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Acta Physica Sinica

Robust quantum computation of the quantum kicked Harper model and dissipative decoherence

Bin Ye, Wenbo Xu, Gu Bin-Jie, (1)江南大学通信与控制工程学院，无锡 214122; (2)江南大学信息工程学院，无锡 214122

Published Jan 1, 2008• 2 citations
Publisher

Abstract

The effects of the dissipative decoherence on the quantum computation of the quantum kicked Harper (QKH) model are investigated using the quantum trajectory approach and the quantum Monte Carlo method. The simulation shows that the dynamical localization for the QKH model and the stochastic web in phase space are destroyed by moderate levels of dissipation. Taking the phase damping channel as the noise model, the universal law for fidelity decay and the timescales of reliable computation are analyzed. Compared with the static imperfections, the timescales for reliable computation of the dissipative system drop more quickly with the increase of the number of qubits when the decay rate is under a certain threshold. While above the threshold, the timescales drop more quickly in the presence of static imperfections.

DOI
10.7498/aps.57.689

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The effects of the dissipative decoherence on the quantum computation of the quantum kicked Harper (QKH) model are investigated using the quantum trajectory approach and the quantum Monte Carlo method. The simulation shows that the dynamical localization for the QKH model and the stochastic web in phase space are destroyed by moderate levels of dissipation. Taking the phase damping channel as the noise model, the universal law for fidelity decay and the timescales of reliable computation are analyzed. Compared with the static imperfections, the timescales for reliable computation of the dissipative system drop more quickly with the increase of the number of qubits when the decay rate is under a certain threshold. While above the threshold, the timescales drop more quickly in the presence of static imperfections.
The stability of quantum computating of the kicked Harper model with various perturbations is investigated. Above a certain threshold of the imperfections, quantum chaos sets in. The effects of the noise errors and the static imperfections on the quantum computation are analyzed by comparing the statistical ergodic properties and the Husimi functions of the perturbed eigenstates with the ideal eigenstates of the Floquet operator. It is shown that the fidelity decay with static imperfections is exponential while it is Gaussian with noise errors. The time scales of reliable computation with these perturbations are obtained through numerical simulations. Due to the errors in classical computation the distance of two initial states increases exponentially, while the stability of quantum computation is independent of the integrable or chaotic nature of the underlying dynamics.
The simulation of complex quantum systems on a quantum computer is studied, taking the kicked Harper model as an example. This well-studied system has a rich variety of dynamical behavior depending on parameters, displays interesting phenomena such as fractal spectra, mixed phase space, dynamical localization, anomalous diffusion, or partial delocalization, and can describe electrons in a magnetic field. Three different quantum algorithms are presented and analyzed, enabling us to simulate efficiently the evolution operator of this system with different precision using different resources. Depending on the parameters chosen, the system is near integrable, localized, or partially delocalized. In each case we identify transport or spectral quantities which can be obtained more efficiently on a quantum computer than on a classical one. In most cases, a polynomial gain compared to classical algorithms is obtained, which can be quadratic or less depending on the parameter regime. We also present the effects of static imperfections on the quantities selected and show that depending on the regime of parameters, very different behaviors are observed. Some quantities can be obtained reliably with moderate levels of imperfection even for large number of qubits, whereas others are exponentially sensitive to the number of qubits. In particular, the imperfection threshold for delocalization becomes exponentially small in the partially delocalized regime. Our results show that interesting behavior can be observed with as little as 7-8 qubits and can be reliably measured in presence of moderate levels of internal imperfections.
Using the methods of quantum trajectories we investigate the effects of dissipative decoherence in a quantum computer algorithm simulating dynamics in various regimes of quantum chaos including dynamical localization, the quantum ergodic regime, and quasi-integrable motion. As an example we use the quantum sawtooth algorithm which can be implemented in a polynomial number of quantum gates. It is shown that the fidelity of quantum computation decays exponentially with time and that the decay rate is proportional to the number of qubits, number of quantum gates, and per gate dissipation rate induced by external decoherence. In the limit of strong dissipation the quantum algorithm generates a quantum attractor which may have a complex or simple structure. We also compare the effects of dissipative decoherence with the effects of static imperfections.
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.
Abstract. Using the methods of quantum trajectories we study effects of dissipative decoherence on the accuracy of the Grover quantum search algorithm. The dependence on the number of qubits and dissipation rate are determined and tested numerically with up to 16 qubits. As a result, our numerical and analytical studies give the universal law for decay of fidelity and probability of searched state which are induced by dissipative decoherence effects. This law is in agreement with the results obtained previously for quantum chaos algorithms. PACS. 03.67.Lx Quantum Computation – 03.65.Yz Decoherence; open systems; quantum statistical meth-ods – 24.10.Cn Many-body theory QICS. 11.40.+s Search – 13.10.+n Effects of noise and imperfections – 11.50.+m Quantum maps, quantum chaos Nowadays the quantum computing attracts a great inter-est of the scientific community [1]. The main reason of that is due to the fact that certain quantum algorithms allow to perform computations much faster than the usual classical algorithms. The famous example is the Shor algo-
We study the effects of static interqubit interactions on the accuracy of various quantum algorithms. Extensive numerical simulations show that their effect is stronger compared to external decoherence. Analytical approach based on Random Matrix Theory is developed. It gives universal law for fidelity decay induced by interqubit static interactions. This determines the time scale for reliable quantum computation in presence of realistic static imperfections and external decoherence. New polynomial algorithms are developed for simulation of complex dynamics in the regime of classical and quantum chaos, and Anderson metal- insulator transition. A generic quantum error correction method is developed; it allows to eliminate coherent effect of static errors. The theoretical results are confirmed by numerical computations with up to 28 qubits.