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 about Does nature always the difficulty of simulating a prepare “easy states” state, just from the state? efficiently? Tuesday, February 10, 15

MOTIVATION Main structural theorem: Rate of convergence Correlations in Gibbs state Characterizes the thermodynamically trivial phase Tuesday, February 10, 15

SETTING Finite lattice system h X Λ Finite local dimension Bounded, local and A commuting Hamiltonian X [ h Z , h Y ] = 0 , H A = h Z ∀ Z, Y Z ∈ A A ⊂ Λ ρ ∝ e − β H Λ Global Gibbs state: Non-commutative spaces: L p Def: Gibbs samplers are primitive semigroups with h f, g i ρ = tr[ ρ 1 / 2 f † ρ 1 / 2 g ] L p inner product Gibbs state as unique stationary state norm L p || f || p p, ρ = tr[ | ρ 1 / 2 p f ρ 1 / 2 p | p ] Tuesday, February 10, 15

GIBBS SAMPLERS Davies generators S B, β Finite system weakly coupled to a markovian thermal bath Bath autocorolation fcn α ( j ) ( ω ) − 1 g α ( j ) ( ω )( S α ( j ) ( ω ) fS † 2 { S α ( j ) ( ω ) S † X L A ( f ) = α ( j ) ( ω ) } ) α ( j ) ,j ∈ A, ω A ⊂ Λ Jump operators: between eigenstates of H Properties: Local (same locality as H) Completely positive Locally reversible: h f, L A ( g ) i ρ = h L A ( f ) , g i ρ Tuesday, February 10, 15

GIBBS SAMPLERS Heat-bath generators X E ρ L A ( f ) = k ( f ) − f local projection onto Gibbs k ∈ A state k ( f ) = tr k [ γ k f γ † E ρ k ] E ρ is a conditional expectation γ k = (tr k [ ρ ]) − 1 / 2 ρ 1 / 2 A Only depends on properties of the state. Properties: Completely positive Local (same locality as H) Locally reversible: h f, L A ( g ) i ρ = h L A ( f ) , g i ρ Tuesday, February 10, 15

RELAXATION TIME || e t L ( � ) − ⇢ || 1 ≤ ✏ Trace norm bound: We want to estimate how rapidly the sampler ⌧ ≥ log( || ⇢ − 1 || / ✏ ) Mixing time: converges to the Gibbs state � || ρ − 1 || ≤ e o ( | Λ | ) τ ∝ | Λ | / λ Λ h f, � L A ( f ) i ρ Reduces to estimating the λ A = inf gap! Var A ( f ) f ∈ A Λ A ( f ) || 2 where Var A = || f − E ρ 2 , ρ Tuesday, February 10, 15

CLUSTERING Λ Def: weak clustering Σ f different norm d ( Σ f , Σ g ) Cov( f, g ) ≤ c || f || 2 , ρ || g || 2 , ρ e − d ( Σ f , Σ g ) / ξ Σ g Cov( f, g ) = h f � h f i ρ , g � h g i ρ i ρ Def: strong clustering Λ A ∩ B Cov A ∪ B ( E A ( f ) , E B ( f )) ≤ c || f || 2 2 , ρ e − w ( A ∩ B ) / ξ B c A c w ( A ∩ B ) Cov A ∪ B ( f, g ) = h f � E A ( f ) , g � E B ( g ) i ρ Tuesday, February 10, 15

DLR THEORY (CLASSICAL) Λ A A = weak clustering Equivalence breaks down for quantum systems! local strong indistinguishability clustering Tuesday, February 10, 15

MAIN THEOREM is gapped L satisfies strong clustering L Tuesday, February 10, 15

PROOF OUTLINE Λ Prop: If the Gibbs state satisfies strong clustering, A ∩ B Cov A ∪ B ( E A ( f ) , E B ( f )) ≤ ✏ || f || 2 2 , ρ B c A c then w ( A ∩ B ) Var A ∪ B ( f ) ≤ (1 + ✏ )(Var A ( f ) + Var B ( f )) √ Assume w ( A ) ≈ w ( B ) ≈ L w ( A ∩ B ) ≈ L can eliminate this term by averaging Var A ∪ B ( f ) ≤ (1 + ✏ )(Var A ( f ) + Var B ( f ))  (1 + ✏ )( � − 1 A h f, � L A ( f ) i ρ + � − 1 B h f, � L B ( f ) i ρ )  (1 + ✏ ) � − 1 A,B ( h f, � L A ∪ B ( f ) i ρ + h f, � L A ∩ B ( f ) i ρ ) applying iteratively completes √ L/ ξ Thus we get: since λ (2 L ) ≈ λ ( L ) ✏ ≤ ce − the proof Tuesday, February 10, 15

PROOF OUTLINE Map Liouvillian onto FF Hamiltonian Use the detectability lemma Can invoke the theory of FF gaped Hamiltonians By constructing an Π l ≈ E = E in E out approximate projector E A ∪ B || ≤ e − l λ / ξ || ˆ E A ˆ E B − ˆ it is not difficult to show that Tuesday, February 10, 15

MAIN THEOREM is gapped L satisfies strong clustering L Tuesday, February 10, 15

APPLICATIONS In 1D strong and weak clustering are equivalent Boundaries can be removed in 1D In 1D Gibbs samplers are always gapped One can use MPS methods in 1D Beyond a universal critical temperature Gibbs samplers are gapped Note: cannot use Araki’s result! Tuesday, February 10, 15

OUTLOOK Consider what this means for topological order at non-zero temperature Extend the results to get Log-Sobolev bounds What can we say about the non- commuting case? Tuesday, February 10, 15

THANK YOU FOR YOUR ATTENTION! Tuesday, February 10, 15