Thermalization Algorithms: Digital vs Analogue
Fernando G.S.L. BrandãoUniversity College London
Joint work with
Michael KastoryanoFreie Universität Berlin
Discrete and analogue Quantum Simulators, Bad Honnef 2014
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, …)
Hamiltonian:
Groundstate:
Thermal state:
Compute: local expectation values (e.g. magnetization), correlation functions (e.g. ), …
Static Properties
Hij
Static PropertiesCan 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)
Static PropertiesCan 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)
(Abrams, Lloyd ‘99)
H(si)ψi
H(s)ψs = E0,sψs
Δ := min Δ(s)H(s)ψs
H(sf)
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
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, Arora et al ‘98)
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, Arora et al ‘98)
• When can we prepare thermal states efficiently?
• Digital vs analogue methods?
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 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
Metropolis SamplingConsider e.g. Ising model:
Coupling to bath modeled by stochastic map Q
The stationary state is the thermal (Gibbs) state:
Metropolis Update:
• (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
i j
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 matrixafter t time steps
E.g. for Metropolis,
stationary distribution
Temporal Mixing
eigenvalueseigenprojectors
Convergence time given by the gap Δ = 1- λ1:
Time of equilibration ≈ n/Δ
We have fast temporal mixing if Δ = n-c
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
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 correlationlength 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 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…
Lindblad Equation:
(most general Markovian and time homogeneous q. master equation)
Quantum Master EquationsCanonical example: cavity QED
Lindblad Equation:
(most general Markovian and time homogeneous q. master equation)
Quantum Master Equations
completely positive trace-preserving map:
fixed point:
How fast does it converge? Determined by gap of of Lindbladian
Canonical example: cavity QED
Lindblad Equation:
Quantum Master EquationsCanonical example: cavity QED
Local master equations: L is k-local if all Ai act on at most k sites
(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
Ai
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 MapsLindbladian:
Lindblad terms:
Hij
Sα (Xα, Yα, Zα)
: spectral density
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 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: max(1/ (Ei – Ej + Ek - El)) = exp(O(n))
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)
Interacting Ham.
O(n)
Energy
density
O(n1/2)
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)
But: for n spin Hamiltonain H: max(1/ (Ei – Ej + Ek - El)) = exp(O(n))
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)
Interacting Ham.
O(n)
Energy
density
O(n1/2)
• 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?
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)
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.
What’s the convergence time? I.e. minimum k s.t.
Seems a hard question!
k
Analogue: “Damped” Davies MapsLindbladian:
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
HijX
Z
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)))
“Damped” Davies Maps are Approximately Local
Define
fact:
acts trivially on A
acts trivially on B
Mixing in Space vs Mixing in Time
thm If for every regions A and B and f acting on
then , for t = O( 2O(l) 2O(ξ) n), l = O(log(n/ε))
A B
Obs: Converse holds true for commuting Hamiltonians
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
fg
Cor For a 1D Hamiltonian, ρβ has correlation length ξ, then
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
fg
Cor For a 1D Hamiltonian, ρβ has correlation length ξ, then
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 and thus with
all
Conditional Covariance and Variance
For a region C:
Ex. If C is the entire lattice,
Conditional Covariance
Conditional Variance
acts trivially on A
acts trivially on B
Mixing in Space vs Mixing in Time
thm If for every regions A and B and f acting on
then , for t = O( 2O(l) 2O(ξ) n), l = O(log(n/ε))
A B
acts trivially on A
acts trivially on B
Mixing in Space vs Mixing in Time
thm If for every regions A and B and f acting on
then , for t = O( 2O(l) 2O(ξ) n), l = O(log(n/ε))
A B
AC : complement of A (yellow + blue)BC : complement of B (ref + blue)
Proof Idea
The relevant gap is(Kastoryano, Temme ‘11, …)
We show that under the clustering condition:
A B
Getting:
V : entire latticeV0 : 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?)
• What are the implications to self-correcting quantum memories?(Fannes, Werner ‘95)