Completely positive master equation for arbitrary driving and small level spacing

Evgeny Mozgunov1 and Daniel Lidar1,2,3,4

1Center for Quantum Information Science & Technology, University of Southern California, Los Angeles, California 90089, USA
2Department of Electrical and Computer Engineering, University of Southern California, Los Angeles, California 90089, USA
3Department of Chemistry, University of Southern California, Los Angeles, California 90089, USA
4Department of Physics and Astronomy, University of Southern California, Los Angeles, California 90089, USA

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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.

Real quantum devices available in the lab today are never perfectly isolated from their environment. ​If the isolation is good enough, the effect of the environment can be either neglected altogether, or approximated by a simple mathematical model. We investigate this intuition in great detail, trying to answer exactly how good the isolation needs to be for a simple mathematical model to be within, say, 5% of what an experimentalist would actually see in the lab. Usually, this question is answered on a case-by-case basis by attempting to fit the experimental results with the model and observing the error of the fit. Here we approach this question in some generality, as a rigorous mathematical theorem that we prove under very weak assumptions.

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.

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► References

[1] R. Alicki and K. Lendi, Quantum Dynamical Semigroups and Applications (Springer Science & Business Media, 2007).

[2] H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems (Oxford University Press, Oxford, 2002).

[3] C.W. Gardiner and P. Zoller, Quantum Noise, Springer Series in Synergetics, Vol. 56 (Springer, Berlin, 2000).

[4] E. B. Davies, ``Markovian master equations,'' Communications in Mathematical Physics 39, 91–110 (1974).

[5] G. Lindblad, ``On the generators of quantum dynamical semigroups,'' Comm. Math. Phys. 48, 119–130 (1976).

[6] 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).

[7] 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).

[8] 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).

[9] 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).

[10] 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).

[11] A. Caldeira and A. Leggett, ``Quantum tunnelling in a dissipative system,'' Annals of Physics 149, 374 – 456 (1983).

[12] 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).

[13] E. Sim, ``Quantum dynamics for a system coupled to slow baths: On-the-fly filtered propagator method,'' The Journal of Chemical Physics 115, 4450–4456 (2001).

[14] K. Kraus, States, Effects, and Operations (Springer, Berlin, 1983).

[15] J. M. Dominy and D. A. Lidar, ``Beyond complete positivity,'' Quant. Inf. Proc. 15, 1349 (2016).

[16] 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).

[17] S. Nakajima, ``On Quantum Theory of Transport Phenomena : Steady Diffusion,'' Prog. Theor. Phys. 20, 948 (1958).

[18] R. Zwanzig, ``Ensemble Method in the Theory of Irreversibility,'' J. Chem. Phys. 33, 1338 (1960).

[19] 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.

[20] D. Bacon, D. A. Lidar, and K. B. Whaley, ``Robustness of decoherence-free subspaces for quantum computation,'' Phys. Rev. A 60, 1944–1955 (1999).

[21] 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).

[22] D. A. Lidar, Z. Bihary, and K. Whaley, ``From completely positive maps to the quantum Markovian semigroup master equation,'' Chem. Phys. 268, 35 (2001).

[23] 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).

[24] A. Shabani and D. A. Lidar, ``Completely positive post-markovian master equation via a measurement approach,'' Phys. Rev. A 71, 020101(R) (2005).

[25] S. Maniscalco and F. Petruccione, ``Non-Markovian dynamics of a qubit,'' Phys. Rev. A 73, 012111 (2006).

[26] J. Piilo, S. Maniscalco, K. Härkönen, and K.-A. Suominen, ``Non-markovian quantum jumps,'' Physical Review Letters 100, 180402– (2008).

[27] H.-P. Breuer and B. Vacchini, ``Quantum semi-markov processes,'' Physical Review Letters 101, 140402– (2008).

[28] R. S. Whitney, ``Staying positive: going beyond lindblad with perturbative master equations,'' Journal of Physics A: Mathematical and Theoretical 41, 175304 (2008).

[29] 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).

[30] D. Chruściński and A. Kossakowski, ``Non-markovian quantum dynamics: Local versus nonlocal,'' Phys. Rev. Lett. 104, 070406 (2010).

[31] T. Albash, S. Boixo, D. A. Lidar, and P. Zanardi, ``Quantum adiabatic Markovian master equations,'' New J. of Phys. 14, 123016 (2012).

[32] E. Mozgunov, ``Local master equation for small temperatures,'' arXiv:1611.04188 (2016).

[33] A. Y. Smirnov and M. H. Amin, ``Theory of open quantum dynamics with hybrid noise,'' New Journal of Physics 20, 103037 (2018).

[34] R. Dann, A. Levy, and R. Kosloff, ``Time-dependent markovian quantum master equation,'' Phys. Rev. A 98, 052129 (2018).

[35] 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).

[36] 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).

