})− TS∗({ρP(j)})

}Convex optimization with linear constraints.




S∗ =∑

j

S(j |{1,2, ..., j − 1} ∩ N (j)) ≥ S



{E({ρP(j)})− TS∗({ρP(j)})




{E({ρP(j)})− TS∗({ρP(j)})





S∗ =∑

j

S(j |{1,2, ..., j − 1} ∩ N (j)) ≥ S



{E({ρP(j)})− TS∗({ρP(j)})




{E({ρP(j)})− TS∗({ρP(j)})





S∗ =∑

j

S(j |{1,2, ..., j − 1} ∩ N (j)) ≥ S



{E({ρP(j)})− TS∗({ρP(j)})




{E({ρP(j)})− TS∗({ρP(j)})





S∗ =∑

j

S(j |{1,2, ..., j − 1} ∩ N (j)) ≥ S



{E({ρP(j)})− TS∗({ρP(j)})




{E({ρP(j)})− TS∗({ρP(j)})





S∗ =∑

j

S(j |{1,2, ..., j − 1} ∩ N (j)) ≥ S



{E({ρP(j)})− TS∗({ρP(j)})




{E({ρP(j)})− TS∗({ρP(j)})





S∗ =∑

j

S(j |{1,2, ..., j − 1} ∩ N (j)) ≥ S



{E({ρP(j)})− TS∗({ρP(j)})




{E({ρP(j)})− TS∗({ρP(j)})




2D anti-ferromagnetic Heisenberg model revisited

! !"# !"$ !"% !"& ' '"# '"$ '"% '"& #!'"(

!'"#(

!'

!!")(

!!"(

!!"#(

*+,,-./,+012-3.45

./,+012-3.45

./,+012-.6789-:;70</7=;>79;</-$6$-

./,+012-3</9,-?7+=<

*+,,-./,+012-@;0A-B,CD,+79E+,-5,+;,F

./,+012-.6789-:;70</7=;>79;</-G6G

!"#$ !"## !"#% !"#&!!"'$

!!"'!(

!!"'!)

!!"'!'

!!"'!*

!!"'!+

!!"'!&

Temperature (1/J)

E0 ! "0.7062...



Improved finite size effects

MEAS requires 8-site diagonalization and matches 16-sitediagonalization.The constraints force the system to behave as if were part of alarger lattice.



















Positive entropy density constraint

Alternative method to estimate ground energy

Minimize Tr(ρH) on finite lattice with translational constraints on ρ.

Our method is more accurate because it imposes positive entropydensity constraints.The free energy at the the temperature where S∗ goes negative isa rigorous lower bound on E0 in the thermodynamical limit.














Closing

Outline




4 Closing


Closing

Tightening the bound

S∗ > S because it neglects some correlations in the systems:Had we worked with a different patch, we would have neglectedother correlations:pS∗1 + (1− p)S∗2 is a valid upper bound to S.In addition, we can impose cross-constraints that will tighten thebound.

Combine patches that explore multiple length-scales.


Closing





Closing





Closing





Closing





Closing

Dual program

Numerical minimization might fail to find the true minimum of F ∗.In that case our result would not be a rigorous lower bound to thesystem’s free energy.Any solution to the dual program would provide a lower bound tothe minimum of F ∗, and hence to the system’s free energy.


Closing

Dual program



Closing

Dual program



Conclusion

Belief propagation is a powerful heuristic to solve statisticalinference problems involving a large number of random variablesin a Markov Random Field.

Fixed points of BP = Stationary point of Bethe free energy.Generalized belief propagation improves the case where thegraph has small loops.

Fixed points of GBP = Stationary point of Kikuchi free energy.

Quantum generalized BP provides reliable estimates because theeffective thermal hamiltonian is short ranged.Markov Entropy Approximation Scheme uses strong sub-additivityto lower bound the system’s free energy.Linearizing the constraints decreases this bound and yields asolvable problem.Numerical results (simple) interpolate nicely between High Tseries and exact diagonalization of larger lattices.


Conclusion






Conclusion






Conclusion






Conclusion






Conclusion






Conclusion






Conclusion






Documents

Markov Entropy Approximation Scheme · 2011-07-22 · Markov Entropy Approximation Scheme David Poulin Département de Physique Université de Sherbrooke Joint work with: Matt Leifer,