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Fernando G.S.L. Brand ão University College London Joint work with Michael Kastoryano Freie Universität Berlin Discrete and analogue Quantum Simulators, Bad Honnef 2014. Thermalization Algorithms : Digital vs Analogue. Dynamical Properties. H ij. Hamiltonian: State at time t :

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slide1

Fernando G.S.L. Brandão

University College London

Joint work with

Michael Kastoryano

FreieUniversität Berlin

Discrete and analogue Quantum Simulators, Bad Honnef 2014

Thermalization Algorithms: Digital vs Analogue

slide2

Dynamical Properties

Hij

Hamiltonian:

State at time t:

Expectation values:

Temporal correlations:

slide3

Quantum Simulators, Dynamical

Digital: Quantum Computer

Can simulate the dynamics of every multi-particle

quantum system

(spin models, fermionic and bosonic models, topological quantum

field theory, ϕ4 quantum field theory, …)

Analog: Optical Lattices, Ion Traps, Circuit cQED, Linear Optics, …

Can simulate the dynamics of particular models

(Bose-Hubbard, spin models, BEC-BCS, dissipative dynamics,

quenched dynamics, …)

slide5

Static Properties

Hij

Hamiltonian:

slide6

Static Properties

Hij

Hamiltonian:

Groundstate:

Thermal state:

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

slide7

Static Properties

Can we prepare groundstates?

Warning: In general it’s impossible to prepare groundstates efficiently, even of one-dimensional translational-invariant models

-- it’s a computational-hard problem

(Gottesman-Irani ‘09)

slide8

Static Properties

Can we prepare groundstates?

Warning: In general it’s impossible to prepare groundstates efficiently, even of one-dimensional translational-invariant models

-- it’s a computational-hard problem

Analogue: adiabatic evolution; works if Δ ≥ n-c

Digital: Phase estimation*; works if can find a “simple” state |0>

such that

*

(Gottesman-Irani ‘09)

H(sf)

H(s)ψs = E0,sψs

Δ := min Δ(s)

H(s)

ψs

H(si)

ψi

(Abrams, Lloyd ‘99)

slide9

Static Properties

Can we prepare thermal states?

Why not? Couple to a bath of the right temperature and wait.

But size of environment might be huge. Maybe not efficient

(Terhal and diVincenzo’00, …)

S

B

slide10

Static Properties

Can we prepare thermal states?

Why not? Couple to a bath of the right temperature and wait.

But size of environment might be huge. Maybe not efficient

(Terhal and diVincenzo’00, …)

S

B

Warning: In general it’s impossible to prepare thermal states efficiently, even at constant temperature and of classical models, but defined on general graphs

Warning 2: Spin glasses not expected to thermalize.

(PCP Theorem, Aroraet al ‘98)

slide11

Static Properties

Can we prepare thermal states?

Why not? Couple to a bath of the right temperature and wait.

But size of environment might be huge. Maybe not efficient

(Terhal and diVincenzo’00, …)

  • When can we prepare thermal states efficiently?
  • Digital vs analogue methods?

S

B

Warning: In general it’s impossible to prepare thermal states efficiently, even at constant temperature and of classical models, but defined on general graphs

Warning 2: Spin glasses not expected to thermalize.

(PCP Theorem, Aroraet al ‘98)

slide12

Summary

1. Glauber Dynamics and Metropolis Sampling

- Temporal vs Spatial Mixing

2. Quantum Master Equations (Davies Maps)

3. Quantum Metropolis Sampling

4. “Damped” Davies Maps

- Lieb-Robinson Bounds

5. Convergence Time of “Damped” Davies Maps

- Quantum Generalization of “Temporal vs Spatial Mixing”

- 1D Systems

slide13

Metropolis Sampling

Consider e.g. Ising model:

Coupling to bath modeled by stochastic map Q

i

j

Metropolis Update:

The stationary state is the thermal (Gibbs) state:

slide14

Metropolis Sampling

Consider e.g. Ising model:

Coupling to bath modeled by stochastic map Q

i

j

Metropolis Update:

The stationary state is the thermal (Gibbs) state:

