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arXiv:quant-ph/0504211

Decoherence models and their effects on quantum maps and algorithms

Mario Leandro Aolita, Marcos Saraceno

Published Apr 27, 2005• 0 citations
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Abstract

In this work we study several models of decoherence and how different quantum maps and algorithms react when perturbed by them. Following closely Ref. [1], generalizations of the three paradigmatic one single qubit quantum channels (these are the depolarizing channel, the phase damping channel and the amplitude damping channel) for the case of an arbitrarily-sized finite-dimensional Hilbert space are presented, as well as other types of noise in phase space. More specifically, Grover's search algorithm's response to decoherence is analyzed; together with those of a family of quantum versions of chaotic and regular classical maps (the baker's map and the cat maps). A relationship between how sensitive to decoherence a quantum map is and the degree of complexity in the dynamics of its associated classical counterpart is observed; resulting in a clear tendency to react the more decoherently the more complex the associated classical dynamics is.

DOI
10.48550/arxiv.quant-ph/0504211

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In this work we study several models of decoherence and how different quantum maps and algorithms react when perturbed by them. Following closely Ref. [1], generalizations of the three paradigmatic one single qubit quantum channels (these are the depolarizing channel, the phase damping channel and the amplitude damping channel) for the case of an arbitrarily-sized finite-dimensional Hilbert space are presented, as well as other types of noise in phase space. More specifically, Grover's search algorithm's response to decoherence is analyzed; together with those of a family of quantum versions of chaotic and regular classical maps (the baker's map and the cat maps). A relationship between how sensitive to decoherence a quantum map is and the degree of complexity in the dynamics of its associated classical counterpart is observed; resulting in a clear tendency to react the more decoherently the more complex the associated classical dynamics is.
We study the behavior of cooperative multiplayer quantum games [35,36] in the presence of decoherence using different quantum channels such as amplitude damping, depolarizing and phase damping. It is seen that the outcomes of the games for the two damping channels with maximum values of decoherence reduce to same value. However, in comparison to phase damping channel, the payoffs of cooperators are strongly damped under the influence\ amplitude damping channel for\ the lower values of decoherence parameter. In the case of depolarizing channel, the game is a no-payoff game irrespective of the degree of entanglement in the initial state for the larger values of decoherence parameter. The decoherence gets the cooperators worse off.
We study the behavior of cooperative multiplayer quantum games [Q. Chen, Y. Wang, J.T. Liu, and K.L. Wang, Phys. Lett. A 327 (2004) 98; A.P. Flitney and L.C.L. Hollenberg, Quantum Inf. Comput. 7 (2007) 111] in the presence of decoherence using different quantum channels such as amplitude damping, depolarizing and phase damping. It is seen that the outcomes of the games for the two damping channels with maximum values of decoherence reduce to same value. However, in comparison to phase damping channel, the payoffs of cooperators are strongly damped under the influence amplitude damping channel for the lower values of decoherence parameter. In the case of depolarizing channel, the game is a no-payoff game irrespective of the degree of entanglement in the initial state for the larger values of decoherence parameter. The decoherence gets the cooperators worse off.
Discrete maps play an important role in the investigation \nof dynamical features of complex classical systems, especially \nwithin the theory of chaos. Similarly, quantum maps have proven \nto be a very useful mathematical tool within the study of complex \nquantum dynamical systems. Using the decoherent histories formulation \nof quantum mechanics we consider a particular framework for studying \nquantum maps which is motivated by the method of classical symbolic \ndynamics. Symbolic dynamics is known to be a very powerful method \nspecifically invented for the purpose of representing classical dynamical \nsystems by a discrete model that is suitable for information theoretic studies. \nOur framework uses the decoherent histories formalism which, \nsimilarly to classical symbolic dynamics, allows one to introduce information \ntheoretic quantities with respect to system dynamics. \nOur research within this framework can be viewed as a contribution \ntowards the development of a general theory of "quantum \nsymbolic dynamics". \n\nWe start by considering a special but very important example \nfor a quantum map: the quantum baker's map, invented for \nthe theoretical investigation of quantum chaos. \nHere we use the decoherent histories formalism to examine \nthe coarse-grained evolution of this map \nwith regard to the question of how classical predictability of \nthe evolution depends on the character of coarse-graining. \n%is affected by the choice of coarse-graining. \nWe demonstrate that hierarchical coarse-grainings display \ninteresting features with respect to this question. \n\nWe proceed with the investigation of decoherence properties \nof arbitrary unitary quantum maps. A number of interesting results \nis obtained within the framework of arbitrarily long histories \nconstructed from a fixed projective partition of a finite \ndimensional Hilbert space. In particular, we derive simple \nnecessary decoherence conditions, which employ only a single \niteration of a given unitary quantum map. Furthermore, \na surprising result is obtained with regard to the \nfundamental question of how the choice of the \ninitial state affects decoherence of histories. \nWithin the considered framework we show that if decoherence \nis established for arbitrary history lengths and all initial \nstates from the smallest natural set of states that \ncan be associated with the framework, then we get decoherence \nof such histories for arbitrary initial states. \nFinally, we make first steps towards proving analogous\nresults for approximate decoherence and suggest \nvarious questions for future research.\n
We study effects of static inter-qubit interactions and random errors in quantum gates on the stability of various quantum algorithms including the Grover quantum search algorithm and the quantum chaos maps. For the Grover algorithm our numerical and analytical results show existence of regular and chaotic phases depending on the static imperfection strength εεε. The critical border εcε_cεc​ between two phases drops polynomially with the number of qubits nqn_qnq​ as εc∼nq−3/2ε_c \sim n_q^{-3/2}εc​∼nq−3/2​. In the regular phase (ε<εc)(ε< ε_c)(ε<εc​) the algorithm remains robust against imperfections showing the efficiency gain εc/εε_c / εεc​/ε for ε>2−nq/2ε> 2^{-n_q/2}ε>2−nq​/2. In the chaotic phase (ε>εc)(ε> ε_c)(ε>εc​) the algorithm is completely destroyed. The results for the Grover algorithm are compared with the imperfection effects for quantum algorithms of quantum chaos maps where the universal law for the fidelity decay is given by the Random Matrix Theory (RMT). We also discuss a new gyroscopic quantum error correction method which allows to reduce the effect of static imperfections. In spite of this decay GYQEC allows to obtain a significant gain in the accuracy of quantum computations.
We study the differences between the processes of decoherence induced by chaotic and regular environments. For this we analyze a family of simple models that contain both regular and chaotic environments. In all cases the system of interest is a ''quantum walker,'' i.e., a quantum particle that can move on a lattice with a finite number of sites. The walker interacts with an environment which has a D-dimensional Hilbert space. The results we obtain suggest that regular and chaotic environments are not distinguishable from each other in a (short) time scale t*, which scales with the dimensionality of the environment as t*{proportional_to}log{sub 2}(D). However, chaotic environments continue to be effective over exponentially longer time scales while regular environments tend to reach saturation much sooner. We present both numerical and analytical results supporting this conclusion. The family of chaotic evolutions we consider includes the so-called quantum multibaker map as a particular case.