Two-qubit system

Equations (4.183) and (4.184) conserve the parity of the number of excitations. Hence, the full two-qubit system can be split into two subsystems... 

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. 

Using the postulate IV, it is possible to construct the Hilbert space for systems containing two or more qubits. For a two-qubit system, the dimension of the Hilbert space is 4 X 4, since it is composed by vectors (kets) and matrices (operators), calculated using the tensor product of each vector and matrix for the individual qubit, as may be seen on Equations (3.4.10) and (3.4.11), where both representations, kets and vectors, are shown ... 

There is an important relation between the two CNOT gates, of a two-qubit system, which is CNOTfl = H CNOT, H H (see [1]). 

First, one can notice that there are no individual sates, of a two-qubit system, a) and fe) such as = a) (g) b). Indeed, if there were such states, one could expand them in the computational basis 0), 1) ... 

There are other possible entangled states of a two qubit system ... 

The set of quantum states [ ), ]< >" ") forms a basis for a two-qubit system, called Bell s basis [14]. 

The next step is to perform an unitary operation U/, which takes the two-qubit system from a generic state, x, y) to the state x,y f x)). This transformation x, y) -> x, y0 f x)) is nothing but the sum of the second qubit, the bottom line of the circuit, with f x), that is the computed function of the first qubit. The binary function, f x), is the one to be... 

A classical search algorithm needs about 0(N) operations in order to find a specified item in a disordered list containing N elements. The quantum search algorithm, created by Grover is quadratically faster than its classical analogous, since only OiVN) operations are needed [19]. In a quanmm computer, the number of elements to be searched is the number of possible states of the system A = 2", where n is the number of qubit system. Grover s algorithm is then considered to be of B-type. For a two-qubit system, with N = 2 = 4... 

In Figure 3.11 the application of the Grover algorithm is illustrated, for (a) a two-qubit system and (b) a ten-qubit system N = 2 = 1024). Notice that the amplitude of the searched state oscillates with the number of times the G operator is applied. Thus, one must know in advance how many solutions exist and also the number of elements in the space where the search is being carried on, for there is a optimum number of runs of the algorithm. These numbers are approximately 1 and 25, for n = 2 and n = 10, respectively. 

The pseudo-pure state preparation by temporal averaging in quadrupolar nuclei can be done in a similar way. To illustrate the procedure lets take a two-qubit system implemented by spin 3/2 nuclei. The corresponding thermal equilibrium density matrix is given by ... 

As discussed in the Chapter 3, the quantum search algorithm is one of the most important for quantum computation. It is used to search for one or more specific quantum states in an uniform superposition. It is often compared to a search of a name (or number) in a disordered list. The main feature of this algorithm is the operation, performed by the oracle , which labels the state (or states) to be searched, by inverting its (their) phase. The second operation is the inversion about the mean value, i.e. the amplitude of each state in the system. These two operations must be applied to the system a certain number of times, which depends on the number of items one is looking for and the total number of elements on the system. For a two qubit system, the number of searches is only 1. Another important application is the ability to use this algorithm for searching the solution of a specific problem, which can be done by preparing the action of the oracle operator. 

As a last topic in this chapter, we review the work of Miquel et al. [29] who implemented a quantum circuit to measure the discrete Wigner function of a two-qubit system through NMR. In Chapter 3 it was made a brief introduction to Wigner functions and the quantum processing of information in phase space. 

P5.2 - Grover search algorithm can be implemented for any item in a two-qubit system using the corresponding optimized pulse sequences [8] ... 

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. 

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). 

Unlike the case of enhancement of yield of product in a chemical reaction, control of qubit state transfers in a quantum computer is useful only if the control does generate sensibly perfect fidelity of population transfer. Fortunately, a typical qubit has a spectrum of states that is much simpler than that of a polyatomic molecule, so that control protocols that focus attention on the dynamics of population transfer in two- and three-level systems are likely to capture the essential dynamics of population transfer in a real qubit system. A large fraction of the theoretical effort devoted to describing such transfers has been confined to those simple cases. To a certain extent, many of these studies are analogous to... 

A one-level system e) that can exchange its population with the bath states [/) represents the case of autoionization or photoionization. However, the above Hamiltonian describes also a qubit, which can undergo transitions between the excited and ground states e) and g), respectively, due to its off-diagonal coupling to the bath. The bath may consist of quantum oscillators (modes) or two-level systems (spins) with different eigenfrequencies. Typical examples are spontaneous emission into photon or phonon continua. In the RWA, which is alleviated in Section 4.4, the present formalism applies to a relaxing qubit, under the substitutions... 

Thus, an exchange between two-qubit entanglement and system-bath entanglement may take place and be dynamically controlled by modulations. If the systems couple to two baths that have common modes, one can observe the transfer of coherence and buildup of entanglement between the two systems via these modes. If, on the other hand, the baths were completely separate, the coherence transfer between each system and its bath can modulate the amount of system-system entanglement, first lost and then regained. 

To gain some insight into this problem we focus here on the analysis of the Berry phase [1] in a weakly dissipative system. It is particularly timely to address this issue now given the recent experimental activities in realization of controlled quantum two-level systems (qubits), and in particular, the interest in observing a Berry phase (BP) (see, e.g., [5]). For instance, the superconducting qubits have a coupling to their environment, which is weak but not negligible [10, 15, 4], and thus it is important to find both the conditions under which the Berry phase can be observed and the nature of that Berry phase. 

Laflamme et al. s three-qubit system [103] is 26. In this molecule the two carbon nuclei are in different chemical environments and therefore represent two qubits, whilst the proton serves as the third. Compound 26 therefore represents the binary numbers 000, 001, 010, 011, 100, 110, and 111. A pulse of radio waves causes the nuclei to be thrown into the entangled superposition state, where they can act as qubits. The NMR machine then initiates the quantum computer program—a series of radiofrequency pulses that act like gates on the qubits and carry out the calculation. The superposition state is then collapsed to give an answer. 

The appropriate technique is the following. First, one encodes the original information a 10) + 6 1) on the two logical states 0/,) = 000) and 1 l) = 111) of a three qubit system. This is simply achieved by adding two physical qubits, initially prepared in the state 0), and by applying some well-chosen unitary transformation C to the compound system of three qubits this operation yields the state a 0/,) I 6 1/,) which is then submitted to the action of the noisy channel. Each of the three physical qubits of the system is likely to independently undergo a bit flip (with probability p). At the end of the channel, one performs the measurement associated with the four projectors... 

In quantum information applications one often treats the system of two qubits being manipulated as part of some information processing. This is modelled by a couple of interacting two-level systems. Following the approach in the present paper, we consider the one system to be strongly damped. In that case it serves as a faked continuum for the other one, and we desire to derive an equation of motion for the originally undamped system. The damping is described by a Markovian term of the Lindblad type. 

The well-characterised spin-qubit system of NV defects can be used for the QIT the quantum processor implementation can be realized by the coherent spin manipulation. The first step for coherent spin manipulation is to prepare a pure state of the internal spin structure. For the [N-V]" centre, spin state initialisation can be easily achieved by optical pumping with a polarised laser beam tuned above the absorption band. Decay from the 3E level via optical emission dominates but conserves spin. On the other hand, decay via the metastable singlet level 1A is slower since it does not conserve spin. Competition between those two processes at room temperature leads to spin polarization of the 3A ground level, making populated mainly the ms = 0 substate [40]. Once... 

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