Rapid mixing and clustering of correlations in open quantum systems Michael Kastoryano Dahlem Center for Complex Quantum Systems, Freie Universit¨ at Berlin QCCC, Prien/Chiemsee Prien/Chiemsee, October 21, 2013 Michael Kastoryano (Berlin) Mixing vs. Clustering Prien/Chiemsee, October 21, 2013 1 / 33
Outline 1 Introduction Setting Motivation 2 Preliminaries Rapid mixing bounds Correlation Measures 3 Rapid mixing implies clustering χ 2 clustering Log-Sobolev clustering and stability Area Law 4 Clustering implies rapid mixing The main theorem Corollaries 5 Outlook Michael Kastoryano (Berlin) Mixing vs. Clustering Prien/Chiemsee, October 21, 2013 2 / 33
Table of Contents Introduction 1 Setting Motivation Preliminaries 2 Rapid mixing bounds Correlation Measures Rapid mixing implies clustering 3 χ 2 clustering Log-Sobolev clustering and stability Area Law Clustering implies rapid mixing 4 The main theorem Corollaries Outlook 5 Michael Kastoryano (Berlin) Mixing vs. Clustering Prien/Chiemsee, October 21, 2013 3 / 33
Setting Finite state space: n × n complex matrices. Michael Kastoryano (Berlin) Mixing vs. Clustering Prien/Chiemsee, October 21, 2013 4 / 33
Setting Finite state space: n × n complex matrices. Markovian Dynamics k − 1 � ∂ t ρ = L ∗ ( ρ ) = i [ H , ρ ] + L k ρ L † 2 { L † k L k , ρ } + k Typically, we will assume that L k and H are bounded (there exists a K < ∞ s.t. || L k || ≤ K for all k ) and geometrically local on a d -dimensional cubic lattice of side length L . Michael Kastoryano (Berlin) Mixing vs. Clustering Prien/Chiemsee, October 21, 2013 4 / 33
Setting Finite state space: n × n complex matrices. Markovian Dynamics k − 1 � ∂ t ρ = L ∗ ( ρ ) = i [ H , ρ ] + L k ρ L † 2 { L † k L k , ρ } + k Typically, we will assume that L k and H are bounded (there exists a K < ∞ s.t. || L k || ≤ K for all k ) and geometrically local on a d -dimensional cubic lattice of side length L . We say that L is primitive if it has has a unique full-rank stationary state σ > 0. Michael Kastoryano (Berlin) Mixing vs. Clustering Prien/Chiemsee, October 21, 2013 4 / 33
Setting Finite state space: n × n complex matrices. Markovian Dynamics k − 1 � ∂ t ρ = L ∗ ( ρ ) = i [ H , ρ ] + L k ρ L † 2 { L † k L k , ρ } + k Typically, we will assume that L k and H are bounded (there exists a K < ∞ s.t. || L k || ≤ K for all k ) and geometrically local on a d -dimensional cubic lattice of side length L . We say that L is primitive if it has has a unique full-rank stationary state σ > 0. We say L is reversible (detailed balance) if L ∗ ( √ σ g √ σ )) = √ σ L ( g ) √ σ. Michael Kastoryano (Berlin) Mixing vs. Clustering Prien/Chiemsee, October 21, 2013 4 / 33
Mixing ⇔ Clustering Mixing times: There exist constant A , b > 0 such that: � e t L ∗ ( ρ 0 ) − σ � 1 ≤ Ae − bt . Michael Kastoryano (Berlin) Mixing vs. Clustering Prien/Chiemsee, October 21, 2013 5 / 33
Mixing ⇔ Clustering Mixing times: There exist constant A , b > 0 such that: � e t L ∗ ( ρ 0 ) − σ � 1 ≤ Ae − bt . Clustering of correlations: There exist constants C , ξ > 0 such that for any subsets of the lattice A , B we get Corr σ ( A : B ) ≤ C poly ( | A | , | B | ) e − d ( A : B ) /ξ , where d ( A : B ) is the distance separating regions A , B . Michael Kastoryano (Berlin) Mixing vs. Clustering Prien/Chiemsee, October 21, 2013 5 / 33
Mixing ⇔ Clustering Mixing times: There exist constant A , b > 0 such that: � e t L ∗ ( ρ 0 ) − σ � 1 ≤ Ae − bt . Clustering of correlations: There exist constants C , ξ > 0 such that for any subsets of the lattice A , B we get Corr σ ( A : B ) ≤ C poly ( | A | , | B | ) e − d ( A : B ) /ξ , where d ( A : B ) is the distance separating regions A , B . The goal of this talk is to explain to what extent these two statements are equivalent. Michael Kastoryano (Berlin) Mixing vs. Clustering Prien/Chiemsee, October 21, 2013 5 / 33
Table of Contents Introduction 1 Setting Motivation Preliminaries 2 Rapid mixing bounds Correlation Measures Rapid mixing implies clustering 3 χ 2 clustering Log-Sobolev clustering and stability Area Law Clustering implies rapid mixing 4 The main theorem Corollaries Outlook 5 Michael Kastoryano (Berlin) Mixing vs. Clustering Prien/Chiemsee, October 21, 2013 6 / 33
Why are these bounds useful? Quantum memories: Davies generators of stabilizer Hamiltonians. Rigorous no-go 1 theorems (New J. Phys. 12 025013 (2010)). Michael Kastoryano (Berlin) Mixing vs. Clustering Prien/Chiemsee, October 21, 2013 7 / 33
