Quantum Gibbs Samplers

Transcription

1 Quantum Gibbs Samplers Fernando G.S.L. Brandão QuArC, Microso8 Research Q- QuArC Retreat, 2015

2 Dynamical Proper6es H ij Hamiltonian: State at Fme t: Compute: ExpectaFon values: Temporal correlafons: H = P hi,ji H ij (t)i = e ith ",...,"i h (t) M (t)i h (t) Me ith Ne ith M (t)i

3 Quantum Simulators, Dynamical Quantum Computer Can simulate the dynamics of every mulf- parfcle quantum system Spin models (Lloyd 96,, Berry, Childs, Kothari 15) Fermionic and bosonic models (Bravyi, Kitaev 00, ) Topological quantum field theory (Freedman, Kitaev, Wang 02) Quantum field theory (Jordan, Lee, Preskill 11) Quantum Chemestry (HasFngs, Wecker, Bauer, Triyer 14) Quantum Simulators Can simulate the dynamics of parfcular models OpFcal La`ces Ion Traps SuperconducFng systems

4 Sta6c Proper6es

5 Sta6c Proper6es H ij Hamiltonian: H = P hi,ji H ij

6 Sta6c Proper6es H ij Hamiltonian: Groundstate: Thermal state: H = P hi,ji H ij H 0 i = E 0 0 i = e H /Z( ) Compute: local expectafon values (e.g. magnefzafon), correlafon funcfons (e.g. hz i Z i+l i),

7 Sta6c Proper6es Can we prepare groundstates? Warning: In general it requires exponenfal Fme to prepare, even for one- dimensional translafonal- invariant models - - it s a computafonal- hard problem (Gocesman- Irani 09)

8 Sta6c Proper6es Can we prepare groundstates? Warning: In general it requires exponenfal Fme to prepare, even for one- dimensional translafonal- invariant models - - it s a computafonal- hard problem (Gocesman- Irani 09) Method 1: AdiabaFc EvoluFon; works if Δ n - c H(s i ) ψ i H(s) ψ s H(s f ) H(s)ψ s = E 0,s ψ s Δ := min Δ(s) Method 2: Phase EsFmaFon; works if can find a simple state 0> such that h 0 0i n c (Abrams, Lloyd 99) *, 0i = X i c i i, 0i! X i c i i,e i i,h i i = E i i i

9 Sta6c Proper6es Can we prepare thermal states? = e H /Z( ) Warning: ExponenFal- Fme in worst case (NP- hard to esfmate energy of general Gibbs states)

10 Sta6c Proper6es Can we prepare thermal states? = e H /Z( ) Warning: ExponenFal- Fme in worst case (NP- hard to esfmate energy of general Gibbs states) Are quantum computers useful in some cases?

11 Plan 1. Classical Glauber dynamics 2. Mixing in space vs mixing in Fme 3. Quantum Master EquaFons 4. Mixing in space vs mixing in Fme for commufng quantum systems 5. Approach to non- commufng 6. PotenFal ApplicaFons

12 Thermaliza6on Can we prepare thermal states? = e H /Z( ) Method 1: Couple to a bath of the right temperature and wait. S B But size of environment might be huge. Maybe not efficient. No guarantee Gibbs state will be reached

13 Method 2: Metropolis Sampling Consider e.g. Ising model: H = X hi,ji J ij s i s j + X i g i s i Coupling to bath modeled by stochasfc map Q i j "..."#"#...i! "..."""#...i Metropolis Update: Q i!j =min(1, exp( (E j E i ))) The stafonary state is the thermal (Gibbs) state: p i = e E i /Z

14 Method 2: Metropolis Sampling Consider e.g. Ising model: H = X hi,ji J ij s i s j + X i g i s i Coupling to bath modeled by stochasfc map Q i j "..."#"#...i! "..."""#...i Metropolis Update: Q i!j =min(1, exp( (E j E i ))) The stafonary state is the thermal (Gibbs) state: p i = e E i /Z (Metropolis et al 53) We devised a general method to calculate the properfes of any substance comprising individual molecules with classical stafsfcs Example of Markov Chain Monte Carlo method. Extremely useful algorithmic technique

15 Classical Glauber Dynamics A stochasfc map R = e G is a Glauber dynamics for a (classical) Hamiltonian if it s generator G is local and the unique fixed point of R is e - βh /Z(β) (+ detailed balance) Ex: Metropolis, Heat- bath generator,.