  • (Metropolis et al ’53) “We devised a general method to calculate the
  • properties of any substance comprising individual molecules with
  • classical statistics”
  • Example of Markov Chain Monte Carlo method.
  • Extremely useful algorithmic technique
slide15

Glauber Dynamics

Metropolis Sampling is an example of Glauber dynamics:

Markov chains (discrete or continuous) on the space of configurations {0, 1}n that have the Gibbs state as the stationary distribution:

transition matrix

after t time steps

stationary distribution

E.g. for Metropolis,

slide16

Temporal Mixing

eigenprojectors

eigenvalues

Convergence time given by the gap Δ = 1- λ1:

Time of equilibration ≈ n/Δ

We have fast temporal mixing if Δ = n-c

slide17

Spatial Mixing

blue: V,

red: boundary

Let be the Gibbs state for

a model in the lattice V with

boundary conditions τ, i.e.

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

0

0

0

0

0

0

0

0

0

0

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

Ex. τ = (0, … 0)

slide18

Spatial Mixing

blue: V,

red: boundary

Let be the Gibbs state for

a model in the lattice V with

boundary conditions τ, i.e.

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

0

0

0

0

0

0

0

0

0

0

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

Ex. τ = (0, … 0)

def: The Gibbs state has correlation length ξ if for every f, g

f

g

slide19

Temporal Mixing vs Spatial Mixing

(Stroock, Zergalinski’92; Martinelli, Olivieri’94, …) For every 1D and 2D model, Gibbs state has constant correlation lengthif, and only, if the Glauber dynamics has a constant gap

constant: independent of the system size

Obs1: Same is true in any fixed dimension using a stronger notion of clustering (weak clustering vs strong clustering)

slide20

Temporal Mixing vs Spatial Mixing

(Stroock, Zergalinski’92; Martinelli, Olivieri’94, …) For every 1D and 2D model, Gibbs state has constant correlation lengthif, and only, if the Glauber dynamics has a constant gap

constant: independent of the system size

Obs1: Same is true in any fixed dimension using a stronger notion of clustering (weak clustering vs strong clustering)

Obs2: Same is true for the log-Sobolev constant of the system

slide21

Temporal Mixing vs Spatial Mixing

(Stroock, Zergalinski’92; Martinelli, Olivieri’94, …) For every 1D and 2D model, Gibbs state has constant correlation lengthif, and only, if the Glauber dynamics has a constant gap

constant: independent of the system size

Obs1: Same is true in any fixed dimension using a stronger notion of clustering (weak clustering vs strong clustering)

Obs2: Same is true for the log-Sobolev constant of the system

Obs3: For many models, when correlation

length diverges, gap is exponentially

small in the system size (e.g. Ising model)

slide22

Temporal Mixing vs Spatial Mixing

(Stroock, Zergalinski’92; Martinelli, Olivieri’94, …) For every 1D and 2D model, Gibbs state has constant correlation lengthif, and only, if the Glauber dynamics has a constant gap

constant: independent of the system size

Obs1: Same is true in any fixed dimension using a stronger notion of clustering (weak clustering vs strong clustering)

Obs2: Same is true for the log-Sobolev constant of the system

Obs3: For many models, when correlation length diverges, gap is exponentially small in the system size (e.g. Ising model)

Obs4: 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, e.g. Dobrushin’s condition)

slide23

Temporal Mixing vs Spatial Mixing

(Stroock, Zergalinski’92; Martinelli, Olivieri’94, …) For every 1D and 2D model, Gibbs state has constant correlation lengthif, and only, if the Glauber dynamics has a constant gap

constant: independent of the system size

Obs1: Same is true in any fixed dimension using a stronger notion of clustering (weak clustering vs strong clustering)

Obs2: Same is true for the log-Sobolev constant of the system

Obs3: For many models, when correlation length diverges, gap is exponentially small in the system size (e.g. Ising model below critical β)

Obs4: 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, e.g. Dobrushin’s condition)

Does something similar hold in the quantum case?

