Two qubit operation

Moving one step ahead to actual computation, within the quantum world everything is unitary, so the dynamics of computation can be described as U ip) —> I ip). An important result is that any unitary operator U may be simulated by a set of one- and two-qubit operations called gates [Barenco 1995 (b)]. Such a decomposition makes implementation feasible, which is good news. To see why this is so powerful, consider an arbitrary (entangled)... 

We first describe the generic setup to obtain a phase gate, or universal two-qubit operation, in Figure 17.3 [10]. We assume that the molecules are individually addressable by optical or microwave fields. (Alternatively, a frequency-based addressing scheme akin to the one in the previous section could be used.) We choose 0) and 11) as, for example, hyperfine states within a zero-dipole-moment manifold, in a level with a long coherence time and ) as a metastable state in a large-dipole-moment manifold. 

An example of a quantum circuit is illustrated in Figure 3.2. The upper line represents the qubit a) and lower one the qubit b). The operations appearing in the figure mean that the S gate is applied to the first qubit, whereas the gate T is applied to the second one. These operations are followed by the application of a two qubit operation U, and finally by the application of a Hadamard gate to the first qubit only. The whole process can be translated in mathematical language as [H 0 1] C/ [S T] af>). 

Figure 3.1 Quantum logic gates symbols for one and two-qubit operations. Adapted with permission from [1].

Figure 7.3 Schematic representation of the operations of some quantum logic gates acting on two qubits. In quantum computation, single qubit rotations (Figure 7.2) and CNOT (controlled-NOT) or INSWAP quantum gates are universal.

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]. 

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. 

In the original KLM scheme, the fundamental element is not the CNOT operation, but the more or less equivalent nonlinear sign-shift (NS) operation from which the two-qubit conditional sign flip gate can be constructed. Similar to the accessibility of the CNOT operation, universal quantum computation becomes possible with such a two-qubit gate [Barenco 1995 (a) Sleator 1995], Recently, our group has experimentally demonstrated the NS operation using photons produced via parametric down-conversion. In contrast to the KLM scheme, our method to observe the NS operates in the polarization basis and therefore does not require interferometric phase stability. 

For any ( > 0, such a state cannot be written as the product of any single qubit states. Two qubits in such a state are strongly (and nonlocally) correlated. By using a sequence of such phase gates with ( ) = k, and simple unitary operations on a single qubit, any quantum computation can be completed. 

FIGURE 17.6 External electric and classical microwave fields can be used to manipulate polar molecules in EZ traps to encode information in rotational states and perform one-qubit operations. Superconducting stripwire resonators can then couple different molecules and carry two-qubit gates. The sites are selected by adjusting the EZ trap voltage appropriately. (From Andr4 A. et al., Nature Phys., 2, 636, 2006 Cote, R., News Views, Nature Phys., 2, 583, 2006. With permission.)... 

The Hilbert space for one qubit has only two dimensions. Quantum information processing requires unitary transformations operating on states of one and two qubits, called logic gates. Some important examples of unitary transformations of one qubit are the Pauli ma-... 

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 ... 

The matrix representation for operators that act in only one qubit of a system containing two qubits can be constructed by calculating the tensorial product between one qubit operator and the 2 x 2 identity matrix ... 

Quantum circuits are diagrams that illustrate the operations necessary to implement a protocol, their time sequence and also the number of qubits present in the system [9]. They are composed of lines, one for each qubit, and symbols, which represent the quanmm logic gates actions in one or more qubits. On Figure 3.1 it is shown the symbols used for one and two-qubit gates. 

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. 

This molecule actually has three coupled spin 1/2 nuclei (the two nuclei plus the nucleus). The chlorine nuclei have effectively negligible couplings with all other nuclei. Such molecule can be used to implement 3-qubit operations, as it has been done in a lot of NMR QIP experiments. 

Now, let us return to the implementation of two-qubit gates. In Chapter 3 we saw that the action of the CNOT gate is invert one of the qubits (the target qubit) provided the other (the control qubit) is in the state 11) . In a two-qubit AB) system this is accomplished by following operators ... 

Note that these operators are not exactly equal to CNOT operators, but they act as CNOT gates for most of two qubit states. 

As a last example of two-qubit gates let us consider and important case where cascaded gates are used to produce the four states of the Bell basis. As discussed in Chapter 3, such states can be created from the computational basis states 00), 01), 110), 111) by the appUcation of the so called EPR generator operator (see Problems 4.3 and 4.4), which is implemented by the pulses corresponding to a Hadamard followed by a CNOT gate ... 

Figure 4.9 Quantum circuit used to create pseudo-pure states in a three qubit system by temporal averaging using two-qubit CNOT and SWAP gates. The pseudo-pure state 000) is obtained after combining the results of the seven (add the identity operator) Uj operations. Adapted with permission from Reference [27] (Copyright 2007 American Physical Society).

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. 

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