Quantum Gibbs Samplers

Fernando G.S.L. BrandãoQuArC, Microsoft Research

Based on joint work with

Michael Kastoryano University of Copenhagen

Quantum Hamiltonian Complexity Reunion WorkshopSimons Institute, Berkeley, 2015

Dynamical Properties

Hij

Hamiltonian:

State at time t:

Compute:

Expectation values:

Temporal correlations:

Quantum Simulators, Dynamical

Quantum Computer Can simulate the dynamics of every multi-particle quantum system

Spin models (Lloyd ‘96, …, Berry, Childs, Kothari ‘15) Fermionic and bosonic models (Bravyi, Kitaev ’00, …) Topological quantum field theory (Freedman, Kitaev, Wang ‘02) Quantum field theory (Jordan, Lee, Preskill ’11, …) Quantum Chemistry (…, Hastings, Wecker, Bauer, Triyer ’14)

Static Properties

Hamiltonian:

Static Properties

Hij

Hamiltonian:

Groundstate:

Thermal state:

Compute: local expectation values (e.g. magnetization), correlation functions (e.g. ), string order

Static Properties

Hij

Static PropertiesCan we prepare thermal states?

Warning: NP-hard to estimate energy of arbitrary classical Gibbs states

Static PropertiesCan we prepare thermal states?

When can we do it efficiently?

Warning: NP-hard to estimate energy of arbitrary classical Gibbs states

Classically: Metropolis SamplingConsider e.g. Ising model:

Coupling to bath modeled by stochastic map Q

The stationary state is the thermal (Gibbs) state:

Metropolis Update:

i j

Glauber Dynamics

A stochastic map R = eG is a Glauber dynamics for a (classical) Hamiltonian if its generator G is local and the unique fixed point of R is e-βH/Z(β) (+ detailed balance) Ex: Metropolis, Heat-bath

Mixing time:

Rapidly mixing 1. tmix < poly(n) 2. tmix < O(n) (constant gap) 3. tmix < O(log(n)) (constant log-Sovolev)

Glauber Dynamics

A stochastic map R = eG is a Glauber dynamics for a (classical) Hamiltonian if its generator G is local and the unique fixed point of R is e-βH/Z(β) (+ detailed balance) Ex: Metropolis, Heat-bath, …

Rapidly mixing Glauber dynamics

Gibbs state with finite correlation

length

Gibbs Sampling in P (vs NP -hard)

(Stroock, Zergalinski ’92; Martinelli, Olivieri ’94, …)

When is Glauber dynamics effective for sampling from Gibbs state?

(Sly ’10, …)

(Proved for hard core model and 2-spin anti- ferromagetic model)

Temporal Mixing

Convergence time bounded by Δ:

Time of equilibration ≈ n/Δ

Detailed balance:

R is self-adjoint with respect to

Spectral gap:

Spatial MixingLet be the Gibbs state for a model in the lattice V with boundary conditions τ, i.e.

blue: V, red: boundary

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

00000

00000

Ex. τ = (0, … 0)

Spatial MixingLet be the Gibbs state for a model in the lattice V with boundary conditions τ, i.e.

blue: V, red: boundary

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

00000

00000

Ex. τ = (0, … 0)def: The Gibbs state has correlation length ξ if for every f, g

fg

Spatial MixingLet be the Gibbs state for a model in the lattice V with boundary conditions τ, i.e.

blue: V, red: boundary

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

00000

00000

Obs1: Equivalent to

Obs2: Equivalent to local indistinguishability (local perturbations perturb locally)

(Stroock, Zergalinski ’92; Martinelli, Olivieri ’94, …) For every classical Hamiltonian, the Gibbs state has finite correlation length if, and only if, the Glauber dynamics has a finite gap

Temporal Mixing <-> Spatial Mixing

Obs1: Same is true for the log-Sobolev constant of the system (convergence in O(log(n)) time)

Obs2: For many models, when correlation length diverges, gap is exponentially small in the system size (e.g. Ising model)

Obs3: Any model in 1D, and any model in arbitrary dim. at high enough temperature, has a finite correlation length(connected to uniqueness of the phase)

(Stroock, Zergalinski ’92; Martinelli, Olivieri ’94, …) For every classical Hamiltonian, the Gibbs state has finite correlation length if, and only if, the Glauber dynamics has a finite gap

Temporal Mixing <-> Spatial Mixing

Obs1: Same is true for the log-Sobolev constant of the system (convergence in O(log(n)) time)

Obs2: For many models, when correlation length diverges, gap is exponentially small in the system size (e.g. Ising model)

Obs3: Any model in 1D, and any model in arbitrary dim. at high enough temperature, has a finite correlation length(connected to uniqueness of the phase)

(Stroock, Zergalinski ’92; Martinelli, Olivieri ’94, …) For every classical Hamiltonian, the Gibbs state has finite correlation length if, and only if, the Glauber dynamics has a finite gap

Temporal Mixing <-> Spatial Mixing

Does something similar hold in the quantum case?

