The Qubit

Notice that both the north pole ([0 1]) and the south pole ([1 0]) are fixed by [T]. This projectivization corresponds to the rotation of the two-sphere around the vertical axis through an angle of a. 

Not every linear-subspace-preserving function on projective space descends from a complex linear operator. However, when we consider the unitary structure in Section 10.3 we find an imperfect but still useful converse — see Proposition 10.9. 

At last, after several chapters of pretending that the state space of a quantum system is linear, we can finally be honest. The state space of each quantum system is a complex projective space. The reader may wish to review Section 1.2 at this point to see that while we were truthful there, we omitted to mention that unit vectors differing by a phase factor represent identical states. (In mathematics, as in life, truthful and honest are not synonyms.) In the next section, we apply our new insight to the spin state space of a spin-1/2 particle. 

In this section we introduce the space of spin states of a spin-1/2 particle, such as an electron. In quantum computation (the investigation of computers whose basic states are quantum, not deterministic), this space of states is called a qubit, pronounced cue-bit. Just as a bit (a choice of 0 or 1) is the smallest unit of information in a deterministic computer, a qubit is the smallest unit of information in a quantum computer. 

The usual presentation of a spin-1/2 particle starts with two physically distinguishable states. These states are usually labeled by kets, such as +z) and —z (in physics texts) or 1) and 0) (in quantum computing texts). The name and asymmetrical notation connote the right half of the complex scalar product (also known as a bracket) used in descriptions of quantum systems, (1). One posits that every quantum state can be written as a superposition of kets  

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Plenary 7(5. N I Koroteev et al, e-mail address Koroteev nik.phys.iusu.su (CARS/CSRS, CAHRS, BioCARS). A survey of the many applications of what we call the Class II spectroscopies from third order and beyond. 2D and 3D Raman imaging. Coherence as stored infonuation, quantum infonuation (the qubit ). Uses tenus CARS/CSRS regardless of order. BioCARS is fourtli order in optically active solutions. 

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... 

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. 

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. 

The RWA in Eq. (4.183) breaks down if the transfer time is similar to or less than the inverse energy separation of the qubit, t < This is a common case for qubits whose resonance frequency is in the microwave (GHz) or radio frequency (RE MHz) range. In such cases, the optimization process must take into account both the dephasing due to the bath as in Eqs. (4.189)-(4.191) and the error due to the non-RWA terms when minimizing the infidelity. 

The goal of any practical modulation scheme is to reduce, or, if possible, eliminate, all the elements of the decoherence matrix in Eq. (4.203) (see Ref. [20] for an alternative solution). However, in order to obtain the optimal modulation [29], one must first know the system-bath coupling spectra of the qubits in question. This information is usually not available a priori and thus most experimentalists have resorted to the suboptimal DD (or bang-bang) modulation, which does not require this knowledge. 

For each modulation parameters pair, [(p,r, one experimentally measure the average decoherence rate of the qubit, R(t) (either decay of an initially excited state, or dephasing of an initial superposition of ground and excited states). Then one runs a simple numerical algorithm that finds the coupling spectrum, G( ), that best fits the observed results. 

Furthermore, if one has a multipartite system, and thus a decoherence matrix, one must perform multiple modulation schemes that address the different qubits, on top of performing the aforementioned single-qubit scheme for all the qubits. This is essential to ascertain the cross-coupling spectra of all the possible qubit pairs. As discussed in Refs [19, 20, 112], these cross-coupling spectra are extremely important in reducing disentanglement and allow, in certain circumstances, to completely eliminate decoherence. 

A novel interplay between entanglement as a QIP resource and entanglement as the source of decoherence was detailed [116]. Two entangled qubits were analyzed, each coupled to a bath via common modes. The non-Markovian timescale was considered, as well as dynamical modulations. It was shown how the entanglement of the qubits could vanish after a finite time (entanglement sudden death, ESD), but later restored by non-Markovian modulation-induced oscillations of the system-bath coherence. 

Figure 10.5. The qubit, a.k.a.. the state space for a spin-1/2 particle, otherwise known as...

We can reconcile the spherical picture with Figure 10.4(a) by noting that while the labeled points each refer to exactly one state of the qubit, each of the unlabeled points corresponds to a whole circle s worth of states, one circle of constant latitude on the sphere P(C ). 

Denoting the phase factor by X instead of g is slightly cleaner notation.) The reader should verify that this is indeed an equivalence relation. Thus the mathematical state space of the qubit implicit in the standard presentation is the set of all possible pairs (c+, c ) such that c+P -F c P = 1 modulo the equivalence relation Notice that the set of all satisfactory c s is just the unit three-sphere inside C. Because the equivalence comes from an action of the group T on this S , we call the state space... 

It is natural to ask which operations descend from V toP(V). IsP(V) a complex vector space Usually not. If V has a complex scalar product, does P( V) have a complex scalar product No. But. as we will see in this section, a complex scalar product on V does endow P(V) with a useful notion of orthogonality. Furthermore, using the complex scalar product on V we can measure angles in P( V). At the end of the section we apply this new technology to the qubit P(C"). 

The representation pi is called the spin-1/2 representation. It arises from the rotation of three-dimensional physical space and its effect on the qubit. In other words, experiments show that if two observers differ by a rotation g, then their observations of states of the qubit differ by a projective unitary transformation [[/] such that [Pg.320]

As a first example, consider the state space of the qubit, PCC ), Let a be any real number. Then the function... 

Complex conjugation is another physical symmetry of the qubit." We will find the following nomenclature useful. 

Proposition 10.7 Suppose S is a physical symmetry of the qubit such that... 

The next proposition classifies the physical symmetries of the qubit. As promised in the introduction to this section, these symmetries consist of the projective unitary symmetries (rotations) and compositions of projective unitary symmetries with complex conjugation (reflections). It follows easily from Proposition 10.1 that for any unitary operator T eU (C ) both [u] [Tv] and [u] i- — fSu)] are well-defined physical symmetries of... 

P(C2). In fact, every physical symmetry of the qubit is of this form. 

Exercise 10.22 Find a group isomorphism between S O (3) and a subgroup of the physical symmetries of the qubit. Use Proposition 10.1 to find a nontrivial group homomorphism from SU (2) into the group of physical symmetries of the qubit. Finally, express the group homomorphism SU(2) —> 50(3) from Section 4.3 in terms of these functions. 

Exercise 10.25 Show that the group of physical symmetries of the qubit is isomorphic to the group 0(3). 

Although atomic nitrogen and phosphorus are usually very reactive species, upon encapsulation they do not react with C60 while their electrons possess a long-lived spin. That spin can play the role of a natural quantum information unit with its two states up and down similar to the classical 1 and 0 states. Going one step further, because of the shielding by C60 the qubit in the fullerene... 

Experimental realization of a quantum computer requires isolated quantum systems that act as the quantum bits (qubits), and the presence of controlled unitary interactions between the qubits. As pointed out by many authors [97-99], if the qubits are not sufficiently isolated from outside influences, decoherences can destroy the quantum interferences that actually form the computation. 

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