Quantum circuits

Quantum systems of any kind can in principle be candidates for quantum hardware, including different kinds of spin qubits we briefly review some of these in the next section. Much effort has been expended on the question of which physical systems are best suited for use in QIP, but no ultimate answer has been found so far. A much quoted list of conditions to build computers was established by DiVincenzo [35], but one has to note that some of these restrictions are specific to the quantum circuit paradigm. 

There are several other ways of approaching QIP that do not use the quantum circuit model. The global control paradigm, embodied in the seminal proposal of... 

Figure 2.11 Sample fragment of a quantum circuit illustrating the effect of some typical quantum gates a SWAP operation between the upper and centre qubits is effected through three consecutive controlled-NOT ...

J., Zueco, D. and Luis, F. (2013) Coupling single molecule magnets to quantum circuits. New J. Phys., 15, 095007. 

Various schemes for hybrid quantum processors based on molecular ensembles as quantum memories and optical interfaces have been proposed. In Ref. [17], a hybrid quantum circuit using ensembles of cold polar molecules with solid-state quantum processors is discussed. As described above, the quantum memory is realized by collective spin states (ensemble qubit), which are coupled to a high-Q stripline cavity via microwave Raman processes. This proposal combines both molecular ensemble and stripline resonator ideas. A variant of this scheme using collective excitations of rotational and spin states of an ensemble of polar molecules prepared in a dipolar... 

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. 

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 3.2 Generic graphical representation of a quantum circuit. Adapted with permission from [1].

After the application of these sequences the system wiU be in the state described by Equation (3.5.11). Therefore, a Swap logic gate must be applied, in order to exchange the sates of individual qubits, accomplishing the QFT. The quantum circuit describing the QFT, may be seen on Figure 3.6. 

For a three-qubit system the QFT operator can be easily implemented using the basic known logic gates H, S and T, apart from the SWAP. The quantum circuit which illustrates this implementation is shown on Figure 3.7. 

The most important quantum algorithm, the Shor algorithm [11], uses the QFT for finding the order of a number, which increases the speed of the factorization process. These are basically implemented by the same quantum circuit and are the main reasons for the exponential gain of speed in comparison with the classical factorizing algorithm. 

In the simplest case of teleport, three qubits are involved, two with Alice (let s label them lira)) and one with Bob (labelled fc . As in the superdense coding process, initially Alice and Bob qubits, a) and b), are in a cat state. Alice wishes to transmit to Bob the unknown state of a third qubit, IV ) = a 0> -I- 8 1>. Of course, she cannot measure ir), for she would only get 0 or 1, with the probabilities a and p, respectively. The quantum circuit that describes the teleport process is illustrated in Figure 3.8, where the top line represents the qubit Alice wants to teleport to Bob ( V >), and the second and third lines, represent the entangled qubit pair, the second one with Alice and the third one with Bob. 

The quantum circuit that describes the Deutsch algorithm is illustrated in Figure 3.9, from which we can see that at the input the qubits are in a quantum state described by ... 

Let us first review an application of QFT a quantum circuit to estimate the phase of a state, (p, which is eigenket of an operator U U u) = The procedure requires two... 

There is a generic quantum circuit that has been used in many algorithms and has various applications [1,23]. It is known as the scattering circuit , because it resembles a scattering process, as illustrated on Figure 3.14. It uses an ancilla qubit which plays the role of the probe particle. By measuring the expectation values (a ) and ay) of the probe, we obtain information about either the interaction (17), or the input state (p). 

For coupled spins 1 /2, the temporal averaging method described above can be generalized to systems with larger number of spins. In these cases, it is necessary to combine 2" - 1 prepared states to create a pseudo-pure state in a system of n spins. The operations for preparing the individual states can be obtained based on CNOT and SWAP gates. For example, for three spins systems the quantum circuits of these operations are shown in Figure 4.9. 

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).

Figure 5.7 (a) General scheme of the quantum circuit implemented by Vandersypen et al. [18] to test Shot s... 

One interesting work is reported by Murali et al. [20], in which a half-adder and subtracter operations are implemented in a quadmpole 7 = 7/2 spin system. They used the nuclei of Cs in a liquid crystal, for testing the half-adder and subtractor quantum circuits, that are illustrated on Figure 5.11. The operations were implemented using sequences of selective tt-pulses, which invert the populations. The algorithms were tested with the sys-... 

Figure 5.10 Quantum circuit and NMR spectra corresponding to the implementation of the Deutsch-Jozsa algorithm in a quadmpole 7 = 3/2 nucleus by Das and Kumar [6). The two qubits are represented by the central and outer transitions. Transitions pointing to the same direction represent constant functions, and to opposite directions balanced ones. Adapted with permission from Ref. [6].

Figure 5.11 Quantum circuits that implement the half-adder and subtractor operations. Adapted with permission from Ref. [20].

Figure 5.14 Two-qubit quantum circuit used to simulate the superconductors BCS Hamiltonian. Boxed letters indicate directions that jr/2 pulses are applied. We kept the author s representation for the free-evolution period (dark circles). Adapted with permission from [28].

The quantum circuit used for this operation is shown in Figure 5.14. The NMR implementation of such circuit presents no particular difficulty, since it involves only simple gates. 

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. 

P5.3 - A reversible half-adder circuit can be implemented in a three-qubit system, according to the quantum circuit shown in Figure 5.11 [20] That circuit can be implemented by NMR in a quadrupole 7 = 7/2 nucleus system. Let Jti-j represent ideal selective n pulses applied to the transition i — j. Show that the sequence... 

Suppose that in a NMR experiment we produce an initial pseudopure state, and apply the quantum circuit that generates a cat state (see Chapter 3). Suppose also that we perform quantum state tomography on this state. We will find a matrix which will be similar to Equation (6.1.2), upon which one has to add the background to build the complete matrix ... 

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