1st step: Need a quantum version of Glauber dynamics…

slide24

Quantum Master Equations

Canonical example: cavity QED

Lindblad Equation:

(most general Markovian and time homogeneous q. master equation)

slide25

Quantum Master Equations

Canonical example: cavity QED

Lindblad Equation:

(most general Markovian and time homogeneous q. master equation)

completely positive trace-preserving map:

fixed point:

How fast does it converge? Determined by gapof of Lindbladian

slide26

Quantum Master Equations

Canonical example: cavity QED

Lindblad Equation:

Local master equations: L is k-local if all Ai act on at most k sites

(Klieschet al ‘11) Time evolution of every k-local Lindbladian on nqubits can be simulated in time poly(n, 2^k) in the circuit model

Ai

slide27

Dissipative Quantum Engineering

Define a master equation whose fixed point is a desired quantum state

(Verstraete, Wolf, Cirac ‘09) Universal quantum computation with local Lindbladian

(Diehl et al ’09, Kraus et al ‘09)Dissipative preparation of entangled states

(Barreiro et al ‘11) Experiment on 5 trapped ions (prepared GHZ state)

Is there a master equation preparing thermal states

of many-body Hamiltonians?

slide28

Davies Maps

Lindbladian:

Lindblad terms:

: spectral density

slide29

Davies Maps

Lindbladian:

Lindblad terms:

: spectral density

Hij

Sα (Xα, Yα, Zα)

Thermal state is the unique fixed point:

(satisfies q. detailed balance: )

slide30

Davies Maps

Interacting Ham.

(Davies ‘74) Rigorous derivation in the

weak-coupling limit:

Coarse grain over time t ≈ λ-2 >> max(1/ (Ei– Ej + Ek - El))

(Ei: eigenvalues of H)

slide31

Davies Maps

Interacting Ham.

(Davies ‘74) Rigorous derivation in the

weak-coupling limit:

Coarse grain over time t ≈ λ-2 >> max(1/ (Ei– Ej + Ek - El))

(Ei: eigenvalues of H)

But: for n spin HamiltonainH: max(1/ (Ei– Ej + Ek - El)) = exp(O(n))

Consequence:

Sα(ω) are non-local (act on nqubits);

cannot be efficiently simulated in the circuit model

(but for commuting Hamiltonian, it is local)

Energy

O(n)

O(n1/2)

density

slide32

Davies Maps

Interacting Ham.

(Davies ‘74) Rigorous derivation in the

weak-coupling limit:

Coarse grain over time t ≈ λ-2 >> max(1/ (Ei– Ej + Ek - El))

(Ei: eigenvalues of H)

But: for n spin HamiltonainH: max(1/ (Ei– Ej + Ek - El)) = exp(O(n))

Consequence:

Sα(ω) are non-local (act on nqubits);

cannot be efficiently simulated in the circuit model

(but for commuting Hamiltonian, it is local)

  • Can we find a local master equation that prepares ρβ?
  • Can we at least find a quantum channel (tpcp map) that can be efficiently implemented on a quantum computer whose fixed point is ρβ?

Energy

O(n)

O(n1/2)

density

slide33

Digital: Quantum Metropolis Sampling

(Temme, Osborne, Vollbrecht, Poulin, Verstraete ‘09)

Classical Metropolis:

slide34

Digital: Quantum Metropolis Sampling

(Temme, Osborne, Vollbrecht, Poulin, Verstraete ‘09)

Classical Metropolis:

Quantum Metropolis:

random U

Prepare (phase estimation)

Make the move with prob.

Gives map Λs.t.

(non trivial; done by Marriott-Watrous trick)

slide35

Digital: Quantum Metropolis Sampling

(Temme, Osborne, Vollbrecht, Poulin, Verstraete ‘09)

What’s the convergence time? I.e. minimum ks.t.

Seems a hard question!

Classical Metropolis:

Quantum Metropolis:

k

random U

Prepare (phase estimation)

Make the move with prob.

Gives map Λs.t.

