LOCAL RECOVERY MAPS AS DUCT TAPE FOR MANY BODY SYSTEMS Michael J. Kastoryano November 14 2016, QuSoft Amsterdam Tuesday, November 15, 16
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 Tuesday, November 15, 16
LOCAL RECOVERY MAPS Strong subadditivity (SSA): 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 ⇔ R AB ( σ ) = ρ 1 / 2 AB ρ − 1 / 2 σρ − 1 / 2 ρ 1 / 2 Petz map 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. Tuesday, November 15, 16
Approximately LOCAL RECOVERY MAPS Strengthening SSA: I ρ ( A : C | B ) ≥ − 2 log 2 F ( ρ , R AB ( ρ AB )) 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 ABC are arbitrary Is the map universal? Is the conditional mutual information necessary? Other properties of the map? Tuesday, November 15, 16
APPLICATIONS Shannon Theory and Entanglement theory Tuesday, November 15, 16
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 Tuesday, November 15, 16
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 Tuesday, November 15, 16
MANY -BODY SETTING Exact recovery H = H ⊗ N For any A, and B shielding A: 2 I ρ ( A : C | B ) = 0 C ` A B Tuesday, November 15, 16
HAMMERSLEY -CLIFFORD 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 Tuesday, November 15, 16
HAMMERSLEY -CLIFFORD 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: Tuesday, November 15, 16
HAMMERSLEY -CLIFFORD 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 local non-commuting H ρ > 0 K. Kato, F Brandao, arXiv:1609.06636 is the ground state of a gaped local non-commuting H ρ = | ψ ih ψ | Tuesday, November 15, 16
AREA LAW Further consequences 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 ) ≤ c | ∂ A | Decaying CMI provides a ` A quantitative MI area law B 1 B 2 B 3 Tuesday, November 15, 16
AREA LAW Further consequences 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 ) ≤ c | ∂ A | Decaying CMI provides a ` A quantitative MI area law B 1 B 2 B 3 Can also show: Small CMI implies efficient MPS/MPO representation! Take-home message: CMI replaces Area Law, HC program replaces the area law conjecture Tuesday, November 15, 16
AREA LAW Further consequences 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 ) ≤ c | ∂ A | Decaying CMI provides a ` A quantitative MI area law What about B 1 dynamics and B 2 B 3 state preparation? Can also show: Small CMI implies efficient MPS/MPO representation! Take-home message: CMI replaces Area Law, HC program replaces the area law conjecture Tuesday, November 15, 16
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 Tuesday, November 15, 16
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 Tuesday, November 15, 16
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? Tuesday, November 15, 16
ANALYTIC RESULTS Note: - Glauber dynamics (Metropolis) is modeled by a semigroup P t = e tL Tuesday, November 15, 16
ANALYTIC RESULTS Note: - Glauber dynamics (Metropolis) is modeled 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 Tuesday, November 15, 16
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 Tuesday, November 15, 16
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) Tuesday, November 15, 16
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 Tuesday, November 15, 16
STATE PREPARATION Based on : MJK, F. Brandao, arXiv:1609.07877 Tuesday, November 15, 16
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. Tuesday, November 15, 16
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 Tuesday, November 15, 16
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 Tuesday, November 15, 16
LOCAL PERTURBATIONS Commuting Hamiltonian Λ ` V e − β ( H A + H B ) = e − β H A e − β H B if [ H A , H B ] = 0 Non-commuting Hamiltonian MB. Hastings, PRB 201102 (2007) 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 || Only works if is local! V Tuesday, November 15, 16
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 Tuesday, November 15, 16
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