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arXiv:1112.3762 · AIP conference proceedings

Combinatorial structure of a holonomic controlled phase gate

Hoshang Heydari

Published Jan 1, 2012• 1 citation
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Abstract

We investigate the combinatorial structures of a holonomic controlled quantum gate based on toric varieties. In particular, we in detail discuss the combinatorial structures of a twoqubit holonomic controlled quantum gate on a two-qubit, a three-qubit, and a four-qubit quantum states. Our results show a very interesting relations between toric varieties of qubit systems and action of a two-qubit holonomic quantum gate on multi-qubit states.

DOI
10.1063/1.3688963

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We investigate the combinatorial structures of a holonomic controlled quantum gate based on toric varieties. In particular, we in detail discuss the combinatorial structures of a twoqubit holonomic controlled quantum gate on a two-qubit, a three-qubit, and a four-qubit quantum states. Our results show a very interesting relations between toric varieties of qubit systems and action of a two-qubit holonomic quantum gate on multi-qubit states.
We investigate the geometrical structure of multipartite states based on the construction of toric varieties. We show that the toric variety represents the space of general pure states and projective toric variety defines the space of separable set of multi-qubits states. We also discuss in details the construction of single-, two-, three-, and multi- qubits states. This construction gives a very simple and elegant visual representation of the geometrical structure of multipartite quantum systems.
In this paper we will investigate the action of holonomic and topological quantum gates on different classes of four qubit states. In particular, we review the construction of holonomic quantum gate based on geometric phase and topological quantum gate based on braid group. Then, we investigate the entanglement properties of three different classes of four‐qubit states based on geometric invariants. The result shows that entanglement properties of the two most generic classes of four‐qubit states can be controlled by holonomic and topological quantum gate..
Universal computation of a quantum system consisting of superpositions of well-separated coherent states of multiple harmonic oscillators can be achieved by three families of adiabatic holonomic gates. The first gate consists of moving a coherent state around a closed path in phase space, resulting in a relative Berry phase between that state and the other states. The second gate consists of "colliding" two coherent states of the same oscillator, resulting in coherent population transfer between them. The third gate is an effective controlled-phase gate on coherent states of two different oscillators. Such gates should be realizable via reservoir engineering of systems that support tunable nonlinearities, such as trapped ions and circuit QED.
In this paper we will investigate the action of holonomic and topological quantum gates on different classes of four qubit states. In particular, we review the construction of holonomic quantum gate based on geometric phase and topological quantum gate based on braid group. Then, we investigate the entanglement properties of three different classes of four‐qubit states based on geometric invariants. The result shows that entanglement properties of the two most generic classes of four‐qubit states can be controlled by holonomic and topological quantum gate..
Abstract Multi-qubit controlled gates are frequently used in quantum information processing. Although they can be built with gates from the universal set of one- and two-qubit gates, this procedure typically becomes very demanding as the number of such gates rapidly grows with the size of the computational problem. Thus, finding a way to realize them with fewer steps and therefore with reduced decoherence errors is important. Particularly, quantum information is entering noisy intermediate-scale quantum era and in this era, quantum computers do not have enough resources for full fault tolerance and therefor can only support the computation with a short duration. This makes the reduction of the steps even more important. Here, we propose the realization of multi-qubit controlled nonadiabatic holonomic gates with connecting systems. Our proposal can efficiently reduce the operation steps of the realization, thereby reducing the affection from decoherence and increasing the precision of the computation. So, it is useful to realize efficient and robust quantum information processors.