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arXiv:quant-ph/0407264

Effects of decoherence and imperfections for quantum algorithms

A. A. Pomeransky, O. V. Zhirov, Dima L. Shepelyansky

Published Jul 30, 2004• 0 citations
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Abstract

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 $ε$. The critical border $ε_c$ between two phases drops polynomially with the number of qubits $n_q$ as $ε_c \sim n_q^{-3/2}$. In the regular phase $(ε< ε_c)$ the algorithm remains robust against imperfections showing the efficiency gain $ε_c / ε$ for $ε> 2^{-n_q/2}$. In the chaotic phase $(ε> ε_c)$ 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.

DOI
10.48550/arxiv.quant-ph/0407264

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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-
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