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arXiv:quant-ph/0701048 · Physical Review A

Network implementation of covariant two-qubit quantum operations

Jaroslav Novotný, G. Alber, Igor Jex

Published Apr 10, 2007• 1 citation
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Abstract

A six-qubit quantum network consisting of conditional unitary gates is presented which is capable of implementing a large class of covariant two-qubit quantum operations. Optimal covariant NOT operations for one- and two-qubit systems are special cases contained in this class. The design of this quantum network exploits basic algebraic properties which also shed light onto these covariant quantum processes.

DOI
10.1103/physreva.75.042309

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A six-qubit quantum network consisting of conditional unitary gates is presented which is capable of implementing a large class of covariant two-qubit quantum operations. Optimal covariant NOT operations for one- and two-qubit systems are special cases contained in this class. The design of this quantum network exploits basic algebraic properties which also shed light onto these covariant quantum processes.
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The structure is investigated of all completely positive quantum operations that transform pure two-qubit input states of a given degree of entanglement in a covariant way. Special cases thereof are quantum NOT operations which transform entangled pure two-qubit input states of a given degree of entanglement into orthogonal states in an optimal way. Based on our general analysis all covariant optimal two-qubit quantum NOT operations are determined. In particular, it is demonstrated that only in the case of maximally entangled input states can these quantum NOT operations be performed perfectly.
The implementation of a quantum computer requires the realization of a large number of N-qubit unitary operations which represent the possible oracles or which are part of the quantum algorithm. Until now there have been no standard ways to uniformly generate whole classes of N-qubit gates. We develop a method to generate arbitrary controlled phase-shift operations with a single network of one-qubit and two-qubit operations. This kind of network can be adapted to various physical implementations of quantum computing and is suitable to realize the Deutsch-Jozsa algorithm as well as Grover's search algorithm.
Abstract. A sequential network of quantum operations is efficiently described by its quantum comb [1], a non-negative operator with suitable normalization constraints. Here we analyze the case of networks enjoying symmetry with respect to the action of a given group of physical transformations, introducing the notion of covariant combs and testers, and proving the basic structure theorems for these objects. As an application, we discuss the optimal alignment of reference frames (without preestablished common references) with multiple rounds of quantum communication, showing that i) allowing an arbitrary amount of classical communication does not improve the alignment, and ii) a single round of quantum communication is sufficient. A quantum comb [1] describes a quantum network with N open slots in which an ordered sequence of variable quantum devices can be inserted, thus programming the quantum operation of the resulting circuit. Mathematically, a comb implements an admissible supermap [2, 3], that transforms an input network of N quantum operations into an output quantum operation. Having at disposal a suitable formalism opens the possibility of optimizing the architecture of quantum circuits for a large number of computational, cryptographic, and game-theoretical tasks, such as discrimination of
Any unitary transformation of quantum computational networks is explicitly decomposed, in an exact and unified form, into a sequence of a limited number of one-qubit quantum gates and the two-qubit diagonal gates that have diagonal unitary representation in usual computational basis. This decomposition may be simplified greatly with the help of the properties of the finite-dimensional multiple-quantum operator algebra spaces of a quantum system and the specific properties of a given quantum algorithm. As elementary building blocks of quantum computation, the two-qubit diagonal gates and one-qubit gates may be constructed physically with one- and two-body interactions in a two-state quantum system and hence could be conveniently realized experimentally. The present work will be helpful for implementing generally any N-qubit quantum computation in those feasible two-state quantum systems and determining conveniently the time evolution of these systems in course of quantum computation.
A quantum logic network to implement quantum telecloning is presented in this paper. The network includes two parts: the first part is used to create the telecloning channel and the second part to teleport the state. It can be used not only to implement universal telecloning for a bipartite entangled state which is completely unknown, but also to implement the phase-covariant telecloning for one that is partially known. Furthermore, the network can also be used to construct a tele-triplicator. It can easily be implemented in experiment because only single- and two-qubit operations are used in the network.