16 Classical Glauber Dynamics A stochasfc map R = e G is a Glauber dynamics for a (classical) Hamiltonian if it s generator G is local and the unique fixed point of R is e - βh /Z(β) (+ detailed balance) Ex: Metropolis, Heat- bath generator,. When is Glauber dynamics effecfve for sampling from Gibbs state? (Stroock, Zergalinski 92; MarFnelli, Olivieri 94, ) Gibbs state with finite correlafon length (Sly 10) Rapidly mixing Glauber dynamics (Proved only for hard core mode and 2- spin anf- ferromagefc model) Gibbs Sampling in P (vs NP - hard)

17 Glauber Dynamics Markov chains (discrete or continuous) on the space of configurations {0, 1} n that have the Gibbs state as the stationary distribution: p t = R t p 0 p 1 (i) =e E i /Z transi6on matrix aner t 6me steps sta6onary distribu6on

18 Temporal Mixing p t = R t P 0 = X i t ip i p 0 eigenvalues eigenprojectors P 0 / p T 1p 1 Convergence Fme given by the gap Δ = 1- λ 1 : kp t p 1 k 1 apple 2 n 2 t log(1/(1 ) 2 n t Time of equilibrafon n/δ We have rapid mixing if Δ = constant

19 Spa6al Mixing µ V Let be the Gibbs state for a model in the la`ce V with boundary condifons τ, i.e. µ V ( )= e H(, ) P e H(, ) blue: V, red: boundary Ex. τ = (0, 0)

20 Spa6al Mixing µ V Let be the Gibbs state for a model in the la`ce V with boundary condifons τ, i.e. µ V ( )= e H(, ) P e H(, ) def: The Gibbs state has correlafon length ξ if for every f, g blue: V, red: boundary Ex. τ = (0, 0) apple sup 2B µ V (fg) µ V (f)µ V (g) kfkkgke dist(f,g)/ f g

21 Temporal Mixing vs Spa6al Mixing (Stroock, Zergalinski 92; MarFnelli, Olivieri 94, ) For every classical Hamiltonian, the Gibbs state has finite correla6on length if, and only if, the Glauber dynamics has a finite gap

22 Temporal Mixing vs Spa6al Mixing (Stroock, Zergalinski 92; MarFnelli, Olivieri 94, ) For every classical Hamiltonian, the Gibbs state has finite correla6on length if, and only if, the Glauber dynamics has a finite gap Obs1: Same is true for the log- Sobolev constant of the system Obs2: For many models, when correlafon length diverges, gap is exponenfally small in the system size (e.g. Ising model) Obs3: Any model in 1D, and any model in arbitrary dim. at high enough temperature, has a finite correlafon length (connected to uniqueness of the phase, e.g. Dobrushin s condifon)

23 Temporal Mixing vs Spa6al Mixing (Stroock, Zergalinski 92; MarFnelli, Olivieri 94, ) For every classical Hamiltonian, the Gibbs state has finite correla6on length if, and only if, the Glauber dynamics has a finite gap Obs1: Same is true for the log- Sobolev constant of the system Obs2: For many models, when correlafon length diverges, gap is exponenfally small in the system size (e.g. Ising model) Obs3: Any model in 1D, and any model in arbitrary dim. at high enough temperature, has a finite correlafon length (connected to uniqueness of the phase, e.g. Dobrushin s condifon) Does something similar hold in the quantum case? 1 st step: Need a quantum version of Glauber dynamics

24 Preparing Quantum Thermal States (Terhal and divincenzo 00, ) Simulate interacfon of system with heat bath no run- Fme esfmate (Poulin, Wocjan 09, ) Grover- type speed- up for preparing Gibbs states exponenfal run- Fme (Temme et al 09) Quantum metropolis: Quantum channel s.t. (i) can be implemented efficiently on a quantum computer and (ii) has Gibbs state as fixed point no run- Fme esfmate (Yung, Aspuru- Guzik 10) Amplitude amplificafon applied to quantum metropolis: Square- root speed- up on spectral gap. no run- Fme esfmate (HasFngs 08; Bilgin, Boixo 10) Poly- Fme quantum/classical algorithm for every 1D model restricted to 1D

25 Quantum Master Equa6ons Canonical example: cavity QED d dt = i[h, ]+2apple a a 1 2 (a a + a a) Lindblad Equa6on: d dt = L( ) = i[h, ]+X i i A i A i 1 2 A i A i 1 2 A i A i (most general Markovian and Fme homogeneous q. master equafon)

26 Quantum Master Equa6ons Canonical example: cavity QED d dt = i[h, ]+2apple a a 1 2 (a a + a a) Lindblad Equa6on: d dt = L( ) = i[h, ]+X i i A i A i 1 2 A i A i 1 2 A i A i (most general Markovian and Fme homogeneous q. master equafon) Generator completely posi6ve trace- preserving map: t ( ) =e tl ( ) fixed point: f = lim t!1 e tl ( ) () L( f )=0 How fast does it converge? Determined by gap of of Lindbladian