Preparing Quantum Thermal StatesWhat’s the quantum analogue of Glauber dynamics?

(Temme et al ‘09) Quantum metropolis: Quantum channel s.t. (i) can be implemented efficiently on a quantum computer and (ii) has Gibbs state as fixed point But generator non-local; hard to use

Preparing Quantum Thermal StatesWhat’s the quantum analogue of Glauber dynamics? Let

Quantum Heat-Bath:

(Another choice: Davies maps)

Preparing Quantum Thermal StatesWhat’s the quantum analogue of Glauber dynamics? Let

Quantum Heat-Bath:

Generator: semigroup:

Convergence:

For commuting Hamiltonians ( ), Λ is local

only acts on a neighborhood around site i

Preparing Quantum Thermal StatesWhat’s the quantum analogue of Glauber dynamics? Let

Quantum Heat-Bath:

Generator: semigroup:

Convergence:

Question: How fast Mt converges to ρ? Can it we related to decay of correlations in ρ? Next:

1. Quantum algorithm inspired on Λ 2. Fast mixing of Λ vs decay of correlations (for commuting only)

Local Indistinguishabilitydef: Hamiltonian H satisfies local indistinguishability if for every regions A, B:

A B B BA

Local Indistinguishabilitydef: Hamiltonian H satisfies local indistinguishability if for every regions A, B:

Obs1: Equivalent to the following type of decay of correlations:

A B B BA

Local Indistinguishabilitydef: Hamiltonian H satisfies local indistinguishability if for every regions A, B:

Obs2: Local expectation values are easy to compute

Obs3: Holds true in 1D at any T (Araki ‘69) and any D at high T (Kliesch et al ‘13)

A B B BA

Local Indistinguishabilitydef: Hamiltonian H satisfies local indistinguishability if for every regions A, B:

Obs 4: Implied by O(log(n)) convergence time of local Liouvillian (Cubitt, Lucia, Michalakis, Perez-Garcia ‘13)

Question: Can we q. sample from the Gibbs state if LI holds?

A B B BA

Local Indistinguishability implies Efficient Preparation

thm: Let H be a commuting local Hamiltonian in d dimensions. Then local indistinguishability implies one can prepare on a q. computer in time exp(O(logd(n)))

Local Recoverability

A B

(Poulin et al ’05, …)There is channel R : B -> AB s.t. R(ρBC) = ρABC

Explicit choice:

Follows from H = HAB + HBC, [HAB, HBC] = 0. Decompose B spaceinto s.t. , with HAB,k acting on A Lk

(Hayden, Jozsa, Petz, Winter ‘04)Equivalent to I(A:C|B) = S(AB) + S(CB) – S(ABC) – S(B) = 0

C Let H be commuting.

Let

Local Recoverability

A B

We will use this reconstruction idea to construct the state from its reductions in small regions (local indistinguishability will guarantee we can patch parts together)

Similar constructions in (Bravyi, Poulin, Terhal ’09, Pastawski, Yoshida ‘14) and (Kitaev ‘13)

C Let H be commuting.

Let

Suppose we have reduced state of on blue region.

Building State from Skeleton

` ` ` ` `

` ` ` ` `

` ` ` ` `

l

l

10l

Suppose we have reduced state of on blue region.

Can prepare the state by applying recovery channels, each acting on O(l2) qubits.

Building State from Skeleton

` ` ` ` `

` ` ` ` `

` ` ` ` `

l

l

10l CB

A

Suppose we have reduced state of on blue region.

Can prepare the state by applying recovery channels, each acting on O(l2) qubits.

Building State from Skeleton

` ` ` `

` ` ` ` `

` ` ` ` `

l

l

10l

Suppose we have reduced state of on blue region.

Can prepare the state by applying recovery channels, each acting on O(l2) qubits.

Building State from Skeleton

` ` ` `

` ` ` ` `

` ` ` ` `

l

l

10l

Suppose we have reduced state of on blue region.

Can prepare the state by applying recovery channels, each acting on O(l2) qubits.

Building State from Skeleton

` ` ` `

` ` ` `

` ` ` ` `

l

l

10l

Preparing the Skeleton

l

l

10l

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` ` ` ` `

Let HBR be the sub-Hamiltonian of H containing the local terms acting only in the blue and red regions. By local indistinguishability:

Choosing l = 4 ξ log(n) gives small error.

l

Preparing the Skeleton

It suffices to construct

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` ` ` ` `

` ` ` ` `

Preparing the Skeleton

It suffices to construct

• First build the regions above (the reduced states can be calculated in time O(exp(l2)) as every 1D system satisfy local indistinghuishability (Araki ‘73))

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Preparing the Skeleton

It suffices to construct

• First build the regions above (the reduced states can be calculated in time O(exp(l2)) as every 1D system satisfy local indistinghuishability (Araki ‘73))

