Thermalization Algorithms : Digital vs Analogue

1. 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

2. Dynamical Properties Hij 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, 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, …)

4. Static Properties

5. Static Properties Hij Hamiltonian:

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

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) H(sf) H(s)ψs = E0,sψs Δ := min Δ(s) H(s) ψs H(si) ψi (Abrams, Lloyd ‘99)

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, Aroraet 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, …) • 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)

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 i j Metropolis Update: The stationary state is the thermal (Gibbs) state:

14. 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

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 stationary distribution E.g. for Metropolis,

16. 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

17. 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)

18. 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

19. 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)

20. 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

21. 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)

22. 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)

23. 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…

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

25. 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

26. 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

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: : spectral density Hij Sα (Xα, Yα, Zα) Thermal state is the unique fixed point: (satisfies q. detailed balance: )

30. 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)

31. 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

32. 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

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 Prepare (phase estimation) Make the move with prob. Gives map Λs.t. (non trivial; done by Marriott-Watrous trick)

35. 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)

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 Hij 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 l l time

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

43. “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)))

44. 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)

45. Convergence Time in 1D Defρβ has correlation length ξif for every f, g f g

46. 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

47. 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)

48. 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

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

50. 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