LOCAL RECOVERY MAPS AS DUCT TAPE FOR MANY BODY SYSTEMS

Transcription

1 LOCAL RECOVERY MAPS AS DUCT TAPE FOR MANY BODY SYSTEMS Michael J. Kastoryano November , QuSoft Amsterdam

2 CONTENTS Local recovery maps Exact recovery and approximate recovery Local recovery for many body systems Hammersley-Clifford and Gibbs sampling State preparation Evaluating local expectation values Efficient state preparation Further Applications

3 LOCAL RECOVERY MAPS Strong subadditivity (SSA): I (A : C B) =S(AB)+S(BC) S(B) S(ABC) 0 Equality I (A : C B) =0, R AB ( BC )= Petz map R AB ( )= 1/2 AB 1/2 B 1/2 B 1/2 AB M. Ohya and D. Petz, (2004) Markov State = j AB L j B R j C there exists a disentangling unitary on B. P. Hayden, et. al., CMP 246 (2004)

4 Approximately LOCAL RECOVERY MAPS Strengthening SSA: I (A : C B) 2 log 2 F (,R AB ( AB )) O. Fawzi and R. Renner, CMP 340 (2015) Rotated Petz map R AB ( )= Z 1 dt (t) 2 +it 1 AB 2 it B 1 2 +it 1 2 it B AB M. Junge, et. al. arxiv: ABC are arbitrary Is the map universal? Is the conditional mutual information necessary? Other properties of the map?

5 APPLICATIONS Shannon Theory and Entanglement theory

6 APPLICATIONS Shannon Theory and Entanglement theory Classical Simulations Tensor networks, stoquastic models Quantum Simulations (sampling) Topological order Quantum error correction Renormalization Group, critical models, AdS/CFT

7 APPLICATIONS Shannon Theory and Entanglement theory Classical Simulations Tensor networks, stoquastic models Quantum Simulations (sampling) Topological order Quantum error correction Renormalization Group, critical models, AdS/CFT

8 MANY-BODY SETTING Exact recovery For any A, and B shielding A: I (A : C B) =0 C H = H N 2 ` A B

9 HAMMERSLEY-CLIFFORD Exact recovery For any A, and B shielding A: I (A : C B) =0 > 0 = ih is the Gibbs state of a local commuting H is the ground state of a local commuting H W. Brown, D. Poulin, arxiv: C H = H N 2 ` A B

10 HAMMERSLEY-CLIFFORD Exact recovery For any A, and B shielding A: I (A : C B) =0 > 0 = ih is the Gibbs state of a local commuting H is the ground state of a local commuting H Approximate recovery W. Brown, D. Poulin, arxiv: C H = H N 2 ` A B For any A, and B shielding A: I (A : C B) apple Ke c`

11 HAMMERSLEY-CLIFFORD Exact recovery For any A, and B shielding A: I (A : C B) =0 > 0 = ih is the Gibbs state of a local commuting H is the ground state of a local commuting H Approximate recovery W. Brown, D. Poulin, arxiv: C H = H N 2 ` A B For any A, and B shielding A: I (A : C B) apple Ke c` > 0 = ih is the Gibbs state of a local non-commuting H K. Kato, F Brandao, arxiv: is the ground state of a gaped local non-commuting H

12 Further consequences AREA LAW I(A : B 1 B n+1 ) I(A : B 1 B n )=I(A : B n+1 B 1 B n ) Mutual info area law: I(A : A c ) apple Decaying CMI provides a quantitative MI area law ` A B 1 B 2 B 3

13 Further consequences AREA LAW I(A : B 1 B n+1 ) I(A : B 1 B n )=I(A : B n+1 B 1 B n ) Mutual info area law: I(A : A c ) apple Decaying CMI provides a quantitative MI area law ` A B 1 B 2 B 3 Can also show: Take-home message: Small CMI implies efficient MPS/MPO representation! CMI replaces Area Law, HC program replaces the area law conjecture

14 Further consequences AREA LAW I(A : B 1 B n+1 ) I(A : B 1 B n )=I(A : B n+1 B 1 B n ) Mutual info area law: I(A : A c ) apple Decaying CMI provides a quantitative MI area law What about ` A B 1 dynamics and B 2 B 3 Can also show: state preparation? Small CMI implies efficient MPS/MPO representation! Take-home message: CMI replaces Area Law, HC program replaces the area law conjecture

