Thermalization Algorithms:
Digital vs Analogue
Fernando G.S.L. Brandão
University College London
Joint work with
Michael Kastoryano
Freie Universität Berlin
Discrete and analogue Quantum Simulators, Bad Honnef 2014
Dynamical Properties
Hij
Hamiltonian:
State at time t:
Expectation values:
Temporal correlations:
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, …)
Static Properties
Static Properties
Hij
Hamiltonian:
Static Properties
Hij
Hamiltonian:
Groundstate:
Thermal state:
Compute: local expectation values (e.g. magnetization),
correlation functions (e.g.
), …
Static Properties
Can we prepare groundstates?
Warning: In general it’s impossible to prepare groundstates
efficiently, even of one-dimensional translational-invariant models
(Gottesman-Irani ‘09)
-- it’s a computational-hard problem
Static Properties
Can we prepare groundstates?
Warning: In general it’s impossible to prepare groundstates
efficiently, even of one-dimensional translational-invariant models
(Gottesman-Irani ‘09)
-- it’s a computational-hard problem
Analogue: adiabatic evolution; works if Δ ≥ n-c
H(sf)
H(si)
ψi
H(s)
ψs
H(s)ψs = E0,sψs
Δ := min Δ(s)
Digital: Phase estimation*; works if can find a “simple” state |0>
such that
*
(Abrams, Lloyd ‘99)
Static Properties
Can we prepare thermal states?
(Terhal and diVincenzo ’00, …)
Why not? Couple to a bath of the right temperature and wait.
S
B
But size of environment might be huge. Maybe not efficient
Static Properties
Can we prepare thermal states?
(Terhal and diVincenzo ’00, …)
Why not? Couple to a bath of the right temperature and wait.
S
B
But size of environment might be huge. Maybe not efficient
Warning: In general it’s impossible to prepare thermal states
efficiently, even at constant temperature and of classical models,
but defined on general graphs
(PCP Theorem, Arora et al ‘98)
Warning 2: Spin glasses not expected to thermalize.
Static Properties
Can we prepare thermal states?
(Terhal and diVincenzo ’00, …)
Whentocan
weof
prepare
thermal
statesand
efficiently?
Why not?•Couple
a bath
the right
temperature
wait.
• Digital vs analogue methods?
S
B
But size of environment might be huge. Maybe not efficient
Warning: In general it’s impossible to prepare thermal states
efficiently, even at constant temperature and of classical models,
but defined on general graphs
(PCP Theorem, Arora et al ‘98)
Warning 2: Spin glasses not expected to thermalize.
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
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 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
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
E.g. for Metropolis,
stationary
distribution
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
Spatial Mixing
Let
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 0 0 0 0 0 0 0 0 0 0
Ex. τ = (0, … 0)
Spatial Mixing
Let
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 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
Temporal Mixing vs Spatial Mixing
(Stroock, Zergalinski ’92; Martinelli, Olivieri ’94, …) For every 1D and 2D
model, Gibbs state has constant correlation length if, 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)
Temporal Mixing vs Spatial Mixing
(Stroock, Zergalinski ’92; Martinelli, Olivieri ’94, …) For every 1D and 2D
model, Gibbs state has constant correlation length if, 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
Temporal Mixing vs Spatial Mixing
(Stroock, Zergalinski ’92; Martinelli, Olivieri ’94, …) For every 1D and 2D
model, Gibbs state has constant correlation length if, 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)
Temporal Mixing vs Spatial Mixing
(Stroock, Zergalinski ’92; Martinelli, Olivieri ’94, …) For every 1D and 2D
model, Gibbs state has constant correlation length if, 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)
Temporal Mixing vs Spatial Mixing
(Stroock, Zergalinski ’92; Martinelli, Olivieri ’94, …) For every 1D and 2D
model, Gibbs state has constant correlation length if, 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
clusteringDoes
(weak
clustering similar
vs strong
clustering)
something
hold
in the quantum case?
Obs2: Same is true for the log-Sobolev constant of the system
1st step: Need a quantum version of Glauber dynamics…
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)
Quantum Master Equations
Canonical example: cavity QED
Lindblad Equation:
(most general Markovian and time homogeneous q. master equation)
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 gap of of Lindbladian
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
Ai
(Kliesch et al ‘11) Time evolution of every k-local Lindbladian on n
qubits can be simulated in time poly(n, 2^k) in the circuit model
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?
Davies Maps
Lindbladian:
Lindblad terms:
: spectral density
Davies Maps
Lindbladian:
Lindblad terms:
: spectral density
Hij
Sα (Xα, Yα, Zα)
Thermal state is the unique fixed point:
(satisfies q. detailed balance:
)
Davies Maps
(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)
Interacting Ham.
