1. 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

2. Dynamical Properties H ij Hamiltonian: State at time t: Expectation values: Temporal correlations:

3. 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, ϕ 4 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, …)

4. Static Properties

5. Hamiltonian: Static Properties H ij

6. Hamiltonian: Groundstate: Thermal state: Compute: local expectation values (e.g. magnetization), correlation functions (e.g. ), … Static Properties H ij

7. 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)

8. 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) (Abrams, Lloyd ‘99) H(s i ) ψ i H(s)ψ s = E 0,s ψ s Δ := min Δ(s) H(s) ψ s H(s f )

9. 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

10. 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)

11. 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?

12. 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

13. Metropolis Sampling Consider e.g. Ising model: Coupling to bath modeled by stochastic map Q The stationary state is the thermal (Gibbs) state: Metropolis Update: i j

14. Metropolis Sampling Consider 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

15. 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

16. Temporal Mixing eigenvalues eigenprojectors Convergence time given by the gap Δ = 1- λ 1 : Time of equilibration ≈ n/Δ We have fast temporal mixing if Δ = n -c

17. 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 0000000000 0000000000 Ex. τ = (0, … 0)

18. 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 0000000000 0000000000 Ex. τ = (0, … 0) def: The Gibbs state has correlation length ξ if for every f, g f g

19. 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)

20. 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

21. 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)

22. 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)

23. 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? 1 st step: Need a quantum version of Glauber dynamics…

24. Lindblad Equation: (most general Markovian and time homogeneous q. master equation) Quantum Master Equations Canonical example: cavity QED

25. 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

26. Lindblad Equation: Quantum Master Equations Canonical example: cavity QED Local master equations: L is k-local if all A i 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 AiAi

27. 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? …

28. Davies Maps Lindbladian: Lindblad terms: : spectral density

29. Davies Maps Lindbladian: Lindblad terms: H ij S α (X α, Y α, Z α ) : spectral density Thermal state is the unique fixed point: (satisfies q. detailed balance: )

30. Davies Maps (Davies ‘74) Rigorous derivation in the weak-coupling limit: Coarse grain over time t ≈ λ -2 >> max(1/ (E i – E j + E k - E l )) (E i : eigenvalues of H) Interacting Ham.

31. Davies Maps (Davies ‘74) Rigorous derivation in the weak-coupling limit: Coarse grain over time t ≈ λ -2 >> max(1/ (E i – E j + E k - E l )) (E i : eigenvalues of H) But: for n spin Hamiltonain H: max(1/ (E i – E j + E k - E l )) = exp(O(n)) Consequence: S α (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(n 1/2 )

32. Davies Maps (Davies ‘74) Rigorous derivation in the weak-coupling limit: Coarse grain over time t ≈ λ -2 >> max(1/ (E i – E j + E k - E l )) (E i : eigenvalues of H) But: for n spin Hamiltonain H: max(1/ (E i – E j + E k - E l )) = exp(O(n)) Consequence: S α (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(n 1/2 ) Can we find a local master equation that prepares ρ β ? Can we at least find a quantum channel (tpcp map) that can be efficiently implemented on a quantum computer whose fixed point is ρ β ?

33. Digital: Quantum Metropolis Sampling (Temme, Osborne, Vollbrecht, Poulin, Verstraete ‘09) Classical Metropolis:

34. 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.

35. 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

36. Davies Maps Lindbladian: Lindblad terms:

37. Analogue: “Damped” Davies Maps Lindbladian: Lindblad terms:

38. 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?

39. 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 H ij X Z

40. Lieb-Robinson Bound II (another formulation) For local Hamiltonian H l l

41. Lieb-Robinson Bound II (another formulation) For local Hamiltonian H time l l

42. Applying Lieb-Robinson Bound to “Damped” Davies Maps Consider: fact: proof: LR bound Damping term

43. has the Gibbs state as its fixed point (up to error 1/poly(n)) and is O(log d (n))-locality for a Hamiltonian on a d-dimensional lattice. Can be simulated on a quantum computer in time exp(O(log d (n))) “Damped” Davies Maps are Approximately Local Define fact:

44. 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( 2 O(l) 2 O(ξ) n), l = O(log(n/ε)) AB Obs: Converse holds true for commuting Hamiltonians A C : complement of A (yellow + blue) B C : complement of B (ref + blue)

45. Convergence Time in 1D Def ρ β has correlation length ξ if for every f, g f g

46. Convergence Time in 1D Def ρ β has correlation length ξ if for every f, g f g Cor For a 1D Hamiltonian, ρ β has correlation length ξ, then for t = O( 2 O(l) 2 O(ξ) n), l = O(log(n/ε))

47. Convergence Time in 1D Def ρ β has correlation length ξ if for every f, g f g Cor For a 1D Hamiltonian, ρ β has correlation length ξ, then for t = O( 2 O(l) 2 O(ξ) n), l = O(log(n/ε)) thm (Araki ‘69) For every 1D Hamiltonian, ρ β has ξ = O(β) Thus: Can prepare 1D Gibbs states in time poly(2 β, n) No phase trans. in 1D

48. Conditional Expectation Let L l *A be the A sub-Lindbladian in Heisenberg picture Note:, Conditional Expectation: fact: proof: commutes with all and thus with all

49. Conditional Covariance and Variance For a region C: Ex. If C is the entire lattice, Conditional Covariance Conditional Variance

50. 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( 2 O(l) 2 O(ξ) n), l = O(log(n/ε)) AB

51. 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( 2 O(l) 2 O(ξ) n), l = O(log(n/ε)) AB A C : complement of A (yellow + blue) B C : complement of B (ref + blue)

52. Proof Idea The relevant gap is (Kastoryano, Temme ‘11, …) We show that under the clustering condition: AB Getting: V : entire lattice V 0 : sublattice of size O(lξ)

53. 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.

54. 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)