QUANTUM GIBBS SAMPLERS: THE COMMUTATIVE CASE M. J. Kastoryano, F. - - PowerPoint PPT Presentation
QUANTUM GIBBS SAMPLERS: THE COMMUTATIVE CASE M. J. Kastoryano, F. G. S. L. Brandao QIP 2015, Sydney Tuesday, February 10, 15 MOTIVATION Simulation of systems in Analysis of thermal equilibrium thermalization in nature Can we say anything
Tuesday, February 10, 15
Can we say anything about the difficulty of simulating a state, just from the state? Does nature always prepare “easy states” efficiently?
Tuesday, February 10, 15
Tuesday, February 10, 15
Finite lattice system
Λ
Finite local dimension Bounded, local and commuting Hamiltonian
[hZ, hY ] = 0, ∀Z, Y
HA = X
Z∈A
hZ
A ⊂ Λ
A
Non-commutative spaces:
ρ ∝ e−βHΛ
Global Gibbs state:
hf, giρ = tr[ρ1/2f †ρ1/2g]
||f||p
p,ρ = tr[|ρ1/2pfρ1/2p|p]
Lp inner product
norm
Lp Lp
Def: Gibbs samplers are primitive semigroups with Gibbs state as unique stationary state
hX
Tuesday, February 10, 15
Davies generators
LA(f) = X
α(j),j∈A,ω
gα(j)(ω)(Sα(j)(ω)fS†
α(j)(ω) − 1
2{Sα(j)(ω)S†
α(j)(ω)})
Finite system weakly coupled to a markovian thermal bath
A ⊂ Λ
Bath autocorolation fcn Jump operators: between eigenstates of H Properties: Locally reversible: Completely positive
hf, LA(g)iρ = hLA(f), giρ
S B, β
Local (same locality as H)
Tuesday, February 10, 15
Heat-bath generators local projection onto Gibbs state
LA(f) = X
k∈A
Eρ
k(f) − f
γk = (trk[ρ])−1/2ρ1/2
Eρ
k(f) = trk[γkfγ† k]
Only depends on properties
is a conditional expectation
Eρ
A
Properties: Locally reversible: Completely positive
hf, LA(g)iρ = hLA(f), giρ
Local (same locality as H)
Tuesday, February 10, 15
VarA = ||f − Eρ
A(f)||2 2,ρ
λA = inf
f∈AΛ
hf, LA(f)iρ VarA(f)
We want to estimate how rapidly the sampler converges to the Gibbs state Trace norm bound: Mixing time:
||etL() − ⇢||1 ≤ ✏
⌧ ≥ log(||⇢−1||/✏)
τ ∝ |Λ|/λΛ
Reduces to estimating the gap! where
Tuesday, February 10, 15
Def: weak clustering
Cov(f, g) ≤ c||f||2,ρ||g||2,ρe−d(Σf ,Σg)/ξ
Σf
Σg
Λ
d(Σf, Σg)
Cov(f, g) = hf hfiρ, g hgiρiρ
Def: strong clustering
CovA∪B(EA(f), EB(f)) ≤ c||f||2
2,ρe−w(A∩B)/ξ
CovA∪B(f, g) = hf EA(f), g EB(g)iρ
Λ
A ∩ B
Ac
Bc
w(A ∩ B)
different norm
Tuesday, February 10, 15
weak clustering local indistinguishability strong clustering Equivalence breaks down for quantum systems!
Λ A A
Tuesday, February 10, 15
Tuesday, February 10, 15
Prop: If the Gibbs state satisfies strong clustering, then
CovA∪B(EA(f), EB(f)) ≤ ✏||f||2
2,ρ
Assume
w(A ∩ B) ≈ √ L
w(A) ≈ w(B) ≈ L VarA∪B(f) ≤ (1 + ✏)(VarA(f) + VarB(f)) (1 + ✏)(−1
A hf, LA(f)iρ + −1 B hf, LB(f)iρ)
(1 + ✏)−1
A,B(hf, LA∪B(f)iρ + hf, LA∩B(f)iρ)
can eliminate this term by averaging Thus we get:
λ(2L) ≈ λ(L)
since
VarA∪B(f) ≤ (1 + ✏)(VarA(f) + VarB(f))
✏ ≤ ce−
√ L/ξ
applying iteratively completes the proof
Λ
A ∩ B
Ac
Bc
w(A ∩ B)
Tuesday, February 10, 15
Map Liouvillian onto FF Hamiltonian Use the detectability lemma Can invoke the theory of FF gaped Hamiltonians By constructing an approximate projector
Πl ≈ E = EinEout
it is not difficult to show that
||ˆ EAˆ EB − ˆ EA∪B|| ≤ e−lλ/ξ
Tuesday, February 10, 15
Tuesday, February 10, 15
Boundaries can be removed in 1D One can use MPS methods in 1D
Note: cannot use Araki’s result!
Tuesday, February 10, 15
Tuesday, February 10, 15
Tuesday, February 10, 15