15 MONTE-CARLO SIMULATIONS Want to evaluate: hqi = X x (x)q(x) / e H classical Gibbs state Idea: - obtain a sample configuration from the distribution - Set up a Markov chain with as an approximate fixed point

16 MONTE-CARLO SIMULATIONS Want to evaluate: hqi = X x (x)q(x) / e H classical Gibbs state Idea: - obtain a sample configuration from the distribution - Set up a Markov chain with as an approximate fixed point Metropolis algorithm: (- start with random configuration) - Flip a spin at random, calculate energy - If energy decreased, accept the flip - If energy increased, accept the flip with probability p flip = e - Repeat until equilibrium is reached E

17 MONTE-CARLO SIMULATIONS Want to evaluate: hqi = X x (x)q(x) / e H classical Gibbs state Idea: - obtain a sample configuration from the distribution - Set up a Markov chain with as an approximate fixed point Metropolis algorithm: (- start with random configuration) - Flip a spin at random, calculate energy - If energy decreased, accept the flip - If energy increased, accept the flip with probability p flip = e E - Repeat until equilibrium is reached Equilibrium?

18 ANALYTIC RESULTS Note: - Glauber dynamics (Metropolis) is modeled by a semigroup P t = e tl

19 ANALYTIC RESULTS Note: - Glauber dynamics (Metropolis) is modeled by a semigroup P t = e tl Fundamental result for Glauber dynamics: has exponentially P t mixes in time O(log(N)) decaying correlations L is gapped F. Martinelli, Lect. Prof. Theor. Stats, Springer A. Guionnet, B. Zegarlinski, Sem. Prob., Springer independent of boundary conditions in 2D independent of specifics of the model no intermediate mixing

20 QUANTUM GIBBS SAMPLERS Commuting Hamiltonian Davies maps are another generalization of Glauber dynamics MJK and K. Temme, arxiv: T t = e tl L = X j2 (R j@ id) R j@ is the Petz recovery map!

21 QUANTUM GIBBS SAMPLERS Commuting Hamiltonian Davies maps are another generalization of Glauber dynamics MJK and K. Temme, arxiv: T t = e tl L = X j2 (R j@ id) R j@ is the Petz recovery map! The exists a partial extension of the statics = dynamics theorem MJK and F. Brandao, CMP 344 (2016)

22 QUANTUM GIBBS SAMPLERS Commuting Hamiltonian Davies maps are another generalization of Glauber dynamics MJK and K. Temme, arxiv: T t = e tl L = X j2 (R j@ id) R j@ is the Petz recovery map! The exists a partial extension of the statics = dynamics theorem MJK and F. Brandao, CMP 344 (2016) Non-commuting Hamiltonian L = X j2 (R j@ id) no longer frustration-free R j@ is the rotated Petz map! Theorem does not hold Davies maps are non-local

23 STATE PREPARATION Based on : MJK, F. Brandao, arxiv:

24 SETTING Lattice: A Hamiltonian: H A = X Z A h Z h j A h Z = 0 for Z K Gibbs states: A = e H A /Tr[e H A ] is the Gibbs state restricted to A Note: Superscript for domain of definition of Gibbs state, while subscript for partial trace.

25 THE MARKOV CONDITION Uniform Markov: C Any subset X = ABC with shielding A from C in X, we have B ` A B I X (A : C B) apple (`) C B ` A B Recall: X = e H X /Tr[e H X ] Also must hold for noncontractible regions

26 CORRELATIONS Uniform Clustering: Any subset X = ABC supp(f) A supp(g) B and with B C ` A Cov X (f,g) apple (`) Cov (f,g) = tr[ fg] tr[ f]tr[ g] C B ` A Note: Uniform Clustering follows from uniform Gap

27 LOCAL PERTURBATIONS Commuting Hamiltonian e (H A +H B) = e H A e H B ` V if [H A,H B ]=0 Non-commuting Hamiltonian MB. Hastings, PRB (2007) General e (H+V ) = O V e H O V O V O`V apple c 1 e c 2` (`) O V apple e V Only works if V is local!

