The Quantum Bit

The quantum analogue to the classical bit is the quantum bit or qubit. In order not to overwhelm the reader with abstract concepts I will introduce the notion of a qubit by an explicit example. In particular, I will discuss the realization of qubits with atoms. This example will be relevant throughout this chapter. 

It is convenient to discuss quantum mechanical states in a mathematically more abstract context, namely as vectors in a Hilbertspace. Some basic and for this chapter relevant properties of this formalism are summarized in the Appendix. Usually the Dirac notation is used to represent state vectors  

Here the sysmbols m, 00101 are used to identify the state vector. The state vector completely describes the state of a quantum system and is the geometric analogue to the wavefunction. 

The connection to quantum computing is now made by identifying state vectors 

6 Quantum Computers First Steps Towards a Realization 

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The basic element of a quantum computer is the quantum bit or qubit. It is the QC counterpart of the Boolean bit, a classical physical system with two well-defined states. A material realization of a qubit is a quantum two-level system, with energy eigenstates, 0) and 1), and an energy gap AE, which can be in any arbitrary superposition cp) = cos(d/2) 0) + exp(i0)sin(0/2) l).These pure superposition states can be visualized by using a Bloch sphere representation (see Figure 7.1). 

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. 

The individual unit of classical information is the bit an object that can take either one of two values, say 0 or 1. The corresponding unit of quantum information is the quantum bit or qubit. It describes a state in the simplest possible quantum system [1,2]. The smallest nontrivial Hilbert space is two-dimensional, and we may denote an orthonormal basis for the vector space as 0> and 11 >. A single bit or qubit can represent at most two numbers, but qubits can be put into infinitely many other states by a superposition ... 

Representative SMS spectra of ferropericlase ((Mgo.75,Feo.25)0) as a function of pressure at room temperature (a) [23] and along with a stainless steel (SS) for CS measurements (b). Dots Experimental measurements black lines modeled spectra with the MOTIF program. The quantum bits at 0, 13, and 45 GPa are generated from the QS of the high-spin Fe " " in the sample, whereas the flat feature of the spectra at 70,79, and 92 GPa indicates disappearance of the QS and the occurrence of the low-spin Fe +. 

Analogously, the unit of information in Quantum Information and Quantum Computation is the quantum bit, or qubit, for short. A qubit can assume the logical values 0 or 1. However, it can also be in a logical state containing any linear combination of them, thanks to laws of quantum mechanics [8], Physically, qubits can be represented by any quantum object with two well defined and distinct eigenstates. Examples of qubits are the photon polarization states, electrons in two-level atoms (as an approximation) and nuclear spins under the influence of a magnetic field. 

An experimental approach to realize quantum logic gates based atomic Ramsey interferometry is presented in Sect. 6.5. The quantum bits are represented by atoms. A beam of atoms is sent through a microwave resonator. The resonator provides the necessary interaction between the atoms to perform quantum logic gates. This is an example for the concept of flying qubits. Experiments along these lines are carried out at the Ecole Normale Superiore in Paris. 

An alternative realization of the flying qubit concept is briefly discussed in Sect. 6.7. The quantum bits here are stored in polarizations of single photons and the coupling between the photons is provided by atoms. This is the only implementation of quantum logic discussed in this chapter that is not based on atoms as the carrier of quantum bits. 

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... 

