FINITE CORRELATION LENGTH IMPLIES EFFICIENT PREPARATION OF GIBBS STATES Fernando GSL Brandao and Michael J. Kastoryano January 20 2017, QIP Seattle
CONTENTS Motivation Local recovery for many body systems Exact and approximate recovery State preparation Evaluating local expectation values Efficient state preparation Further Applications
MOTIVATION Finite temperature quantum simulations Strongly correlated/frustrated materials Quantum SDP solvers, Quantum machine learning New tools for the analysis of many body systems Local recovery in many body systems Exotic phases/topological order
STRONG SUB-ADDITIVITY Strong sub-additivity (SSA): A C B I ρ ( A : C | B ) = S ( AB ) + S ( BC ) − S ( B ) − S ( ABC ) ≥ 0
STRONG SUB-ADDITIVITY Strong sub-additivity (SSA): A C B I ρ ( A : C | B ) = S ( AB ) + S ( BC ) − S ( B ) − S ( ABC ) ≥ 0 Area Law for mixed states: I ( A : A c ) ≡ S ( A ) + S ( A c ) − S ( AA c ) ≤ c | ∂ A |
STRONG SUB-ADDITIVITY Strong sub-additivity (SSA): A C B I ρ ( A : C | B ) = S ( AB ) + S ( BC ) − S ( B ) − S ( ABC ) ≥ 0 Area Law for mixed states: I ( A : A c ) ≡ S ( A ) + S ( A c ) − S ( AA c ) ≤ c | ∂ A | B ` Quantitative extension to the Area Law A I ( A : B 1 · · · B n +1 ) − I ( A : B 1 · · · B n ) = I ( A : B n +1 | B 1 · · · B n ) Tells us how rapidly the area law is saturated
LOCAL RECOVERY MAPS Strong subadditivity (SSA): A C B I ρ ( A : C | B ) = S ( AB ) + S ( BC ) − S ( B ) − S ( ABC ) ≥ 0 Equality R AB ( ρ BC ) = ρ I ρ ( A : C | B ) = 0 ⇔ Petz map R AB ( σ ) = ρ 1 / 2 AB ρ − 1 / 2 σρ − 1 / 2 ρ 1 / 2 B B AB M. Ohya and D. Petz, (2004) Markov State ρ = ⊕ j ρ AB L j ⊗ ρ B R j C P. Hayden, et. al., CMP 246 (2004) there exists a disentangling unitary on B.
Approximately LOCAL RECOVERY MAPS Strengthening SSA: I ρ ( A : C | B ) ≥ − 2 log F ( ρ , R AB ( ρ BC )) O. Fawzi and R. Renner, CMP 340 (2015) Rotated Petz map Z 1 − 1 − 1 1 2 + it 2 − it 2 + it 2 − it R AB ( σ ) = dt β ( t ) ρ AB ρ σρ ρ B B AB M. Junge, et. al. arXiv:1509.07127 D. Sutter, et. al. arXiv:1604.03023 ABC are arbitrary Related to theory of approximate error correction (subspaces) S. Flammia et. al. , arXiv:1610.06169
CLASSIFICATION Exact recovery H = H ⊗ N For any A, and B shielding A: 2 I ρ ( A : C | B ) = 0 C is the Gibbs state of a local ρ > 0 commuting H ` A W. Brown, D. Poulin, arXiv:1206.0755 is the ground state of a ρ = | ψ ih ψ | B local commuting H
CLASSIFICATION Exact recovery H = H ⊗ N For any A, and B shielding A: 2 I ρ ( A : C | B ) = 0 C is the Gibbs state of a local ρ > 0 commuting H ` A W. Brown, D. Poulin, arXiv:1206.0755 is the ground state of a ρ = | ψ ih ψ | B local commuting H Approximate recovery I ⇢ ( A : C | B ) ≤ Ke − c ` For any A, and B shielding A: is the Gibbs state of a quasi-local Hamiltonian ρ > 0 K. Kato, F Brandao, arXiv:1609.06636 is the ground state of a gaped quasi-local Hamiltonian ρ = | ψ ih ψ |
Dynamics?
MONTE-CARLO SIMULATIONS Want to evaluate: X π ∝ e − β H h Q i = π ( x ) Q ( x ) classical Gibbs state x Idea: - obtain a sample configuration from the distribution π - Set up a Markov chain with as an approximate π fixed point
MONTE-CARLO SIMULATIONS Want to evaluate: X π ∝ e − β H h Q i = π ( x ) Q ( x ) classical Gibbs state x Idea: - obtain a sample configuration from the distribution π - Set up a Markov chain with as an approximate π 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 − β ∆ E - Repeat until equilibrium is reached
MONTE-CARLO SIMULATIONS Want to evaluate: X π ∝ e − β H h Q i = π ( x ) Q ( x ) classical Gibbs state x Idea: - obtain a sample configuration from the distribution π - Set up a Markov chain with as an approximate π 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 − β ∆ E - Repeat until equilibrium is reached Equilibrium?
