Qubits properties

From the above, it is apparent that most members of the extensive [Ln2] family described fulfil the basic requirements to behave as 2qubit CNOT or -y/ SWAP quantum gates. As part of the process to analyse this possibility, the qubit properties of the individual Ln(III) ions needed to be evaluated. 

A wide variety of proof-of-principle systems have been proposed, synthesized and studied in the field of molecular spin qubits. In fact, due to the fast development of the field, several chemical quantum computation reviews using magnetic molecules as spin qubits have been published over the past decade, covering both experimental and theoretical results [67-69]. Only in a minority of experiments implementing non-trivial one- or two-qubit gates has been carried out, so in this aspect this family is clearly not yet competitive with other hardware candidates.1 Of course, the main interest of the molecular approach that makes it qualitatively different is that molecules can be chemically engineered to tailor their properties and acquire new functionalities. 

In this chapter, we focus on the use of lanthanides as spin-based hardware for QC. The remainder of this introductory section provides some essential concepts and definitions and then it succinctly describes some of the existing proposals for QC. The second section provides a brief overview of results obtained with spin-based systems other than lanthanides. The following two sections review experiments made on qubits and quantum gates, respectively, based on lanthanides, highlighting their specific properties and advantages for QC applications. 

Single atomic ions confined in radio frequency traps and cooled by laser beams (Figure 7.4a) formed the basis for the first proposal of a CNOT quantum gate with an explicit physical system [14]. The first experimental realization of a CNOT quantum gate was in fact demonstrated on a system inspired by this scheme [37]. In this proposal, two internal electronic states of alkaline-earth or transition metal ions (e.g. Ba2+ or Yb3+) define the qubit basis. These states have excellent coherence properties, with T2 and T2 in the range of seconds [15]. Each qubit can be... 

Lanthanide ions have emerged as a very promising category of chemically accessible realizations of spin-based qubits. Their suitability for this task, which results from their physical, chemical and quantum mechanical properties, is discussed in the following sections. 

Some properties of these ions make them particularly appealing as solid state spin qubits. The fact that they can be diluted into diamagnetic crystals offers a simple method to optimize their quantum coherence. Similarly to trapped ions, they are simple each qubit is embodied by a single atom. Yet, their immediate... 

In order to evaluate the individual properties as qubit of each Ln(III) ion within the pertinent [Ln2] complexes, it is necessary to study each ion without the interference of the other. This could be done if it were possible to prepare dinuclear complexes containing the l.n(III) to be studied in the desired position accompanied by a diamagnetic Ln(III) centre (or analogue) at the other site. This was possible thanks to the exceptional structural characteristics of the [Ln2] family. 

Figure 7.17 (a) Magnetic properties of [LaTb] and [Tb2] in the form of yT versus T plot per mole of Tb(lll). (b) Schematic representation of the qubit definition, weak coupling and asymmetry, as derived from magnetic and heat capacity data. 

Lanthanide ions offer several salient properties that make them especially attractive as qubit candidates (i) their magnetic states provide proper definitions of the qubit basis (ii) they show reasonably long coherence times (iii) important qubit parameters, such as the energy gap AE and the Rabi frequency 2R, can be chemically tuned by the design of the lanthanide co-ordination shell and (iv) the same molecular structure can be realized with many different lanthanide ions (e.g. with or without nuclear spin), thus providing further versatility for the design of spin qubits or hybrid spin registers. 

Research on modeling of endohedral fullerenes within single-walled carbon nanotubes (SWNTs) has received increased attention towards the understanding of their electronic and structural properties [304-307]. However, very recently particular emphasis was given to the endohedral fullerenes N C60 [308-313] and P C60 [314] due to the electron spin on the nitrogen or phosphorus site, respectively. Having an extremely long decoherence time the unpaired electron spin could be used as a qubit within a quantum computer. 

While considering ensembles of identical particles, one has to take into account their quantum statistical properties. This imposes additional symmetry restrictions on the ensemble dynamics and requires the inclusion of additional symmetry operations into the group of operations. This situation is discussed in the second paper of the chapter for Fermi systems, where the additional operation is antisymmetrization over the wave functions of individual fermionic qubits. 

In conclusions, the possibility to implement qubit devices by using properties of non-conventional superconductors has been discussed. Different tech-... 

Finally, here it comes the really weird property. It concerns quantum states which can be produced in more-than-one qubit systems, like ... 

The influence of the measurement result of a qubit affecting the state of another, as happens in an entangled state, is called non-locality. This strange property was pointed out for the first time in a very influential paper, published in 1935, by Albert Einstein, Boris Podolsky and Nathan Rose [13]. The paper aimed to demonstrate that Quantum Mechanics was an incomplete theory. According to the authors, a theory to be considered complete should contain what they defined reality elements. A reality element would be, still according to the authors, any physical quantity whose value could be predicted before performing a measurement on the system. For example, when a measurement of the observable cty is performed on a qubit, in an entangled cat state, the result determines the state of the other qubit, which could then be predicted before a measurement. Hence the observable Uy is a reality element. However, before the measurement is performed on the first qubit. 

Teleport is a process through which the state of a qubit is transferred to another, using the non-local properties of entangled states [17]. Differently from superdense coding, no qubit is transferred in teleport, but only a quantum state. 

Furthermore, once a measurement is made and the eigenvalue X is determined, the remaining I qubits will be projected onto the corresponding eigenvector. Therefore, others properties of the quantum system under study can be obtained by simply continuing the calculation, and many important physical information can then be extracted. 

We have seen that qubits can be accomplished by different quantum properties of a system. The basic requirement is that they must be well characterized and susceptible to manipulation by an external perturbation, so that the input states can be adequately prepared and controlled to produce the desired calculation. Besides, the physical representation of the qubit in quantum information processing must be unequivocal. This is certainly a requirement that NMR systems fulflll. In fact, a natural implementation of a qubit is an isolated spin 1/2 in a magnetic field [1]. In the operator basis, the general state of this spin can be represented by IV ) = a -l-l/2) - - j8 —1/2) (Figure 4.1). Labeling the states -l-l/2) as 0) and I-1/2) as 1), each state of the system can be represented by a single label, 0) or 11), which means one-qubit of information. 

Besides the inherent properties that a quantum computer, including the quantum gates, must have, there are technical issues regarding implemented and proposed architectures. These issues include scalability, decoherence and computational speed. Scalability refers to the ability to increase the number of qubits for calculation. Coherence is the ability for the qubit to retain its encoded information and thus decoherence is a loss of encoded information. Computational speed refers to the general number of quantum gates that can be applied before decoherence destroys the quantum state information. As will be alluded to below, it is not so much the difference between atoms and molecules in quantum computing architectures that determines feasibility but more so the choice of quantum state for qubit representation. 

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