Markovian master equations are a ubiquitous tool in the study of open quantum systems, but deriving them from first principles involves a series of compromises. On the one hand, the Redfield equation is valid for fast environments (whose correlation function decays much faster than the system relaxation time) regardless of the relative strength of the coupling to the system Hamiltonian, but is notoriously non-completely-positive. On the other hand, the Davies equation preserves complete positivity but is valid only in the ultra-weak coupling limit and for systems with a finite level spacing, which makes it incompatible with arbitrarily fast time-dependent driving.
Here we show that a recently derived Markovian coarse-grained master equation (CGME), already known to be completely positive, has a much expanded range of applicability compared to the Davies equation, and moreover, is locally generated and can be generalized to accommodate arbitrarily fast driving. This generalization, which we refer to as the time-dependent CGME, is thus suitable for the analysis of fast operations in gate-model quantum computing, such as quantum error correction and dynamical decoupling. Our derivation proceeds directly from the Redfield equation and allows us to place rigorous error bounds on all three equations: Redfield, Davies, and coarse-grained. Our main result is thus a completely positive Markovian master equation that is a controlled approximation to the true evolution for any time-dependence of the system Hamiltonian, and works for systems with arbitrarily small level spacing. We illustrate this with an analysis showing that dynamical decoupling can extend coherence times even in a strictly Markovian setting.
Our result states that as long as the dynamics of the environment is much faster than its effect on the system, so the environment has time to equilibrate regardless of what the system is doing, then there is a mathematical model that can be trusted. The generality of our result is such that it applies to any number of qubits, as well as other quantum systems such as atoms and quantum dots. The control applied to the quantum system by the experimentalist is included in our model and can have a nontrivial interplay with the open system effects. Our model captures this interplay in many relevant cases, both for future theoretical results in quantum information, and for the simulation of quantum experiments.
 H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems (Oxford University Press, Oxford, 2002).
 C.W. Gardiner and P. Zoller, Quantum Noise, Springer Series in Synergetics, Vol. 56 (Springer, Berlin, 2000).
 T. Albash, W. Vinci, A. Mishra, P. A. Warburton, and D. A. Lidar, “Consistency tests of classical and quantum models for a quantum annealer,” Phys. Rev. A 91, 042314- (2015a).
 S. Boixo, V. N. Smelyanskiy, A. Shabani, S. V. Isakov, M. Dykman, V. S. Denchev, M. H. Amin, A. Y. Smirnov, M. Mohseni, and H. Neven, “Computational multiqubit tunnelling in programmable quantum annealers,” Nat Commun 7 (2016).
 T. Albash, I. Hen, F. M. Spedalieri, and D. A. Lidar, “Reexamination of the evidence for entanglement in a quantum annealer,” Physical Review A 92, 062328- (2015b).
 A. Mishra, T. Albash, and D. A. Lidar, “Finite temperature quantum annealing solving exponentially small gap problem with non-monotonic success probability,” Nature Communications 9, 2917 (2018).
 R. Feynman and F. Vernon, “The theory of a general quantum system interacting with a linear dissipative system,” Annals of Physics 24, 118 – 173 (1963).
 D. E. Makarov and N. Makri, “Path integrals for dissipative systems by tensor multiplication. condensed phase quantum dynamics for arbitrarily long time,” Chemical Physics Letters 221, 482 – 491 (1994).
 V. Gorini, A. Frigerio, M. Verri, A. Kossakowski, and E. C. G. Sudarshan, “Properties of quantum Markovian master equations,” Reports on Mathematical Physics 13, 149-173 (1978).
 A. Redfield, “The theory of relaxation processes,” in Advances in Magnetic Resonance, Advances in Magnetic and Optical Resonance, Vol. 1, edited by J. S. Waugh (Academic Press, 1965) pp. 1 – 32.
 A. J. van Wonderen and K. Lendi, “Virtues and limitations of markovian master equations with a time-dependent generator,” J. Stat. Phys. 100, 633-658 (2000).
 D. A. Lidar, Z. Bihary, and K. Whaley, “From completely positive maps to the quantum Markovian semigroup master equation,” Chem. Phys. 268, 35 (2001).
 S. Daffer, K. Wodkiewicz, J.D. Cresser, J.K. McIver, “Depolarizing channel as a completely positive map with memory,” Phys. Rev. A 70, 010304(R) (2004).
 J. Piilo, S. Maniscalco, K. Härkönen, and K.-A. Suominen, “Non-markovian quantum jumps,” Physical Review Letters 100, 180402- (2008).
 H.-P. Breuer and B. Vacchini, “Quantum semi-markov processes,” Physical Review Letters 101, 140402- (2008).
 R. S. Whitney, “Staying positive: going beyond lindblad with perturbative master equations,” Journal of Physics A: Mathematical and Theoretical 41, 175304 (2008).
 L.-A. Wu, G. Kurizki, and P. Brumer, “Master equation and control of an open quantum system with leakage,” Physical Review Letters 102, 080405- (2009).
 D. Chruściński and A. Kossakowski, “Non-markovian quantum dynamics: Local versus nonlocal,” Phys. Rev. Lett. 104, 070406 (2010).
 T. Albash, S. Boixo, D. A. Lidar, and P. Zanardi, “Quantum adiabatic Markovian master equations,” New J. of Phys. 14, 123016 (2012).