(non trivial; done by Marriott-Watrous trick)

slide36

Davies Maps

Lindbladian:

Lindblad terms:

slide37

Analogue: “Damped” Davies Maps

Lindbladian:

Lindblad terms:

slide38

Analogue: “Damped” Davies Maps

Lindbladian:

Lindblad terms:

Thermal state is the unique fixed point:

(satisfies q. detailed balance:

follows from: )

What is the locality of this Lindbladian?

slide39

Lieb-Robinson Bound

In non-relativistic quantum mechanics there is no strict speed of light limit. But there is an approximate version

(Lieb-Robinson ‘72) For local Hamiltonian H

Hij

X

Z

slide40

Lieb-Robinson Bound II

(another formulation) For local Hamiltonian H

l

l

slide41

Lieb-Robinson Bound II

(another formulation) For local Hamiltonian H

l

l

time

slide42

Applying Lieb-Robinson Bound

to “Damped” Davies Maps

Consider:

fact:

proof:

LR bound

Damping term

slide43

“Damped” Davies Maps are Approximately Local

Define

fact:

has the Gibbs state as its fixed point (up to error

1/poly(n)) and isO(logd(n))-locality for a Hamiltonian on a

d-dimensional lattice.

Can be simulated on a quantum computer in time exp(O(logd(n)))

slide44

Mixing in Space vs Mixing in Time

thmIf for every regions A and Band f acting on

then , for t = O(2O(l) 2O(ξ) n), l = O(log(n/ε))

Obs: Converse holds true for commuting Hamiltonians

acts trivially on A

acts trivially on B

A

B

AC : complement of A (yellow + blue)

BC : complement of B (ref + blue)

slide45

Convergence Time in 1D

Defρβ has correlation length ξif for every f, g

f

g

slide46

Convergence Time in 1D

Defρβ has correlation length ξif for every f, g

CorFor a 1D Hamiltonian, ρβ has correlation length ξ, then

for t = O( 2O(l) 2O(ξ) n), l = O(log(n/ε))

f

g

slide47

Convergence Time in 1D

Defρβ has correlation length ξif for every f, g

CorFor a 1D Hamiltonian, ρβ has correlation length ξ, then

for t = O( 2O(l) 2O(ξ) n), l = O(log(n/ε))

f

g

thm (Araki ‘69) For every 1D Hamiltonian, ρβ has ξ = O(β)

No phase trans. in 1D

Thus: Can prepare 1D Gibbs states in time poly(2β, n)

slide48

Conditional Expectation

Let Ll*A be the A sub-Lindbladian in Heisenberg picture

Note: ,

Conditional Expectation:

fact:

proof: commutes with all and thus with

all

slide49

Conditional Covariance and Variance

Conditional Covariance

For a region C:

Ex. If C is the entire lattice,

Conditional Variance

slide50

Mixing in Space vs Mixing in Time

thmIf for every regions A and Band f acting on

then , for t = O(2O(l) 2O(ξ) n), l = O(log(n/ε))

acts trivially on A

acts trivially on B

A

B

slide51

Mixing in Space vs Mixing in Time

thmIf for every regions A and Band f acting on

then , for t = O(2O(l) 2O(ξ) n), l = O(log(n/ε))

acts trivially on A

acts trivially on B

A

B

AC : complement of A (yellow + blue)

BC : complement of B (ref + blue)

slide52

Proof Idea

The relevant gap is

(Kastoryano, Temme ‘11, …)

We show that under the clustering condition:

Getting:

V : entire lattice

V0 : sublattice of size O(lξ)

A

B

slide53

Conclusions and Open Questions

  • “Davies like” master equations + Lieb-Robinson bound give interesting approach for preparing thermal states efficiently.
  • Connections between clustering properties of the thermal states (mixing in space) and fast convergence of the master equation (mixing in time), also in the quantum case.
slide54

Conclusions and Open Questions

  • “Davies like” master equations + Lieb-Robinson bound give interesting approach for preparing thermal states efficiently.
  • Connections between clustering properties of the thermal states (mixing in space) and fast convergence of the master equation (mixing in time), also in the quantum case.
  • Openquestions:
  • Can we get O(log(n))-local Gibbs sampler in any dimension?
  • (true in 2D if can improve Lieb-Robinson bound to Gaussian
  • decay).
  • How about really local samplers? Connected to stability question of “Damped Davies” maps.
  • Can we prove in generality equivalence of spatial mixing vs temporal mixing? How about in 2D? (how to fix the boundary in the q. case?)
  • What are the implications to self-correcting quantum memories?

(Fannes, Werner ‘95)