28 APPROXIMATIONS Uniform Markov C I X (A : C B) apple (`) ` A B Uniform clustering C Cov X (f,g) apple (`) B ` A Local perturbations e (H+V ) O`V e H O`V apple c 1 e c 2` (`) ` V

29 LOCAL INDISTINGUISHABILITY Result 1: Any subset with B Any subset X = ABC with shielding A from in, if is and supp(g) C X B uniformly clustering, supp(f) A X = ABC C ` A Cov X (f,g) apple (`) tr BC [ ABC ] tr B [ AB ] 1 apple c AB ( (`)+ (`)) B Consequence: Efficient evaluation of local expectation values ho A i =tr[ O A ] tr[ AB O A ]

30 LOCAL INDISTINGUISHABILITY Result 1: Proof idea: Any subset with B Any subset X = ABC with shielding A from in, if is and supp(g) C X B uniformly clustering, supp(f) A X = ABC C ` A Cov X (f,g) apple (`) tr BC [ ABC ] tr B [ AB ] 1 apple c AB ( (`)+ (`)) B Remove pieces of the boundary of B one by one telescopic sum tr BC [ X AB C ] 1 apple X j tr BC [ X j+1 X j ] 1 Bound each term tr BC [ X j+1 X j ] 1 sup g A tr[g A (O`j X j O`, j X j ] =Cov X j (g,o`, A j O`j)

31 Main Result: STATE PREPARATION If is uniformly Any subset X = clustering ABC and uniformly Markov, then there with exists a depth D +1 and supp(g) circuit of Bquantum channels F = F D+1 F 1 of local range, such that supp(f) A O(log(L)) Cov X (f,g) apple (`) F( ) 1 apple cl D ( (`)+ (`)+ (`)) MJK, F. Brandao, arxiv:

32 Main Result: STATE PREPARATION If is uniformly Any subset X = clustering ABC and uniformly Markov, then there with exists a depth D +1 and supp(g) circuit of Bquantum channels F = F D+1 F 1 of local range, such that supp(f) A O(log(L)) Cov X (f,g) apple (`) F( ) 1 apple cl D ( (`)+ (`)+ (`)) Corollary: MJK, F. Brandao, arxiv: If is uniformly Any subset X = clustering ABC and uniformly Markov, then there with exists a depth M = and supp(g) O(log(L)) circuit B of strictly local quantum channels, such that supp(f) A F = F M F 1 Cov X (f,g) apple (`) F( ) 1 apple cl D ( (`)+ (`)+ (`))

33 PROOF OUTLINE Step 1: A Cover the lattice in concentric + squares A A A + A A ` By the Markov condition R A + ( A c) 1 apple N A ( (`)+ (`)) By Local indistinguishability tr A [ Ac A c ] A c] 1 apple N A (`) Local cpt map F A R A + tr A F A ( Ac ) 1 apple N A ( (`)+ (`)+ (`)) If we can build the lattice A c reconstruct the original lattice. with holes, then we can

34 PROOF OUTLINE Step 2: Break up the connecting regions B B B + B + By the Markov condition A B B ` c R A B + ( Ac B c ) Ac 1 apple N B ( (`)+ (`)) By Local indistinguishability tr B [ (A B )c ] Ac B c ] 1 apple N B (`) Local cpt map c F B R A B + tr B F B F A ( (A B )c ) 1 apple (N A + N B )( (`)+ (`)+ (`)) If we can build the lattice the original lattice. (A B ) c, then we can reconstruct

35 PROOF OUTLINE Step 3: Project onto C C By locality F C ( ) = c tr C [ ] Finally F C F B F A ( ) 1 apple (N C + N A + N B )( (`)+ (`)+ (`)) The entire lattice can be built from a local circuit of cpt maps.

36 GROUND STATES Proof ingredients (uniform) Local indistinguishability (uniform) Markov condition Local definition of states For injective PEPS, proof can be reproduced exactly. We can show that the conditions of the theorem hold it the topological entanglement entropy is zero.

37 SPECTRAL GAP We showed: F C F B F A ( ) 1 apple L D e `/ Define F A = e tl A L A = X j (F Ai id) If F A, F B, F C had the same fixed point, then is gaped, by the reverse detectability lemma. L = L A + L B + L C A. Anshu, et. al., Phys. Rev. B 93, (2016) The same strategy works for proving gaps of parent Hamiltonians of injective PEPS New strategy for proving the gap of the 2D AKLT model!!! All about boundary conditions

38 OUTLOOK Spectral gap analysis, entanglement spectrum New classification for many-body systems Approximate Quantum error correction New codes? Tradeoff bounds S. Flammia, J. Haah, MJK, I. Kim, arxiv: Renormalization Group, critical models, AdS/CFT

39 THANK YOU!