Bit Quantum Gates

It is now shown how the CM phonon mode is used to perform 2-bit quantum gates between any pair of ions. Before the gate all the quantum information is stored in the internal levels of the ions. The CM phonon mode is cooled to its motional ground state. Suppose we wish to perform a 2-bit quantum gate between the ith and the jth. ion in the trap. 

We will show how to perform the conditional sign change gate. This quantum gate is characterized by the following mapping 

In the ion trap QC the quantum gate (6.18) is performed in three steps. Let us first outline these steps before we discuss them in more detail. Fig. 6.6 shows the necessary operations on the three quantum systems involved in the gate ion i, ion j, and the CM phonon mode. Step (i) the qubit stored in the ith atom is transferred to the CM phonon mode by an laser pulse with an appropriate frequency and duration. That is, if ion i is in the excited state e)i a vibration in the string of ions is induced. On the other hand, if ion i is in the ground state p)i the ions remain in place. This is done in order to make the qubit accessible to all the other ions. Since all ions participate in the CM motion they all can see the quantum 

6 Quantum Computers First Steps Towards a Realization 

We will now elaborate on the three steps summarized above in more detail. Step (i) the qubit stored in the ith ion has to be transferred to the CM phonon mode. This requires to provide a coupling between the internal levels of the ion (where the qubit is stored before the operation) with the external motion (in which all ions participate due to the electrostatic coupling between them). This coupling can be provided by an appropriate laser. We have already seen in Sect. 6.2.2 that laser light can couple the internal levels of an atom or an ion. But a laser field carries also momentum. If a photon is absorbed by an ion momentum conservation requires that the photon momentum is transferred to the CM motion of the ion. This coupling provided by the laser allows now the required transfer of the qubit 

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The most simple computation we can perform is an operation that changes the coefficients of a single qubit. Under ideal conditions, that is in the absence of unwanted couplings to the environment, the coefficients are transformed reversibly (by a unitary transformation). This is a consequence of the Schrodinger equation which governs the dynamics of (nonrelativistic) quantum systems. We will call these operations 1-bit quantum gates in the following. 

These states i) and are quantum superpositions of the basis states 0) and 1). The only difference between these two state is the relative sign between the two basis states. If we perform a measurement in the state of the qubit in basis I 0), I 1) each basis state will be measured with equal probability in both cases. With respect to this measurement these two states are therefore indistinguishable. They are not the same however. Suppose we perform a 1-bit quantum gate that transforms the two basis states according to the following mapping ... 

This simple 1-bit quantum gate T applied to a basis state creates an equally weighted superposition of both basis states. The effect of this quantum gate on the two states I i) and J 2) can be easily obtained by replacing the basis states in Eqs. (6.6) and (6.7) by the corresponding superposition states on the right hand side of Eq. (6.8). This is due to the linearity of quantum mechanics. The transformation performed on these two states i) and 2) reads ... 

This is not yet sufficient to performing any arbitrary possible 1-bit quantum gate. What we need in addition to the rotations (6.10) are phase shifts conditional to the state of the atom ... 

With resonant and non-resonant laser interaction we can in principle perform any arbitrary 1-bit quantum gate on atoms. Atomic qubits are technologically feasible and have already been realized experimentally for example in the context of single trapped ions (Bollinger et al. 1991). The basics of this technology will be discussed in Section 6.4.5. 

In summary, we have now demonstrated how a conditional sign-change gate on two qubits can be performed in an ion trap QC. This gate is, along with arbitrary 1-bit quantum gates, sufficient to perform any arbitrary quantum computation. Let us summarize the three necessary transformations in the following table ... 

A simplified version of a 2-bit quantum gate based on the ion trap QC proposal has already been demonstrated experimentally at NIST in Boulder (Monroe et al. 1995). In their experiment a single Be+-ion was stored in an ion trap. The motion of the ion was cooled by laser cooling techniques to the motional ground state 0)cm (for trapping and cooling technology see Sect. 6.4.5). 

In the NIST experiment the four classical basis states were prepared and the the controlled-NOT gate was applied. Afterwards the state of the system was measured with the quantum jump technique (see Sect. 6.4.4). The 2-bit quantum gate yielded the expected result with a reliability of 90%. The sources of errors that are responsible for this probability of failure will be discussed in Sect. 6.8.4. 

In this section I will discuss a recently proposed error correction scheme designed particularly for the ion trap QC (Cirac et al. 1996). The scheme corrects for an important source of errors during the execution of 2-bit quantum gates. Because the scheme is not intended to correct for the most general error it can be implemented efficiently with regard to time and memory overhead. It is likely that this scheme can be tested as soon as a prototype ion trap QC is available. 

It is easy to verify that the states after the transformation are normalized. This time evolution operator is an example for a 1-bit quantum gate. A quantum superposition with arbitrary coefficients a and j3 evolves according to the linearity of U T) as follows ... 

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