 L. C. Venuti and D. A. Lidar, “Error reduction in quantum annealing using boundary cancellation: Only the end matters,” Phys. Rev. A 98, 022315 (2018).
 G. McCauley, B. Cruikshank, D. I. Bondar, and K. Jacobs, “Completely positive, accurate master equation for weakly-damped quantum systems across all regimes,” arXiv:1906.08279 (2019).
 F. Benatti, R. Floreanini, and U. Marzolino, “Environment-induced entanglement in a refined weak-coupling limit,” EPL (Europhysics Letters) 88, 20011 (2009).
 C. Majenz, T. Albash, H.-P. Breuer, and D. A. Lidar, “Coarse graining can beat the rotating-wave approximation in quantum markovian master equations,” Phys. Rev. A 88, 012103- (2013).
 T. S. Cubitt, A. Lucia, S. Michalakis, and D. Perez-Garcia, “Stability of local quantum dissipative systems,” Communications in Mathematical Physics 337, 1275-1315 (2015).
 R. Alicki, D. A. Lidar, and P. Zanardi, “Internal consistency of fault-tolerant quantum error correction in light of rigorous derivations of the quantum markovian limit,” Phys. Rev. A 73, 052311 (2006).
 M. Žnidarič, “Dephasing-induced diffusive transport in the anisotropic heisenberg model,” New Journal of Physics 12, 043001 (2010).
 R. Bhatia, Matrix Analysis, Graduate Texts in Mathematics No. 169 (Springer-Verlag, New York, 1997).
 F. Verstraete, J. J. Garcia-Ripoll, and J. I. Cirac, “Matrix product density operators: Simulation of finite-temperature and dissipative systems,” Phys. Rev. Lett. 93, 207204 (2004).
 J. Haah, M. Hastings, R. Kothari, and G. H. Low, “Quantum algorithm for simulating real time evolution of lattice hamiltonians,” in 2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS) (2018) pp. 350-360.
 H. Pichler, A. J. Daley, and P. Zoller, “Nonequilibrium dynamics of bosonic atoms in optical lattices: Decoherence of many-body states due to spontaneous emission,” Phys. Rev. A 82, 063605 (2010).
 L.-M Duan and G.-C. Guo, “Reducing decoherence in quantum-computer memory with all quantum bits coupling to the same environment,” Phys. Rev. A 57, 737 (1998).
 D. A. Lidar and K. B. Whaley, “Decoherence-free subspaces and subsystems,” in Irreversible Quantum Dynamics, Lecture Notes in Physics, Vol. 622, edited by F. Benatti and R. Floreanini (Springer, Berlin, 2003) p. 83.
 L. Viola, E. M. Fortunato, M. A. Pravia, E. Knill, R. Laflamme, and D. G. Cory, “Experimental realization of noiseless subsystems for quantum information processing,” Science 293, 2059-2063 (2001).
 D. Kielpinski, V. Meyer, M. A. Rowe, C. A. Sackett, W. M. Itano, C. Monroe, and D. J. Wineland, “A decoherence-free quantum memory using trapped ions,” Science 291, 1013-1015 (2001).
 J. E. Ollerenshaw, D. A. Lidar, and L. E. Kay, “Magnetic resonance realization of decoherence-free quantum computation,” Phys. Rev. Lett. 91, 217904 (2003).
 D. Suter and G. A. Álvarez, “Colloquium: Protecting quantum information against environmental noise,” Reviews of Modern Physics 88, 041001- (2016).
 J. E. Gough and H. I. Nurdin, “Can quantum markov evolutions ever be dynamically decoupled?” in 2017 IEEE 56th Annual Conference on Decision and Control (CDC) (2017) pp. 6155-6160.
 C. Addis, F. Ciccarello, M. Cascio, G. M. Palma, and S. Maniscalco, “Dynamical decoupling efficiency versus quantum non-markovianity,” New Journal of Physics 17, 123004 (2015).
 C. Arenz, D. Burgarth, P. Facchi, and R. Hillier, “Dynamical decoupling of unbounded hamiltonians,” Journal of Mathematical Physics, Journal of Mathematical Physics 59, 032203 (2018).
 L. Li, M. J. W. Hall, and H. M. Wiseman, “Concepts of quantum non-markovianity: A hierarchy,” Concepts of quantum non-Markovianity: A hierarchy, Physics Reports 759, 1-51 (2018).
 I. de Vega, M. C. Bañuls, and A. Pérez, “Effects of dissipation on an adiabatic quantum search algorithm,” New J. of Phys. 12, 123010 (2010).
 L. Isserlis, “On certain probable errors and correlation coefficients of multiple frequency distributions with skew regression,” Biometrika 11, 185 (1916).
 R. Alicki, M. Fannes, and M. Horodecki, “A statistical mechanics view on kitaev’s proposal for quantum memories,” Journal of Physics A: Mathematical and Theoretical 40, 6451-6467 (2007).
 B. Altshuler, H. Krovi, and J. Roland, “Anderson localization makes adiabatic quantum optimization fail,” Proceedings of the National Academy of Sciences 107, 12446-12450 (2010).
Could not fetch Crossref cited-by data during last attempt 2020-02-06 15:54:51: Could not fetch cited-by data for 10.22331/q-2020-02-06-227 from Crossref. This is normal if the DOI was registered recently. On SAO/NASA ADS no data on citing works was found (last attempt 2020-02-06 15:54:51).
This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.