The overall OD vibrational distribution from the HOD photodissociation resembles that from the D2O photodissociation. Similarly, the OH vibrational distribution from the HOD photodissociation is similar to that from the H2O photodissociation. There are, however, notable differences for the OD products from HOD and D2O, similarly for the OH products from HOD and H2O. It is also clear that rotational temperatures are all quite cold for all OH (OD) products. From the above experimental results, the branching ratio of the H and D product channels from the HOD photodissociation can be estimated, since the mixed sample of H2O and D2O with 1 1 ratio can quickly reach equilibrium with the exact ratios of H2O, HOD and D2O known to be 1 2 1. Because the absorption spectrum of H2O at 157nm is a broadband transition, we can reasonably assume that the absorption cross-sections are the same for the three water isotopomer molecules. It is also quite obvious that the quantum yield of these molecules at 157 nm excitation should be unity since the A1B surface is purely repulsive and is not coupled to any other electronic surfaces. From the above measurement of the H-atom products from the mixed sample, the ratio of the H-atom products from HOD and H2O is determined to be 1.27. If we assume the quantum yield for H2O at 157 is unity, the quantum yield for the H production should be 0.64 (i.e. 1.27 divided by 2) since the HOD concentration is twice that of H2O in the mixed sample. Similarly, from the above measurement of the D-atom product from the mixed sample, we can actually determine the ratio of the D-atom products from HOD and D2O to be 0.52. Using the same assumption that the quantum yield of the D2O photodissociation at 157 nm is unity, the quantum yield of the D-atom production from the HOD photodissociation at 157 nm is determined to be 0.26. Therefore the total quantum yield for the H and D products from HOD is 0.64 + 0.26 = 0.90. This is a little bit smaller ( 10%) than 1 since the total quantum yield of the H and D productions from the HOD photodissociation should be unity because no other dissociation channel is present for the HOD photodissociation other than the H and D atom elimination processes. There are a couple of sources of error, however, in this estimation (a) the assumption that the absorption cross-sections of all three water isotopomers at 157 nm are exactly the same, and (b) the accuracy of the volume mixture in the... 

In this equation, C andT refer to control and target qubits, respectively. The resulting state (output of the qugate) is said to be an entangled state of the two qubits, that is, a state that cannot be written as a product of states for each qubit [30]. The occurrence of such entangled states is another characteristic trait of QC, at the basis of secure quantum communication or cryptography. It also implies that, as opposed to what happens with a classical bit, an arbitrary quantum bit cannot be copied (the COPY classical operation is, in fact, based on the application of a succession of classical CNOT gates) [4]. 

In the early development of the atomic model scientists initially thought that, they could define the sub-atomic particles by the laws of classical physics—that is, they were tiny bits of matter. However, they later discovered that this particle view of the atom could not explain many of the observations that scientists were making. About this time, a model (the quantum mechanical model) that attributed the properties of both matter and waves to particles began to gain favor. This model described the behavior of electrons in terms of waves (electromagnetic radiation). 

It is a bit of a lie to say, as we did in previous chapters, that complex scalar product spaces are state spaces for quantum mechanical systems. Certainly every nonzero vector in a complex scalar product space determines a quantum mechanical state however, the converse is not true. If two vectors differ only by a phase factor, or if two vectors normaUze to the same vector, then they will determine the same physical state. This is one of the fundamental assumptions of quantum mechanics. The quantum model we used in Chapters 2 through 9 ignored this subtlety. However, to understand spin we must face this issue. 

The moral of our story of the quantum chemical reaction dynamicist should be perfectly clear — at least two hydrogens are the dynamicists best friend. Indeed, our current supercomputers may seem to be a bit less super. 

When the transverse hopping is diffusive, the effect of tb can only be felt through virtual (perturbative) processes that are faster than the quantum coherence time 1 / 0( ) and in which energy conservation is not required (uncertainty principle). These processes involve pairs of correlated particles in all channels. For instance, a correlated electron-hole pair on one chain can be broken temporarily, one particle hopping to a nearest-neighbor chain (a virtual process) followed a bit later by the other particle, there... 

Such an entity represents the basic unit of information in a quantum computer—a quantum bit or qubit. Unlike a classical bit, which can store only a single value—a 0 or 1—a qubit can store both 0 and 1 at the same time. The state of a two-qubit register could be written... 

This work has been supported by the Russian Foundation for Basic Research grant 08-02-99042-r-ofi, a grant of the Program "Quantum Nanostructures" of the Presidium of RAS and a grant "Quantum bit on base of micro- and nanostructures with metal conductivity" of the Program "Technology Basis of New Computing Methods" of ITCS department of RAS. 

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