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arXiv:1412.7350 · Physical Review A

Optimizing for an arbitrary perfect entangler. II. Application

Michael H. Goerz, Giulia Gualdi, Daniel M. Reich, Christiane P. Koch, Felix Motzoi, K. Birgitta Whaley, Jiří Vala, Matthias M. Müller, Simone Montangero, Tommaso Calarco

Published Jun 8, 2015• 49 citations
arXivHTMLPDF

Abstract

The difficulty of an optimization task in quantum information science depends on the proper mathematical expression of the physical target. Here we demonstrate the power of optimization functionals targeting an arbitrary perfect two-qubit entangler, which allow generation of a maximally entangled state from some initial product state. We show for two quantum information platforms of current interest, i.e., nitrogen vacancy centers in diamond and superconducting Josephson junctions, that an arbitrary perfect entangler can be reached faster and with higher fidelity than both specific two-qubit gates and local equivalence classes of two-qubit gates. Our results are obtained using two independent optimization approaches, underscoring the critical role of the optimization target.

DOI
10.1103/physreva.91.062307

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The difficulty of an optimization task in quantum information science depends on the proper mathematical expression of the physical target. Here we demonstrate the power of optimization functionals targeting an arbitrary perfect two-qubit entangler, which allow generation of a maximally entangled state from some initial product state. We show for two quantum information platforms of current interest, i.e., nitrogen vacancy centers in diamond and superconducting Josephson junctions, that an arbitrary perfect entangler can be reached faster and with higher fidelity than both specific two-qubit gates and local equivalence classes of two-qubit gates. Our results are obtained using two independent optimization approaches, underscoring the critical role of the optimization target.
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Here we consider perfect entanglers from another perspective. It is shown that there are some special perfect entanglers which can maximally entangle a full product basis. We explicitly construct a one-parameter family of such entanglers together with the proper product basis that they maximally entangle. This special family of perfect entanglers contains some well-known operators such as controlled-NOT (CNOT) and double-CNOT, but not SWAP\sqrt{\mathrm{SWAP}}SWAP​. In addition, it is shown that all perfect entanglers with entangling power equal to the maximal value 29\frac{2}{9}92​ are also special perfect entanglers. It is proved that the one-parameter family is the only possible set of special perfect entanglers. Also we provide an analytic way to implement any arbitrary two-qubit gate, given a proper special perfect entangler supplemented with single-qubit gates. Such gates are shown to provide a minimum universal gate construction in that just two of them are necessary and sufficient in implementation of a generic two-qubit gate.
Optimal implementation of quantum gates is crucial for designing a quantum computer. The necessary condition for optimal construction of a two-qubit unitary operation is obtained. It can be proved that the BBB gate is the unique gate that can construct a two-qubit universal circuit with only two applications, i.e., this condition is also sufficient in the case of two applications of the elementary two-qubit gate. It is also shown that many perfect entanglers cannot simulate an arbitrary two-qubit gate with only three applications. We show how a two-qubit gate's construction power is related to the construction power of its mirror gate, and introduce a specific kind of two-qubit entangling gate, i.e., a super controlled gate, which can construct an arbitrary two-qubit gate by three applications.