Davies Maps
(Davies ‘74) Rigorous derivation in the
Interacting Ham.
weak-coupling limit:
Coarse grain over time t ≈ λ-2 >> max(1/ (Ei – Ej + Ek - El))
(Ei: eigenvalues of H)
But: for n spin Hamiltonain H:
Energy
O(n)
max(1/ (Ei – Ej + Ek - El)) = exp(O(n))
O(n1/2)
density
Consequence:
Sα(ω) are non-local (act on n qubits);
cannot be efficiently simulated in the circuit model
(but for commuting Hamiltonian, it is local)
Davies Maps
(Davies ‘74) Rigorous derivation in the
Interacting Ham.
weak-coupling limit:
• Can we find a local master equation that prepares ρβ?
Coarse grain over time t ≈ λ-2 >> max(1/ (Ei – Ej + Ek - El))
• Can we at least find a quantum
channelof
(tpcp
(Ei: eigenvalues
H) map)
that can be efficiently implemented on a quantum
But: for ncomputer
spin Hamiltonain
whose H:
fixedmax(1/
point is(Eρi β–?Ej + Ek - El)) = exp(O(n))
Energy
O(n)
O(n1/2)
density
Consequence:
Sα(ω) are non-local (act on n qubits);
cannot be efficiently simulated in the circuit model
(but for commuting Hamiltonian, it is local)
Digital: Quantum Metropolis Sampling
(Temme, Osborne, Vollbrecht, Poulin, Verstraete ‘09)
Classical Metropolis:
Digital: Quantum Metropolis Sampling
(Temme, Osborne, Vollbrecht, Poulin, Verstraete ‘09)
Classical Metropolis:
Quantum Metropolis:
random U
1. Prepare
(phase estimation)
2.
3. Make the move
with prob.
(non trivial; done by Marriott-Watrous trick)
Gives map Λ s.t.
Digital: Quantum Metropolis Sampling
(Temme, Osborne, Vollbrecht, Poulin, Verstraete ‘09)
What’s the convergence time? I.e. minimum k s.t.
Classical Metropolis:
k
a hard question!
QuantumSeems
Metropolis:
random U
1. Prepare
(phase estimation)
2.
3. Make the move
with prob.
(non trivial; done by Marriott-Watrous trick)
Gives map Λ s.t.
Davies Maps
Lindbladian:
Lindblad terms:
Analogue: “Damped” Davies Maps
Lindbladian:
Lindblad terms:
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?
)
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
Lieb-Robinson Bound II
(another formulation) For local Hamiltonian H
l
l
Lieb-Robinson Bound II
(another formulation) For local Hamiltonian H
l
time
l
Applying Lieb-Robinson Bound
to “Damped” Davies Maps
Consider:
fact:
proof:
LR bound
Damping term
“Damped” Davies Maps are
Approximately Local
Define
fact:
has the Gibbs state as its fixed point (up to error
1/poly(n)) and is O(logd(n))-locality for a Hamiltonian on a
d-dimensional lattice.
Can be simulated on a quantum computer in time exp(O(logd(n)))
Mixing in Space vs Mixing in Time
thm If for every regions A and B and f acting on
, for t = O( 2O(l) 2O(ξ) n), l = O(log(n/ε))
then
Obs: Converse holds true for
commuting Hamiltonians
acts trivially on A
A
B
acts trivially on B
AC : complement of A (yellow + blue)
BC : complement of B (ref + blue)
Convergence Time in 1D
Def ρβ has correlation length ξ if for every f, g
f
g
Convergence Time in 1D
Def ρβ has correlation length ξ if for every f, g
Cor For a 1D Hamiltonian, ρβ has
correlation length ξ, then
f
g
for t = O( 2O(l) 2O(ξ) n), l = O(log(n/ε))
Convergence Time in 1D
Def ρβ has correlation length ξ if for every f, g
Cor For a 1D Hamiltonian, ρβ has
correlation length ξ, then
f
g
for t = O( 2O(l) 2O(ξ) n), l = O(log(n/ε))
thm (Araki ‘69) For every 1D Hamiltonian, ρβ has ξ = O(β)
Thus: Can prepare 1D Gibbs states in time poly(2β, n)
No phase
trans. in 1D
Conditional Expectation
Let Ll*A be the A sub-Lindbladian in Heisenberg picture
Note:
,
Conditional Expectation:
fact:
proof:
commutes with all
all
and thus with
Conditional Covariance and
Variance
Conditional Covariance
For a region C:
Ex. If C is the entire lattice,
Conditional Variance
Mixing in Space vs Mixing in Time
thm If for every regions A and B and f acting on
, for t = O( 2O(l) 2O(ξ) n), l = O(log(n/ε))
then
acts trivially on A
A
B
acts trivially on B
Mixing in Space vs Mixing in Time
thm If for every regions A and B and f acting on
, for t = O( 2O(l) 2O(ξ) n), l = O(log(n/ε))
then
acts trivially on A
A
B
acts trivially on B
AC : complement of A (yellow + blue)
BC : complement of B (ref + blue)
Proof Idea
(Kastoryano, Temme ‘11, …)
The relevant gap is
We show that under the clustering condition:
Getting:
A
B
V : entire lattice
V0 : sublattice of size O(lξ)
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.
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.
Open questions:
• 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?)
(Fannes, Werner ‘95)
• What are the implications to self-correcting quantum memories?