Decoherence

QCs will be exposed to a variety of errors. It is generally believed that the most important source of errors is the coupling of the QC to its environment (Chuang et al. 1995, Unruh 1995, Palma et al. 1996). This coupling will destroy quantum superpositions required for quantum computations and is usually referred to as decoherence (Zurek 1991). 

The interaction process of the QC with the environment can be understood as follows. We consider a single qubit interacting with its environment. The environment can be viewed as a high-dimensional quantum system. Initially the quantum bit and the environment are in a product (or non-entangled) state 

It is now easy to see that this process will in general spoil the quantum superposition in the qubit. To demonstrate this let us consider the following superposition state before the interaction with the environment  

6 Quantum Computers First Steps Towards a Realization 

The amplitudes of (0) ( 1)) have interfered constructively (destructively). However, if we perform the same operation after the interaction with the environment has taken place we find 

Thus far we have dealt with the idealized case of isolated molecules that are neither -subject to external collisions nor display spontaneous emission. Further, we have V assumed that the molecule is initially in a pure state (i.e., described by a wave function) and that the externally imposed electric field is coherent, that is, that the j field is described by a well-defined function of time [e.g., Eq. (1.35)]. Under these. circumstances the molecule is in a pure state before and after laser excitation and S remains so throughout its evolution. However, if the molecule is initially in a mixed4 state (e.g., due to prior collisional relaxation), or if the incident radiation field is notlf fully coherent (e.g., due to random fluctuations of the laser phase or of the laser amplitude), or if collisions cause the loss of quantum phase after excitation, then J phase information is degraded, interference phenomena are muted, and laser controi. is jeopardized. f 

Loss of quantum information (either of the phase or of the amplitude of a state) i to the interaction of a system with its environment is termed decoherence. Exampli include the obvious case where a system is actually embedded in an external onment, for example, a molecule in solution, or more subtle cases, for ex where the system is chosen as the center of mass of a body and the environme the 1023 variables associated with the motion of the atoms that comprise the  

The current view is that certain forms of decoherence can cause the Ioss ( quantum interference in just such a way that the system then obeys cla mechanics [158]. This view does not obviate the possibility that classical meet is, in fact, the limit of quantum mechanics when ft — 0 (i.e, when the system  

Consider then a system s interacting with an environment. The total Hamiltonian Hiot is of the form 

Since it is the dynamics of the system that is of interest, it would be convenient to preaverage over the environment variables and obtain an equation of motion for ps(t), the system component of the density matrix. Formal work of this kind [161,. .. 162] yields the so-called generalized master equation. Deriving the generalized f master equation, and extracting the various approximations utilized, goes well astray of the central focus of this book. For this reason we just sketch the models id direct the reader to suitable review articles [161, 162] that provide an appropriate pview. 

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Preskill J 1999 Battling decoherence the fault-tolerant quantum computer P/rys. Today June... 

Because of our inability to analyze the interaction of microscopic QM systems and macroscopic measuring devices to a sufficient degree, we make use of a set of empirical rules that are known as measurement theory. Some day, measurement theory will become a proven set of theorems in QM,, as the proponents of the decoherence theory, among others, claim. Until such time, it is beneficial to introduce the measurement process, and the principles associated with it, separately from the dynamics described by the Schrbdinger equation. 

Once instrumental effects on M have been accounted for, useful information about the physical system may be deduced from the modulation depth. For isolated molecules, averaging over scattering angles and summing over continuum indices will reduce the ratio R. Further loss of modulation depth may be caused by decoherence in dissipative systems (vide infra), making this quantity a potentially useful observable for deducing structural and dynamical effects. 

The previous sections focused on the case of isolated atoms or molecules, where coherence is fully maintained on relevant time scales, corresponding to molecular beam experiments. Here we proceed to extend the discussion to dense environments, where both population decay and pure dephasing [77] arise from interaction of a subsystem with a dissipative environment. Our interest is in the information content of the channel phase. It is relevant to note, however, that whereas the controllability of isolated molecules is both remarkable [24, 25, 27] and well understood [26], much less is known about the controllability of systems where dissipation is significant [78]. Although this question is not the thrust of the present chapter, this section bears implications to the problem of coherent control in the presence of dissipation, inasmuch as the channel phase serves as a sensitive measure of the extent of decoherence. 

Thus in the zero dephasing case, 8s reduces to the Breit-Wigner phase of the intermediate state resonance, elaborated on in the previous sections. In the dissipative environment, it is sensitive also to decay and decoherence mechanisms, as illustrated later. 

It is interesting to note (see Eqs. (59)—(61)) that pure decoherence introduces dependence of the channel phase on the final state energies. This dependence can be utilized to obtain new insights into the resonance properties, as illustrated in Fig. 16. Here we explore the photon energy dependence of 8/ for final state... 

Damping effects, transition state trajectory deterministic driving, 209-213 stochastically moving manifolds, 215-222 De Broglie wavelength, two-pathway excitation, coherence spectroscopy 165-166 Decoherence, two-pathway excitation, coherence spectroscopy ... 

On the other hand, lanthanides with 100% isotopical purity such as terbium or holmium are preferred to simplify the operation and minimize decoherence in spin qubits. In this respect, the existence, for some lanthanides, of a manifold of electronuclear states can provide additional resources for the implementation of multiple qubit states within the same molecule [31]. All atoms in the first coordination sphere should be oxygen, and the sample should be deuter-ated if the compound contains hydrogen, to avoid interaction with other nuclei spins. Again, POM chemistry has been shown to provide ideal examples of this kind. 

The final section deals with known examples of molecular spin qubits based on lanthanide SIMs. Distinction is made between single-qubit molecules and molecules which embody more than one qubit. This section includes some comments about decoherence in these molecular systems and strategies to control it. 

Before reviewing existing examples, a very brief explanation on the mechanisms of decoherence for molecular spin qubits is necessary more details are available elsewhere [67]. Broadly speaking, the three decoherence sources for these systems are spin bath decoherence, oscillator bath decoherence and pairwise dipolar decoherence, and can be regulated by a combination of temperature, magnetic field and chemical design of the system [70]. The spin bath mainly consists of nuclear spins, but in general it also includes any localized excitations that can couple to the... 

Hanson, R., Gywat, O. and Awschalom, D.D. (2006) Room-temperature manipulation and decoherence of a single spin... 

Morello, A., Stamp, P.C.E. and Tupitsyn, I. (2006) Pairwise decoherence in coupled spin qubit networks. Phys. Rev. 

Another practical tool is dynamical decoupling, a technique that uses sequences of fast qubit rotations to mitigate the effects of decoherence. The pulse sequences are designed such that the interactions of each qubit with its environment tend to average out [34, 35]. While still a major concern, decoherence may thus not be the strong impediment it originally seemed to represent for the advent of QC. 

Figure 7.9 (a) Echo-detected X-band EPR spectrum of a powder sample of Gd001 Y099W30 at T = 6 K. (b) Decoherence times 7, (solid symbols) and T2 (open symbols) measured at fi0H = 0.347 T, as a function of the atomic concentration of Gd ions. 

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