Rapid Thermalization of Spin Chain Commuting Hamiltonians.

Ivan Bardet, ∗ Ángela Capel, 3, 4, † Li Gao, 3, 5, ‡ Angelo Lucia, 7, § David Pérez-García, ¶ and Cambyse Rouzé ∗∗ Inria Paris, 75012 Paris, France Department of Mathematics, Technische Universität München, 85748 Garching, Germany Munich Center for Quantum Science and Technology (MCQST), 80799 München, Germany Fachbereich Mathematik, Universität Tübingen, 72076 Tübingen, Germany Department of Mathematics, University of Houston, 77204 Houston, USA Departamento de Análisis Matemático y Matemática Aplicada, Universidad Complutense de Madrid, 28040 Madrid, Spain Instituto de Ciencias Matemáticas, 28049 Madrid, Spain

[1]  Victor V. Albert,et al.  Symmetries and conserved quantities in Lindblad master equations , 2013, 1310.1523.

[2]  J. Ignacio Cirac,et al.  Noise-driven dynamics and phase transitions in fermionic systems , 2012, 1207.1653.

[3]  E. Davies,et al.  Markovian master equations , 1974 .

[4]  A. Lucia,et al.  On the modified logarithmic Sobolev inequality for the heat-bath dynamics for 1D systems , 2019, Journal of Mathematical Physics.

[5]  R. Holley,et al.  Rapid Convergence to Equilibrium in One Dimensional Stochastic Ising Models , 1985 .

[6]  Li Gao,et al.  Fisher Information and Logarithmic Sobolev Inequality for Matrix-Valued Functions , 2018, Annales Henri Poincaré.

[7]  B. Zegarliński,et al.  Log-Sobolev inequalities for infinite one dimensional lattice systems , 1990 .

[8]  F. Verstraete,et al.  Quantum computation and quantum-state engineering driven by dissipation , 2009 .

[9]  Leonard Gross,et al.  Hypercontractivity and logarithmic Sobolev inequalities for the Clifford-Dirichlet form , 1975 .

[10]  David Perez-Garcia,et al.  Irreducible forms of matrix product states: Theory and applications , 2017, 1708.00029.

[11]  Umesh V. Vazirani,et al.  The detectability lemma and quantum gap amplification , 2008, STOC '09.

[12]  Salman Beigi,et al.  Quantum Reverse Hypercontractivity: Its Tensorization and Application to Strong Converses , 2018, ArXiv.

[13]  Krysta Marie Svore,et al.  Quantum Speed-Ups for Solving Semidefinite Programs , 2017, 2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS).

[14]  A. Parkins,et al.  Dissipation-driven quantum phase transitions in collective spin systems , 2008, 0805.1256.

[15]  Á. Rivas,et al.  Symmetry-protected topological phases at finite temperature , 2015, 1502.01355.

[16]  M. Fannes,et al.  On thermalization in Kitaev's 2D model , 2008, 0810.4584.

[17]  T. Cubitt,et al.  Rapid mixing and stability of quantum dissipative systems , 2014, 1409.7809.

[18]  P. Zoller,et al.  Topology by dissipation , 2013, 1302.5135.

[19]  P. Zoller,et al.  Topology by dissipation in atomic quantum wires , 2011, 1105.5947.

[20]  D. Perez-Garcia,et al.  Matrix Product Density Operators: Renormalization Fixed Points and Boundary Theories , 2016, 1606.00608.

[21]  S. Diehl,et al.  Symmetry Classes of Open Fermionic Quantum Matter , 2020, Physical Review X.

[22]  E. Knill,et al.  Quantum simulations of classical annealing processes. , 2008, Physical review letters.

[23]  A. Wallraff,et al.  Observation of topological Uhlmann phases with superconducting qubits , 2016, 1607.08778.

[24]  X. Wen.  Quantum orders and symmetric spin liquids , 2001, cond-mat/0107071.

[25]  M. Junge,et al.  Mixed-norm Inequalities and Operator Space Lp Embedding Theory , 2010 .

[26]  Cambyse Rouz'e,et al.  Complete Entropic Inequalities for Quantum Markov Chains , 2021, Archive for Rational Mechanics and Analysis.

[27]  Xiao-Gang Wen,et al.  Tensor-Entanglement-Filtering Renormalization Approach and Symmetry Protected Topological Order , 2009, 0903.1069.

[28]  David Pérez-García,et al.  Quantum conditional relative entropy and quasi-factorization of the relative entropy , 2018, Journal of Physics A: Mathematical and Theoretical.

[29]  T. Prosen,et al.  A note on symmetry reductions of the Lindblad equation: transport in constrained open spin chains , 2012, 1203.0943.

[30]  Anna Maria Paganoni,et al.  Entropy inequalities for unbounded spin systems , 2002 .

[31]  Ivan Bardet,et al.  Hypercontractivity and Logarithmic Sobolev Inequality for Non-primitive Quantum Markov Semigroups and Estimation of Decoherence Rates , 2018, Annales Henri Poincaré.

[32]  Angelo Lucia,et al.  Thermalization in Kitaev's quantum double models via Tensor Network techniques , 2021, 2107.01628.

[33]  Frank Pollmann,et al.  Entanglement spectrum of a topological phase in one dimension , 2009, 0910.1811.

[34]  Ivan Bardet,et al.  Approximate Tensorization of the Relative Entropy for Noncommuting Conditional Expectations , 2021, Annales Henri Poincaré.

