CNOT gate

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 former experiments [Sanaka 2002 O Brien 2003 Pittman 2003] destructive linear optical gate operations have been realized. As they necessarily destroy the output state, such schemes are not classically feed-forwardable. The first realization of a CNOT gate, which operates on two polarization qubits carried by independent photons and that satisfies the feed-forwardability criterion, was done by this experiment. 

A CNOT gate flips the second (target) bit if and only if the first one (control) has the logical value 1 and the control bit remains unaffected. The scheme we use to achieve the CNOT gate was first proposed in Ref. [Pittman 2001] and is shown in figure 2. 

Figure 2 A scheme to obtain a photonic realization of CNOT gate with two independent qubits. The qubits are encoded in the polarization of the photons. The scheme makes use of linear optical components, polarization entanglement and postselection. When one and only one photon is detected at the polarization sensitive detectors in the spatial modes 63 and 64 and in the polarization H, the scheme works as a CNOT gate.

We thus have the control bit encoded in 0,4 and in 61, the photon in 0,4 will be the control input to the destructive CNOT gate, and will thus be destroyed, while the second photon in bl will be the output control qubit. 

Figure 5 Demonstration of the ability of the CNOT gate to transform a separable state into an entangled state. In a) the coincidence ratio between the different terms HH,. .., VV is measured, proving the birefringence of the PBS has been sufficiently compensated in b) the superposition between HH and VV is proved to be coherent, by showing via Ou-Hong Mandel dip at 45° that the desired (H + V) state of the target bit emerges much more often than the spurious state (H — V). The fidelity is 81% 2% in the first case and 77% 3% for the second.

Figure 3J SWAP gate built from three CNOT gates. Adapted with permission from (1).

The logic gate called SWAP is built from a circuit containing only cnot gates, as shown on Figure 3.3. The first CNOT is the controlled by the first qubit (CNOTa), and second one is controlled by the second qubit (CNOT ). 

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

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. 

A two qubit-gate very used in quantum algorithms is the SWAP gate. It can be directly implemented by the pulses corresponding to three successive CNOT gates. 

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 ... 

As a specific example of generating pseudo-pure states in NMR spin systems, lets consider a case of two /-coupled spin 1/2 (see Section 4.1). The equilibrium density matrix of this system can be written as (4.3.3). Remember that the pulse sequence for the CNOT gate, the operations Uq, Ui, and U2 can be written as. 

Notice that the z rotations in the CNOT gates are not necessary for producing U, and U2. This is a typical example of the pulse simplification discussed in the last section. 

Starting from the equilibrium state, the CNOT gates do not introduce any off-diagonal elements into the deviation density matrix, so we can regard (4.3.21) as two subsets of pseudo-pure states, the state 00)o with label state 0) and the state ll)i with state label... 

Figure 4.12 Deviation density matrix truth table for CNOT gates implemented in a quadrupolar spin 3/2 system. Adapted with permission from Reference [31] (Copyright 2007 Elsevier).

An illustration of the use of the density matrix tomography process is shown in Figure 4.12. It is the measured truth table for a CNOT gate with control on qubits A and B, obtained from experimentally determined density matrices. 

The matrices for the CNOT gates for quadrupolar spin 3/2 system describe in the text are ... 

In general the Hadamard, Phase and NOT gates are 1-qubit operations, the CNOT gate is a 2-qubit operation, and the Toffoli is a 3-qubit operation... 

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