[37] F. Benatti, R. Floreanini, and U. Marzolino, ``Environment-induced entanglement in a refined weak-coupling limit,'' EPL (Europhysics Letters) 88, 20011 (2009).

[38] F. Benatti, R. Floreanini, and U. Marzolino, ``Entangling two unequal atoms through a common bath,'' Phys. Rev. A 81, 012105 (2010).

[39] M. Merkli, ``Quantum markovian master equations: Resonance theory shows validity for all time scales,'' Annals of Physics 412, 167996 (2020).

[40] 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).

[41] 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).

[42] E. Knill, ``Quantum computing with realistically noisy devices,'' Nature 434, 39–44 (2005).

[43] 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).

[44] D. A. Lidar, ``Lecture notes on the theory of open quantum systems,'' arXiv preprint arXiv:1902.00967 (2019).

[45] T. Albash and D. A. Lidar, ``Decoherence in adiabatic quantum computation,'' Phys. Rev. A 91, 062320– (2015).

[46] M. Žnidarič, ``Dephasing-induced diffusive transport in the anisotropic heisenberg model,'' New Journal of Physics 12, 043001 (2010).

[47] M. V. Medvedyeva, T. Prosen, and M. Žnidarič, ``Influence of dephasing on many-body localization,'' Phys. Rev. B 93, 094205 (2016).

[48] R. Bhatia, Matrix Analysis, Graduate Texts in Mathematics No. 169 (Springer-Verlag, New York, 1997).

[49] P. Gaspard and M. Nagaoka, ``Slippage of initial conditions for the redfield master equation,'' Journal of Chemical Physics 111, 5668–5675 (1999).

[50] G. Vidal, ``Efficient classical simulation of slightly entangled quantum computations,'' Phys. Rev. Lett. 91, 147902 (2003).

[51] 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).

[52] E. H. Lieb and D.W. Robinson, ``The finite group velocity of quantum spin systems,'' Commun. Math. Phys. 28, 251 (1972).

[53] 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.

[54] 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).

[55] 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).

[56] P. Zanardi and M. Rasetti, ``Noiseless quantum codes,'' Phys. Rev. Lett. 79, 3306–3309 (1997).

[57] D. A. Lidar, I. L. Chuang, and K. B. Whaley, ``Decoherence-free subspaces for quantum computation,'' Phys. Rev. Lett. 81, 2594–2597 (1998).

[58] 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.

[59] P.G. Kwiat, A.J. Berglund, J.B. Altepeter, and A.G. White, ``Experimental Verification of Decoherence-Free Subspaces,'' Science 290, 498 (2000).

[60] 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).

[61] 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).

[62] J. E. Ollerenshaw, D. A. Lidar, and L. E. Kay, ``Magnetic resonance realization of decoherence-free quantum computation,'' Phys. Rev. Lett. 91, 217904 (2003).

[63] L. Viola and S. Lloyd, ``Dynamical suppression of decoherence in two-state quantum systems,'' Phys. Rev. A 58, 2733–2744 (1998).

[64] P. Zanardi, ``Symmetrizing evolutions,'' Physics Letters A 258, 77–82 (1999).

[65] D. Lidar and T. Brun, eds., Quantum Error Correction (Cambridge University Press, Cambridge, UK, 2013).

[66] D. Suter and G. A. Álvarez, ``Colloquium: Protecting quantum information against environmental noise,'' Reviews of Modern Physics 88, 041001– (2016).

[67] H. K. Ng, D. A. Lidar, and J. Preskill, ``Combining dynamical decoupling with fault-tolerant quantum computation,'' Phys. Rev. A 84, 012305– (2011).

[68] K. Szczygielski and R. Alicki, ``Markovian theory of dynamical decoupling by periodic control,'' Physical Review A 92, 022349– (2015).

[69] 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.

[70] 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).

[71] 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).

[72] 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).

[73] 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).

[74] https:/​/​​mvjenia/​CGMEcode, code for the numerical section of the paper.

[75] L. Isserlis, ``On certain probable errors and correlation coefficients of multiple frequency distributions with skew regression,'' Biometrika 11, 185 (1916).

[76] T. Albash, D. A. Lidar, M. Marvian, and P. Zanardi, ``Fluctuation theorems for quantum processes,'' Phys. Rev. E 88, 032146– (2013).

[77] T. Albash and D. A. Lidar, ``Adiabatic quantum computation,'' Reviews of Modern Physics 90, 015002– (2018).

[78] 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).

[79] H. Bombin, ``Topological subsystem codes,'' Physical Review A 81, 032301– (2010).

[80] 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).

[81] M. Reed and B. Simon, Methods of Modern Mathematical Physics: Fourier analysis, self-adjointness, Vol. 2 (Academic Press, 1975).

[82] H. Alzer, ``On some inequalities for the incomplete gamma function,'' Mathematics of Computation 66, 771 (1997).

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