• Second apply channels to fill in the gaps

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Non-Commuting Hamiltonians?def: Hamiltonian H satisfies Markov property if for every A, B, I(A : C|Bl) < c 2-l/ξ

Markov property related to fast saturation of area law:

(Wolf, Verstraete, Hastings, Cirac ‘07) I(A:Bl) < 2 Area(A)/T

I(A : C | Bl) = I(A : BlC) – I(A:Bl)

I(A : C|Bl) < 2-l/ξ if I(A : Bl+1) < I(A : Bl) + c 2-2l/ξ

A Bl

C

l

l

I(A:Bl)

Non-Commuting Hamiltonians?def: Hamiltonian H satisfies Markov property if for every A, B, I(A : C|Bl) < c 2-l/ξ

A Bl

C

l

thm: Let H be a local Hamiltonian in d dimensions. Then local indistinguishability + Markov property imply one can prepare in time exp(O(logd(n)))

Similar proof. New ingredient: Small conditional mutual information implies approximate recoverability by inequality due to Fawzi and Renner. Recovery map found efficiently by SDP.

Going forward

Previous approach has several drawbacks:

1. Only quasipolynomial run-time

2. Local indistinguishability might be too restrictive

3. Algorithm does not mimic nature

Rest of the talk: Fast mixing of quantum heat-bath vs mixing in space (but is less general; will achieve wish list only for 2-local commuting Hamiltonians )

Gap

The relevant gap is given by

L2 weighted inner product:

Variance:

Gap equal to spectral gap of

Mixing time of order

Previous Results

(Alicki, Fannes, Horodecki ‘08) Λ = Ω(1) for 2D toric code

(Alicki, Horodecki, Horodecki, Horodecki ‘08) Λ = exp(-Ω(n)) for 4D toric code

(Temme ‘14) Λ > exp(- βε)/n, with ε the energy barrier, for stabilizer Hamiltonians

Gap has been estimated for:

Equivalence of Clustering in Space and Time for Quantum Commuting

thm For commuting Hamiltonians in a finite dimensional lattice, the Davies generator has a constant gap if, and only if, the Gibbs state satisfies strong clustering of correlations

(Kastoryano, B. 1409.3435)

Equivalence of Clustering in Space and Time for Quantum Commuting

Strong Clustering holds true in:

• 1D at any temperature• Any D at sufficiently high temperature (critical T determined only by dim and interaction range)

thm For commuting Hamiltonians in a finite dimensional lattice, the Davies generator has a constant gap if, and only if, the Gibbs state satisfies strong clustering of correlations

Equivalence of Clustering in Space and Time for Quantum Commuting

Strong Clustering holds true in:

• 1D at any temperature• Any D at sufficiently high temperature (critical T determined only by dim and interaction range)

Caveat: At high temperature cluster expansion works well for computing local expectation values. (Open: How the two threshold T’s compare?)

Q advantage: we get the full Gibbs state (e.g. could perform swap test of purifications of two Gibbs states. Good for anything?)

thm For commuting Hamiltonians in a finite dimensional lattice, the Davies generator has a constant gap if, and only if, the Gibbs state satisfies strong clustering of correlations

Strong Clustering

A B

def: Strong clustering holds if there is ξ>0 s.t. for every A and B and operator f acting on

Conditional Covariance:

Conditional Expectation:

Xc : complement of X

d(X, Y) : distance between regions X and Y

Strong Clustering

A B

def: Strong clustering holds if there is ξ>0 s.t. for every A and B and operator f acting on

Fact 1:

prop: For 2-local commuting Hamiltonians, strong clustering is equivalent to

Based on C* algebraic trick of decomposing local spaces into irreducible spaces (Bravyi, Vyalyi ‘03)

Strong Clustering -> GapWe show that under the clustering condition:

A B

Getting:

V : entire latticeV0 : sublattice of size O(ξ)

Key lemma: If

then

Follows idea of a proof of the classical analogue for Glauber dynamics (Bertini et al ‘00)

Gap -> Strong Clustering

Employs the following mapping between Liouvillians for commuting Hamiltonians and local Hamiltonians on a larger space:

Apply the detectability lemma (Aharonov et al ‘10) to prove gap -> strong clustering (strengthening proof previous proof that gap -> clustering)

ConclusionsEquivalence clustering in time vs clustering in space for commuting quantum models:

• Local indistinguishability -> quasipolynomial-time q. sampling • Strong clustering <-> gapped q. sampler Is strong clustering related to “weak clustering” beyond 2-local?

Equivalence for non-commuting?

• Local indsitinguishability: when Markov property holds?• Strong clustering: how to define it? Needs at least (quasi)-local

Liouvillian. Can define in 1D and any D at high temperature. Unknown in general…

What’s the complexity of approximately computing tr(ρT H)? Is it in QMA? Is it QMA-hard?