Completely positive, trace preserving (CPT) maps and Lindblad master equations are both widely used to describe the dynamics of open quantum systems. The connection between these two descriptions is a classic topic in mathematical physics. One direction was solved by the now famous result due to Lindblad, Kossakowski, Gorini and Sudarshan, who gave a complete characterisation of the master equations that generate completely positive semi-groups. However, the other direction has remained open: given a CPT map, is there a Lindblad master equation that generates it (and if so, can we find its form)? This is sometimes known as the Markovianity problem. Physically, it is asking how one can deduce underlying physical processes from experimental observations.
We give a complexity theoretic answer to this problem: it is NP-hard. We also give an explicit algorithm that reduces the problem to integer semi-definite programming, a well-known NP problem. Together, these results imply that resolving the question of which CPT maps can be generated by master equations is tantamount to solving P = NP: any efficiently computable criterion for Markovianity would imply P = NP; whereas a proof that P = NP would imply that our algorithm already gives an efficiently computable criterion. Thus, unless P does equal NP, there cannot exist any simple criterion for determining when a CPT map has a master equation description.
However, we also show that if the system dimension is fixed (relevant for current quantum process tomography experiments), then our algorithm scales efficiently in the required precision, allowing an underlying Lindblad master equation to be determined efficiently from even a single snapshot in this case.
Our work also leads to similar complexity-theoretic answers to a related long-standing open problem in probability theory.
Master Equation Quantum Channel Open Quantum System Stochastic Matrice Embedding Problem
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in to check access.
Boulant, N., Havel, T.F., Pravia, M.A., Cory, D.G.: Robust method for estimating the Lindblad operators of a dissipative quantum process from measurements of the density operator at multiple time points. Phys. Rev. A 67 042322 (2003)Google Scholar
Boulant N., Emerson J., Havel T.F., Cory D.G.: Incoherent noise and quantum information processing. J. Chem. Phys. 121(7), 2955 (2004)ADSCrossRefGoogle Scholar
Howard M. et al.: Quantum process tomography and linblad estimation of a solid-state qubit. New J. Phys 8, 33 (2006)ADSCrossRefGoogle Scholar
Weinstein Y.S. et al.: Quantum process tomography of the quantum fourier transform. J. Chem. Phys 121, 6117 (2004)ADSCrossRefGoogle Scholar
Lidar D.A., Bihary Z., Whaley K.B.: From completely positive maps to the quantum Markovian semigroup master equation. Chem. Phys 268, 35 (2001)ADSCrossRefGoogle Scholar
Mukherjea A.: The role of nonnegative idempotent matrices in certain problems in probability. In: Charles R. Johnson, ed., Matrix theory and applications. Providence, RI: Amer. Math. Soc., 1990Google Scholar
Cubitt, T.S.: The embedding problem for stochastic matrices is NP-hard. Manuscript in preparationGoogle Scholar
Nielsen M.A., Knill E., Laflamme R.: Complete quantum teleportation using nuclear magnetic resonance. Nature 396, 52 (1998)ADSCrossRefGoogle Scholar
Vandersypen L.M.K., Chuang I.L.: NMR techniques for quantum control and computation. Rev. Mod. Phys 76, 1037 (2004)ADSCrossRefGoogle Scholar
Emerson J. et al.: Symmetrized characterization of noisy quantum processes. Science, 317, 1893 (2007)ADSCrossRefGoogle Scholar
Gurvits, L.: Classical deterministic complexity of Edmonds problem and quantum entanglement. In: Proceedings of the thirty-fifth ACM symposium on Theory of computing. New York: ACM Press, 2003, pp. 10–19Google Scholar
Porkolab, L., Khachiyan, L.: Computing integral points in convex semi-algebraic sets. In: Proceedings of the 38th Annual Symposium on Foundations of Computer Science (FOCS ’97), Discataway, NJ: IEEE 1997, p. 162Google Scholar
Porkolab, L.: On the Complexity of Real and Integer Semidefinite Programming. PhD thesis, Rutgers, (1996)Google Scholar