ANALYTIC RESULTS Note: - Glauber dynamics (Metropolis) is modelled by a semigroup P t = e tL
ANALYTIC RESULTS Note: - Glauber dynamics (Metropolis) is modelled by a semigroup P t = e tL Fundamental result for Glauber dynamics: mixes in time has exponentially O (log( N )) P t π decaying correlations is gapped L 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
QUANTUM GIBBS SAMPLERS Commuting Hamiltonian T t = e t L X L = ( R j ∂ − id ) Davies maps are another j ∈ Λ generalization of Glauber R j ∂ is the Petz recovery map! dynamics MJK and K. Temme, arXiv:1505.07811 The exists a partial extension of the statics = dynamics theorem MJK and F. Brandao, CMP 344 (2016)
QUANTUM GIBBS SAMPLERS Commuting Hamiltonian T t = e t L X L = ( R j ∂ − id ) Davies maps are another j ∈ Λ generalization of Glauber R j ∂ is the Petz recovery map! dynamics MJK and K. Temme, arXiv:1505.07811 The exists a partial extension of the statics = dynamics theorem MJK and F. Brandao, CMP 344 (2016) Non-commuting Hamiltonian X L = ( R j ∂ − id ) no longer frustration-free j ∈ Λ Theorem does not hold R j ∂ is the rotated Petz map! Davies maps are non-local
New approach
SETTING Λ Lattice: A ⊂ Λ h j A Hamiltonian: X H A = h Z Z ⊂ A h Z = 0 for | Z | ≥ K ρ A = e − β H A / Tr[ e − β H A ] Gibbs states: is the Gibbs state restricted to A Superscript for domain of definition of Gibbs state, Note: while subscript for partial trace.
THE MARKOV CONDITION Uniform Markov: C Any subset with X = ABC ⊂ Λ B ` A shielding from in , we A C X have B I ρ X ( A : C | B ) ≤ � ( ` ) Λ B C A B ` ρ X = e − β H X / Tr[ e − β H X ] Recall: Also must hold for non- contractible regions
CORRELATIONS Uniform Clustering: C Any subset with ` X = ABC ⊂ Λ B and A supp( f ) ⊂ A supp( g ) ⊂ B Cov ρ X ( f, g ) ≤ ✏ ( ` ) Λ A ` B C Cov ρ ( f, g ) = | tr[ ρ fg ] − tr[ ρ f ]tr[ ρ g ] | Uniform Clustering Note: follows from uniform Gap
LOCAL PERTURBATIONS Commuting Hamiltonian e − β ( H A + H B ) = e − β H A e − β H B Λ if [ H A , H B ] = 0 ` V Non-commuting Hamiltonian General e − β ( H + V ) = O V e − β H O † V V || ≤ c 1 e − c 2 ` ≡ � ( ` ) || O V − O ` || O V || ≤ e β || V || MB. Hastings, PRB 201102 (2007) Only works if is local! V
APPROXIMATIONS Uniform Markov C ` A I ρ X ( A : C | B ) ≤ � ( ` ) B Uniform clustering C ` Cov ρ X ( f, g ) ≤ ✏ ( ` ) B A Local perturbations Λ || e − � ( H + V ) − O ` V || ≤ c 1 e − c 2 ` ≡ � ( ` ) V e − � H O ` ` V
LOCAL INDISTINGUISHABILITY Result 1: Any subset with C X = ABC ⊂ Λ B Any subset with X = ABC ⊂ Λ shielding from in , if is A C X ρ and supp( f ) ⊂ A supp( g ) ⊂ B ` uniformly clustering, A Cov ρ X ( f, g ) ≤ ✏ ( ` ) B || tr BC [ ⇢ ABC ] − tr B [ ⇢ AB ] || 1 ≤ c | AB | ( ✏ ( ` ) + � ( ` )) Λ Consequence: Efficient evaluation of local expectation values h O A i = tr[ ρ Λ O A ] ⇡ tr[ ρ AB O A ]
LOCAL INDISTINGUISHABILITY Proof idea: Result 1: Any subset with C X = ABC ⊂ Λ B Any subset with X = ABC ⊂ Λ shielding from in , if is A C X ρ and supp( f ) ⊂ A supp( g ) ⊂ B ` uniformly clustering, A Cov ρ X ( f, g ) ≤ ✏ ( ` ) B || tr BC [ ⇢ ABC ] − tr B [ ⇢ AB ] || 1 ≤ c | AB | ( ✏ ( ` ) + � ( ` )) Remove pieces of the boundary of one by one B telescopic sum || tr BC [ ρ X − ρ AB ⊗ ρ C ] || 1 ≤ X || tr BC [ ρ X j +1 − ρ X j ] || 1 j Bound each term j ρ X j O ` , † || tr BC [ ρ X j +1 − ρ X j ] || 1 ≈ sup | tr[ g A ( O ` − ρ X j ] | j g A = Cov ⇢ Xj ( g A , O ` , † j O ` j )
STATE PREPARATION Main Result: If is uniformly clustering and uniformly Markov, then there ρ Any subset with X = ABC ⊂ Λ exists a depth circuit of quantum channels of F = F D +1 · · · F 1 D + 1 and supp( f ) ⊂ A supp( g ) ⊂ B local range , such that O (log( L )) Cov ρ X ( f, g ) ≤ ✏ ( ` ) || F ( ) − ⇢ || 1 ≤ cL D ( ✏ ( ` ) + � ( ` ) + � ( ` )) MJK, F. Brandao, arXiv:1609.07877
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