Constructing Quantum Circuits for Maximally Entangled multi-qubit states

Constructing Quantum Circuits for Maximally Entangled multi-qubit states

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Unformatted text preview: arXiv:1003.3142v1 [quant-ph] 16 Mar 2010 Constructing quantum circuits for maximally entangled multi-qubit states using the genetic algorithm Zheyong Fan 1 ‡ , Hugo de Garis 2 , Ben Goertzel 2 , 3 , Zhongzhou Ren 1 , and Huabi Zeng 1 1 Department of Physics, Nanjing University, Nanjing, 210093, China 2 Artificial Intelligence Institute, Computer Science Department, Xiamen University, Xiamen, Fujian Province, China 3 Novamente LLC Abstract. Numerical optimization methods such as hillclimbing and simulated annealing have been applied to search for highly entangled multi-qubit states. Here the genetic algorithm is applied to this optimization problem – to search not only for highly entangled states, but also for the corresponding quantum circuits creating these states. Simple quantum circuits for maximally (highly) entangled states are discovered for 3, 4, 5, and 6-qubit systems; and extension of the method to systems with more qubits is discussed. Among other results we have found explicit quantum circuits for maximally entangled 5 and 6-qubit circuits, with only 8 and 13 quantum gates respectively. One significant advantage of our method over previous ones is that it allows very simple construction of quantum circuits based on the quantum states found. ‡ Email: [email protected] Constructing quantum circuits for maximally entangled multi-qubit states using the genetic algorithm 2 1. Introduction Quantum entanglement [1, 2], enabling states with correlations that have no classical analogue, is one of the central concepts differentiating quantum information from classical information. Entanglement is essential to many quantum information protocols such as quantum key distribution, dense coding, and quantum teleportation, and is also thought to play important roles in quantum computational speedup [3]. Abstractly, a bipartite pure state is called entangled if it is not decomposable to a tensor product of two states of the two subsystems. Quantitatively, there are measures [4] which tell us how entangled a state is. Maximally or highly entangled states are of particular interest because their entanglement provides a valuable resource which can be used to perform tasks that are otherwise difficult or impossible. Maximally entangled quantum states of small systems are well known. For 2-qubit states, the Bell state | Bell ) = ( | 00 ) + | 11 ) ) / √ 2 is maximally entangled on all counts: maximal entanglement entropy, maximal violation of the Bell inequality, and complete mixture of its one-party reduced states. The 3-qubit generalization of the Bell state is the GHZ state | GHZ3 ) = ( | 000 ) + | 111 ) ) / √ 2, which is also maximally entangled. One can easily generalize the GHZ state to general n-qubit states, | GHZ n ) = ( | 00 ··· )±| 11 ··· 1 ) ) / √ 2. However, for 4 or more qubits, these states are not maximally entangled [5], and in fact display below-average entanglement, as shown by the numerical calculations of Borras et al...
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Constructing Quantum Circuits for Maximally Entangled multi-qubit states

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