Quantum logic gates

The generic state of a qubit is represented by a linear combination of the two eigenkets  

This representation allows a geometric visualization of the qubit quantum state as a point on the surface of a unit radius sphere, called Bloch sphere. The most important points on Bloch sphere are shown on the table below, adapted from Ref. [1]. 

The power of quantum computation comes from the existence of superposition of states of qubits, particularly entanglement, and the ability to manipulate them through unitary transformations, as will be seen in the next sections. 

The Hilbert space for one qubit has only two dimensions. Quantum information processing requires unitary transformations operating on states of one and two qubits, called logic gates. Some important examples of unitary transformations of one qubit are the Pauli ma- 

These matrices, plus the 2 x 2 identity matrix, form a basis in the 2x2 matrix space, so that any operation of one qubit can be decomposed as a linear combination of the four matrices. Notice that X = Y = Y and Z = Z, and also that XX = 1, YY = 1 and ZZ = 1. The action of each one of these operations on a generic quantum state are written below  

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Research on multi-qubit molecules starts with the synthesis and characterization of systems that seem to embody more than one qubit, for example, systems with weakly coupled electron spins. Indeed, many molecular structures include several weakly coupled magnetic ions [76-78]. On a smaller scale, the capability of implementing a Controlled-NOT quantum logic gate using molecular clusters... 

Figure 7.3 Schematic representation of the operations of some quantum logic gates acting on two qubits. In quantum computation, single qubit rotations (Figure 7.2) and CNOT (controlled-NOT) or INSWAP quantum gates are universal.

Furthermore, we have shown in Refs. [28, 112] that in mulfiqubit quantum computation, it is vital to consider all possible modulations and fields, and not only those that directly perform a quantum logic gate. Thus, while usually one only requires the optimization of the fields applied to perform the gate, we have stressed that the gate fidelity can be greatly increased if one also applies and optimizes all other possible fields. 

SLI is not specific to molecular eigenstates, but universal to the superposition of any eigenstates in a variety of quantum systems. It is thus expected as a new tool for quantum logic gates not only in MEIP but also for other systems such as atoms, ions, and quantum dots. SLI also provides a new method to manipulate WPs with fs laser pulses in general applications of coherent control. 

A quantum computational network can be decomposed into quantum logic gates [94, 95], analogously to the situation for classical computers. Quantum logic gates provide fundamental examples of conditional quantum dynamics, in which one subsystem undergoes a coherent evolution, which depends on the quantum state of another subsystem. 

Gershenfeld and Chuang s two-qubit system [101] uses an NMR machine and the protons in 25. They demonstrated a nonlinear interaction between spins, a prerequisite for quantum logic gates. This was realized through the controlled-NOT operation (CNOT) which conditionally flips one spin based on the value of another [102], This gate can be considered as a quantum XOR gate. 

Bonding Patterns that Neutral and Charged Gold Clusters form with Small Ammonia Clusters and which implement Quantum Logic Gates... 

Thus far, studies of coherent optical processes in a PBG have assumed fixed (static) values of the atomic transition frequency [Quang 1997], However, in order to operate quantum logic gates, based on pairwise entanglement of atoms by field-induced dipole-dipole interactions [Brennen 1999 Petrosyan 2002 Opatrny 2003], one should be able to switch the interaction on- and off-, most conveniently by AC Stark-shifts of the transition frequency of one atom relative to the other, thereby changing its detuning from the PBG edge. 

The key to making a quantum logic gate is to provide conditional dynamics that is, we desire to perform on one physical subsystem a unitary transformation which is conditioned upon the quantum state of another subsystem [46]. In the context of cavity QED, the required conditional dynamics at the quantum level has recently been demonstrated [50,51]. For trapped ions, conditional dynamics at the quantum level has been demonstrated in verifications of zero-point laser cooling where absorption on the red sideband depended on the motional quantum state of the ion [11,12]. Recently, we have demonstrated a CN logic gate in this experiment, we also had the ability to prepare arbitrary input states to the gate (the keyboard operation of step (2a) below). 

Fig. 2. Schematic setup for the SQDS. This system can be regarded as a pair of two-ways interferometers coupled at their central stage. The system acts as a quantum logic gate for quanton and detector qubits. The quantization of the Phase Shifter accounts for the conditional dynamics between them.

In this section, we briefly describe the foundations of quantum information and computing. We list some physical systems that have been explored to create them and show that polar molecules represent a very attractive possibility for the realization of quantum logic gates. 

Here, as in the book of Michael Nielsen and Isaac Chuang, we will use two notations for the Pauli matrices X, Y and Z whenever they represent quantum logic gates, and the usual Ox, Oy and in a more physical context. 

Figure 3.1 Quantum logic gates symbols for one and two-qubit operations. Adapted with permission from [1].

Quantum logic gates generated by radiofrequency pulses... 

A.K. Khitrin, B.M. Fung, Nuclear magnetic resonance quantum logic gates using quadrupolar nuclei, J. Chem. Phys. 22 (2000) 6963-6965. 

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