27 Quantum Master Equa6ons Canonical example: cavity QED d dt = i[h, ]+2apple a a 1 2 (a a + a a) Lindblad Equa6on: d dt = L( ) = i[h, ]+X i i A i A i 1 2 A i A i 1 2 A i A i Local master equa6ons: L is k- local if all A i act on at most k sites A i (Kliesch et al 11) Time evolufon of every k- local Lindbladian on n qubits can be simulated in Fme poly(n, 2^k) in the circuit model

28 Davies Maps (aka Quantum Glauber Dynamics) Lindbladian: L( ) = X g (!) S (!) [,S (!)] + e! S (!)[,S (!) ]+h.c.,!>0 Lindblad terms: g (!) : spectral density Z S (!) = e i!t e ith S e ith dt S (!) k,l = (E k E l!)he k S E l i

29 Davies Maps (aka Quantum Glauber Dynamics) Lindbladian: L( ) = X g (!) S (!) [,S (!)] + e! S (!)[,S (!) ]+h.c.,!>0 Lindblad terms: g (!) : spectral density Z S (!) = e i!t e ith S e ith dt S (!) k,l = (E k E l!)he k S E l i H ij H = P hi,ji H ij S α (X α, Y α, Z α ) Thermal state is the unique fixed point: L( )=0 (safsfies q. detailed balance: L = L, (X) := 1/2 X 1/2 )

30 Why Davies Maps? (Davies 74) Rigorous derivafon in the weak- coupling limit: H SB = InteracFng Ham. X S B Coarse grain over Fme t λ - 2 >> max(1/ (E i E j + E k - E l )) (E i : eigenvalues of H)

31 Why Davies Maps? (Davies 74) Rigorous derivafon in the weak- coupling limit: H SB = InteracFng Ham. X S B Coarse grain over Fme t λ - 2 >> max(1/ (E i E j + E k - E l )) (E i : eigenvalues of H) But: for n spin Hamiltonian 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); S (!) = Z e i!t e ith S e ith dt But for commu6ng Hamiltonian H, it is local Z S (!) = e i!t e ) S e ) dt

32 Spectral Gap The relevant gap is given by (L) :=inf f hf,l(f)i var (f) L 2 weighted inner product: hf,gi := tr( 1/2 f 1/2 g ) Variance: var := hf,fi tr( f) 2 (Kastoryano, Temme 11, ) Mixing Fme of order ke tl ( ) n (L)T k 1 apple 2 2n/T e (L)t

33 Previous Results (Alicki, Fannes, Horodecki 08) Λ = Ω(1) for 2D toric code (Alicki, Horodecki, Horodecki, Horodecki 08) Λ = exp(- Ω(n)) for 4D toric code (Temme 14) Λ > exp(- βε)/n, with ε the energy barrier, for stabilizer Hamiltonians

34 Equivalence of Clustering in Space and Time for Quantum Commu6ng thm For commufng Hamiltonians in a finite dimensional la`ce, the Davies generator has a constant gap if, and only if, the Gibbs state safsfies strong clustering of correla6ons (Kastoryano, B )

35 Equivalence of Clustering in Space and Time for Quantum Commu6ng thm For commufng Hamiltonians in a finite dimensional la`ce, the Davies generator has a constant gap if, and only if, the Gibbs state safsfies strong clustering of correla6ons Strong Clustering holds true in: 1D at any temperature Any D at sufficiently high temperature (crifcal T determined only by dim and interacfon range)

36 Equivalence of Clustering in Space and Time for Quantum Commu6ng thm For commufng Hamiltonians in a finite dimensional la`ce, the Davies generator has a constant gap if, and only if, the Gibbs state safsfies strong clustering of correla6ons Strong Clustering holds true in: 1D at any temperature Any D at sufficiently high temperature (crifcal T determined only by dim and interacfon range) Gives first polynomial- Fme quantum algorithm for preparing Gibbs states of commufng models at high temperature. Caveat: At high temperature cluster expansion works well for compufng local expectafon values. (Open: How the two threshold T s compare?) Q advantage: we get the full Gibbs state (e.g. could perform swap test of purificafons of two Gibbs states. Good for anything?)

37 Strong Clustering [ B) B def: Strong clustering holds if there is ξ>0 s.t. for every A and B and operator f acfng on A [ B [ (@(A [ B)) apple cov A[B (E A (f), E B (f)) hf,fi e d(ac,b c )/ Condi6onal Covariance: cov X (f,g) := hf E X (f),g E X (g)i Condi6onal Expecta6on: X c : complement of X E X (f) := lim t!1 e tl X (f) L X := X k2x L k d(x, Y) : distance between regions X and Y

38 Strong Clustering [ B) B def: Strong clustering holds if there is ξ>0 s.t. for every A and B and operator f acfng on A [ B [ (@(A [ B)) apple cov A[B (E A (f), E B (f)) hf,fi e d(ac,b c )/ Fact: E X (f) =I X f X c Strong clustering follows if for every f, g in AUB and M>0 [ B) cov(f,g) M apple e d(f,g)/ with tr (A[B) c(m I A[B ) tr(m I A[B ) condifonal state given M on boundary.