[35]  G. Pisier.  Non-commutative vector valued Lp-spaces and completely p-summing maps , 1993, math/9306206.

[36]  N. Wiebe,et al.  Tomography and generative training with quantum Boltzmann machines , 2016, 1612.05204.

[37]  Stephen D. Bartlett,et al.  Symmetry-protected topological order at nonzero temperature , 2016, 1611.05450.

[38]  V. Umanità,et al.  Decoherence for Quantum Markov Semi-Groups on Matrix Algebras , 2013 .

[39]  Fernando G. S. L. Brandão,et al.  Quantum Gibbs Samplers: The Commuting Case , 2014, Communications in Mathematical Physics.

[40]  H. Araki.  Gibbs states of a one dimensional quantum lattice , 1969 .

[41]  Carlos Palazuelos,et al.  Survey on Nonlocal Games and Operator Space Theory , 2015, 1512.00419.

[42]  R Raussendorf,et al.  A one-way quantum computer. , 2001, Physical review letters.

[43]  E. M.,et al.  Statistical Mechanics , 2021, Manual for Theoretical Chemistry.

[44]  J. Cirac,et al.  Dissipative phase transition in a central spin system , 2012, 1205.3341.

[45]  M. Sanz,et al.  Matrix product states: Symmetries and two-body Hamiltonians , 2009, 0901.2223.

[46]  K. Temme,et al.  Necessity of an energy barrier for self-correction of Abelian quantum doubles , 2016 .

[47]  D. Pérez-García,et al.  Classification of phases for mixed states via fast dissipative evolution , 2018, Quantum.

[48]  F. Verstraete,et al.  Quantum Metropolis sampling , 2009, Nature.

[49]  M. A. Martin-Delgado,et al.  Thermal instability of protected end states in a one-dimensional topological insulator , 2012, 1207.2198.

[50]  Daniel Stilck Francca,et al.  The modified logarithmic Sobolev inequality for quantum spin systems: classical and commuting nearest neighbour interactions , 2020, 2009.11817.

[51]  Á. Cuevas.  Quantum logarithmic Sobolev inequalities for quantum many-body systems: An approach via quasi-factorization of the relative entropy , 2019 .

[52]  Xiao-Gang Wen,et al.  Classification of gapped symmetric phases in one-dimensional spin systems , 2010, 1008.3745.

[53]  Phase diagram and critical exponents of a dissipative Ising spin chain in a transverse magnetic field. , 2004, Physical review letters.

[54]  Yoshiko Ogata,et al.  Classification of gapped ground state phases in quantum spin systems , 2021, 2110.04675.

[55]  Germany,et al.  Quantum states and phases in driven open quantum systems with cold atoms , 2008, 0803.1482.

[56]  Alexei Kitaev,et al.  Topological phases of fermions in one dimension , 2010, 1008.4138.

[57]  F. Cesi.  Quasi-factorization of the entropy and logarithmic Sobolev inequalities for Gibbs random fields , 2001 .

[58]  Hans-J. Briegel,et al.  Computational model underlying the one-way quantum computer , 2002, Quantum Inf. Comput..

[59]  Quasi-factorization and Multiplicative Comparison of Subalgebra-Relative Entropy , 2019, 1912.00983.

[60]  K. Temme,et al.  Quantum logarithmic Sobolev inequalities and rapid mixing , 2012, 1207.3261.

[61]  David Perez-Garcia,et al.  Stability of Local Quantum Dissipative Systems , 2013, 1303.4744.

[62]  Sergey Bravyi,et al.  Topological quantum order: Stability under local perturbations , 2010, 1001.0344.

[63]  Andreas Bluhm,et al.  Exponential decay of mutual information for Gibbs states of local Hamiltonians , 2021 .

[64]  F. Verstraete,et al.  Matrix product states and projected entangled pair states: Concepts, symmetries, theorems , 2020, Reviews of Modern Physics.

[65]  N. Cooper,et al.  Interacting symmetry-protected topological phases out of equilibrium , 2019, Physical Review Research.

[66]  H. Briegel,et al.  Persistent entanglement in arrays of interacting particles. , 2000, Physical review letters.

[67]  David Pérez-García,et al.  Order parameter for symmetry-protected phases in one dimension. , 2012, Physical review letters.

[68]  Daniel W. Stroock,et al.  Uniform andL2 convergence in one dimensional stochastic Ising models , 1989 .

[69]  F. Pastawski,et al.  Hypercontractivity of quasi-free quantum semigroups , 2014, 1403.5224.

[70]  Vlatko Vedral,et al.  Quantum phase transition between cluster and antiferromagnetic states , 2011, 1103.0251.

[71]  Ievgeniia Oshurko.  Quantum Machine Learning , 2020, Quantum Computing.

[72]  U. Vazirani,et al.  Quantum Hamiltonian complexity and the detectability lemma , 2010, 1011.3445.

[73]  David Pérez-García,et al.  Classifying quantum phases using matrix product states and projected entangled pair states , 2011 .

[74]  D. Loss,et al.  Majorana qubit decoherence by quasiparticle poisoning , 2012, 1204.3326.

[75]  鈴木 増雄.  Time-Dependent Statistics of the Ising Model , 1965 .

[76]  N. Cooper,et al.  Fragility of time-reversal symmetry protected topological phases , 2020 .

[77]  Frank Pollmann,et al.  Symmetry protection of topological phases in one-dimensional quantum spin systems , 2009, 0909.4059.