Why are these bounds useful? Quantum memories: Davies generators of stabilizer Hamiltonians. Rigorous no-go 1 theorems (New J. Phys. 12 025013 (2010)). Stability of Liouvillian dynamics (arXiv:1303.4744, arXiv:1303.6304). 2 Michael Kastoryano (Berlin) Mixing vs. Clustering Prien/Chiemsee, October 21, 2013 7 / 33
Why are these bounds useful? Quantum memories: Davies generators of stabilizer Hamiltonians. Rigorous no-go 1 theorems (New J. Phys. 12 025013 (2010)). Stability of Liouvillian dynamics (arXiv:1303.4744, arXiv:1303.6304). 2 Topology in open systems, or at non-zero temperature. 3 Michael Kastoryano (Berlin) Mixing vs. Clustering Prien/Chiemsee, October 21, 2013 7 / 33
Why are these bounds useful? Quantum memories: Davies generators of stabilizer Hamiltonians. Rigorous no-go 1 theorems (New J. Phys. 12 025013 (2010)). Stability of Liouvillian dynamics (arXiv:1303.4744, arXiv:1303.6304). 2 Topology in open systems, or at non-zero temperature. 3 (Runtimes of dissipative algorithms and state preparation (Nature Phys. 5 , 633 4 (2009) ). ) Michael Kastoryano (Berlin) Mixing vs. Clustering Prien/Chiemsee, October 21, 2013 7 / 33
Why are these bounds useful? Quantum memories: Davies generators of stabilizer Hamiltonians. Rigorous no-go 1 theorems (New J. Phys. 12 025013 (2010)). Stability of Liouvillian dynamics (arXiv:1303.4744, arXiv:1303.6304). 2 Topology in open systems, or at non-zero temperature. 3 (Runtimes of dissipative algorithms and state preparation (Nature Phys. 5 , 633 4 (2009) ). ) (Bounds on the thermalization times of quantum systems, i.e. efficient Gibbs 5 samplers?) Michael Kastoryano (Berlin) Mixing vs. Clustering Prien/Chiemsee, October 21, 2013 7 / 33
Table of Contents Introduction 1 Setting Motivation Preliminaries 2 Rapid mixing bounds Correlation Measures Rapid mixing implies clustering 3 χ 2 clustering Log-Sobolev clustering and stability Area Law Clustering implies rapid mixing 4 The main theorem Corollaries Outlook 5 Michael Kastoryano (Berlin) Mixing vs. Clustering Prien/Chiemsee, October 21, 2013 8 / 33
Rapid mixing: χ 2 bound χ 2 bound: Let L be a primitive reversible Liouvillian with stationary state σ > 0, then � e t L ∗ ( ρ 0 ) − σ � 1 ≤ � � σ − 1 � e − λ t , for any initial state ρ 0 . Michael Kastoryano (Berlin) Mixing vs. Clustering Prien/Chiemsee, October 21, 2013 9 / 33
Rapid mixing: χ 2 bound χ 2 bound: Let L be a primitive reversible Liouvillian with stationary state σ > 0, then � e t L ∗ ( ρ 0 ) − σ � 1 ≤ � � σ − 1 � e − λ t , for any initial state ρ 0 . Proof sketch: write ρ t = e t L ∗ ( ρ 0 ) , then � ρ t − σ � 2 1 ≤ χ 2 ( ρ t , σ ) ≤ χ 2 ( ρ 0 , σ ) e − 2 t λ , � ( ρ − σ ) σ 1 / 2 ( ρ − σ ) σ 1 / 2 � is the χ 2 divergence, and it satisfies where χ 2 ( ρ, σ ) = tr χ 2 ( ρ, σ ) ≤ � σ − 1 � . Note that if L is reversible, then λ is just the spectral gap of L . For a system of N spins (qubits) � σ − 1 � ≥ 2 N . Michael Kastoryano (Berlin) Mixing vs. Clustering Prien/Chiemsee, October 21, 2013 9 / 33
Ultra-rapid mixing: Log-Sobolev bound Log-Sobolev bound: Let L be a primitive reversible Liouvillian with stationary state σ > 0, then � e t L ( ρ 0 ) − σ � 1 ≤ � 2 log ( � σ − 1 � ) e − 2 α t , for any initial state ρ 0 . Michael Kastoryano (Berlin) Mixing vs. Clustering Prien/Chiemsee, October 21, 2013 10 / 33
Ultra-rapid mixing: Log-Sobolev bound Log-Sobolev bound: Let L be a primitive reversible Liouvillian with stationary state σ > 0, then � e t L ( ρ 0 ) − σ � 1 ≤ � 2 log ( � σ − 1 � ) e − 2 α t , for any initial state ρ 0 . Same proof but with χ 2 ( ρ, σ ) replaced by S ( ρ � σ ) = tr [ ρ ( log ρ − log σ )] . The Log-Sobolev constant α can only be obtained by a complicated variational formula ⇒ equivalent to Hypercontractivity of the semigroup. The bound provides an exponentially improved pre-factor! Importantly, α ≤ λ See J. Math. Phys. 54, 052202 (2013) for more details. Michael Kastoryano (Berlin) Mixing vs. Clustering Prien/Chiemsee, October 21, 2013 10 / 33
Table of Contents Introduction 1 Setting Motivation Preliminaries 2 Rapid mixing bounds Correlation Measures Rapid mixing implies clustering 3 χ 2 clustering Log-Sobolev clustering and stability Area Law Clustering implies rapid mixing 4 The main theorem Corollaries Outlook 5 Michael Kastoryano (Berlin) Mixing vs. Clustering Prien/Chiemsee, October 21, 2013 11 / 33
Recommend
More recommend