39 Strong Clustering - > Gap We show that under the clustering condifon: (A [ B) & min( (A), (B)) A B Ge`ng: (V ) & (V 0 ) V : enfre la`ce V 0 : subla`ce of size O(ξ) Follows idea of a proof of the classical analogue for Glauber dynamics (BerFni et al 00) Key lemma: If cov A[B (E A (f), E B (f)) apple "var A[B (f) var A[B (f) apple (1 2") 1 (var A (f) + var B (f)) then

40 Gap - > Strong Clustering Employs the following mapping between Liouvillians for commufng Hamiltonians and local Hamiltonians on a larger space: ˆL := 1/4 L( 1/4 f 1/4 ) 1/4 Apply the detectability lemma (Aharonov et al 10) to prove gap - > strong clustering (strengthening proof previous proof that gap - > clustering)

41 Non- Commu6ng Hamiltonians? Davies Master EquaFon is Non- Local Ques6on: Can it be implemented efficiently on a quantum computer?

42 Non- Commu6ng Hamiltonians? Davies Master EquaFon is Non- Local Beyond master equa6ons: def: Hamiltonian H safsfies local indisfnguishability (LI) if for every A B tr B c(e H/T /Z(T )) tr B c(e H A/T /Z A (T )) 1 apple e d(a,bc )/ Λ A B

43 Non- Commu6ng Hamiltonians? Davies Master EquaFon is Non- Local Beyond master equa6ons: def: Hamiltonian H safsfies local indisfnguishability (LI) if for every A B tr B c(e H/T /Z(T )) tr B c(e H A/T /Z A (T )) thm Suppose H (local Ham in d dim) safsfies LI and for every region A, 1 apple e d(a,bc )/ l I(A : B l ) I(A : B l+1 ) e cl A B l Then e - H/T /Z(T) can be created by a quantum circuit of size exp(o(log d (n))) Based on recent lower bound by Fawzi and Renner on condifonal mutual informafon

44 Non- Commu6ng Hamiltonians? Davies Master EquaFon is Non- Local Beyond master equa6ons: def: Hamiltonian H safsfies local indisfnguishability (LI) if for every A B tr B c(e H/T /Z(T )) tr B c(e H A/T /Z A (T )) thm Suppose H (local Ham in d dim) safsfies LI and for every region A, 1 apple e d(a,bc )/ l I(A : B l ) I(A : B l+1 ) e cl A B l Then e - H/T /Z(T) can be created by a quantum circuit of size exp(o(log d (n))) When is the condifon on mutual informafon true? (always true for commufng models) Can we improve to poly(n) Fme?

45 Applica6ons of Q. Gibbs Samplers 1. Machine Learning? (Nate s talk) 2. Quantum Algorithms for semidefinite programming? Minimize tr(y ) Subject to: tr(a i Y ) 1 Y 0 A i,y 2 Mat(d, d) O(d 4 log(1/ε))- Fme algorithm (interior point methods) For sparse A i, Y, is there a polylog(d) quantum algorithm? (think of HHL quantum algorithm for linear equafons)

46 Applica6ons of Q. Gibbs Samplers 1. Machine Learning? (Nate s talk) 2. Quantum Algorithms for semidefinite programming? Minimize tr(y ) Subject to: tr(a i Y ) 1 Y 0 A i,y 2 Mat(d, d) O(d 4 log(1/ε))- Fme algorithm (interior point methods) For sparse A i, Y, is there a polylog(d) quantum algorithm? (think of HHL quantum algorithm for linear equafons) No, unless NP in BQP

47 Applica6ons of Q. Gibbs Samplers 1. Machine Learning? (Nate s talk) 2. Quantum Algorithms for semidefinite programming? Minimize tr(y ) Subject to: tr(a i Y ) 1 Y 0 A i,y 2 Mat(d, d) (Peng, Tangwongsan 12) For A i > 0 ( A <1), can reduce the problem to polylog(d) evaluafons of tr e P i ia i A j e P i ia i (for λ i < log(d)/ε)

48 Applica6ons of Q. Gibbs Samplers 1. Machine Learning? (Nate s talk) 2. Quantum Algorithms for semidefinite programming? Minimize tr(y ) Subject to: tr(a i Y ) 1 Y 0 A i,y 2 Mat(d, d) (Peng, Tangwongsan 12) For A i > 0 ( A <1), can reduce the problem to polylog(d) evaluafons of tr e P i ia i A j e P i ia i (for λ i < log(d)/ε) Are there interesfng choices of {A i } for which quantum gives an advantage to compute the Gibbs state above?