In the previous sections we have demonstrated good gate fidelity within an one-dimensional calculation. The applicability of such results, however, is
not clear for the following reasons. At each intersection, to connect the wave functions in different segments of quantum wires the Griffith [35] bound-ary conditions are imposed, which treat each segment as simple straight line without taking its geometry into consideration. Taking the geometry of quantum wires into account, we expect the transmission through an intersec-tion maximizes only when the incident energy of electrons can couple to the quasi-bound states of the intersection. The transmission probability depends on incident energy complicatedly while that does not depend on incident en-ergy if the Griffith boundary conditions are imposed. On the other hand, the construction of the σz and T gates in our architecture relies on how the phases of the transmission amplitudes increase with the magnetic flux. And the phase dependence is determined by the relative position of the incoming and outgoing leads along the ring. In real situation the leads and the arms of the rings have non-negligible width. Can we still establish a perfect phase property when the width effect is taken into account? If not, are we able to construct such gates like σz and T gates which require the phase-shifting property of the transmission amplitudes?
We perform a two-dimensional calculation based on the lattice Green’s function method [41] to address the above questions. This method is suitable for our use because of its ability to calculate the transmission property for an arbitrarily shaped conductor. Another appealing feature of this method is that some types of magnetic fields may be incorporated quite simply into the formalism, and therefore it is well suited to study various effects of mag-netotransport [41]. Moreover, we can obtain the Green’s function recursively and then extract the scattering matrix elements from the Green’s function as in [42]. The discretized mesh used in our calculation is shown in Fig.
3.4, and the single particle Green’s function is solved on this mesh under the
Figure 3.4: Actually simulated meshes in our calculation. The numbers along the axes are the index of the mesh. In this calculation 121 × 238 meshes are included. Red arrows indicating the relevant length of this system. The dashed line in the center separates the left half region and the right half one, across it the magnetic field is taken to be abruptly different. The effective mass of the electron is 0.09 m0.
effective mass approximation. We have made an infinite barrier approxima-tion, that is, the potential energy in the light region is set to be zero and that in the dark region is set to be infinite. The straight light segments in the center connect respectively to the semi-infinite incoming and outgoing leads, in which the magnetic field is assumed to be not present. However, at actually simulated mesh shown here the magnetic field is present, and is taken to be abruptly different in the left half region and the right half region (separated in the figure by a dashed line).
In a realistic two-dimensional calculation the quantum waveguides are not singlemoded. We define the logic state of the qubit as the presence of a single electron in the first transverse mode of a rail, and we restrict the incident energy in the range between the first and second transverse mode so as to minimize the error due to leakage to other transverse modes. The
9.5 10 10.5 11 11.5
Figure 3.5: Transmission probabilities from the 0-rail of the input to the 0-rail (left panel) and 1-rail (right panel) of the output at a zero magnetic field. The plot energy range is between the first and second transverse mode, 3.02eV and 11.49eV in this system. The light and dark line correspond to T0 and T1, respectively.
probabilities of transmission T0 and T1 (T0 = |T11|2, T1 = |T12|2) at a zero magnetic field are shown in Fig. 3.6 in the energy range between the first and second transverse mode. The zero energy point is set at the conduction band edge. The incorporation of the geometry of the intersection greatly complicates the energy response of the coupled ring system, as contrasted to the one-dimensional situation where T0and T1are periodic to kL/π. Another notable feature of this system is that the incident electron from the input 0-rail can better couple to the output 1-rail rather than to the output 0-rail (we will be more precise to this claim later).
8.44
Figure 3.6: Transmission probabilities from the rail of the input to the 0-rail (left panel) and 1-0-rail (right panel) of the output. φ1∗φ2 6 0, |φ1| = |φ2| . In Fig. 3.6 we calculate T0 and T1 with respect to the incident electron energy and the normalized magnetic flux φ. Here the normalized flux φ is defined by φi = Φi/Φ0, where Φi is the average of the magnetic flux enclosed by the outer or inner circle, through the left (i = 1) or right (i = 2) ring. In the two unit of φ1of the plot region shown here, the transmission probabilities are almost periodic to φ1 (φ2 has the opposite sign of φ1). Having the ratio of wire width to ring radius as 101, this system well retains the periodic oscillation of the transmission probability with respect to the enclosed magnetic flux.
To observe how the non-negligible widths of the quantum wires affects the phase property of T11 and T12, the phase relationship at E = 8.548(eV ) is illustrated Fig. in 3.7 to φ1 (φ2 has the opposite sign of φ1). The argument of T11 increases about 0.103π (or 9.7491 π) at a unit of φ1, while that of T12 increases half as fast. Although in this two-dimensional system we tried to keep our original design (i.e., we deploy the two leads along the ring at a distance of 162 ring circumference), the increment of the arguments is different from what we have designed in the one-dimensional case. Also, the increment of the arguments at a unit of φ1 is no longer exactly the same at different
5 10 15 20 line resp.). The dots mark the increment of the arguments at each unit of φ1
and the straight lines are linear fittings of the increment.
E and φ1, but in the figure shown the increment varies less than 1.5% from a linear extrapolation. Since the increment of the arguments is not even rational numbers, it is not apparent here whether the phase property of this two-dimensional system could facilitate the construction of the σz and T gates. From the working points shown in Table 3.2, however, we indeed see the working points of the σz and T gates deviate from the working point of the I gate only by almost integer units of φ1 and φ2. How could non-rational increment of the arguments adds up to a phase difference of π/4 or π between T11 and T12? It may be explained as the following. Not only does the increment of the arguments increases linearly at each integer point of φ1, the arguments themselves increase linearly in between integer points of φ1. Note that the former claim holds at other energies but the latter does not. So the working points of the σz and T gates will not deviate very far from that of the I gate by exactly integer shifts of φ1 and φ2, and the fidelity of the I gate can be retained or even improved for the two gates. Another interesting thing to note is that the σx gate has a fidelity even higher than that of the I gate and reaches the threshold fidelity of quantum computation.
By applying a gate voltage to the central region as what we have done in the
Table 3.2: Error rate for the set of AB quantum gates in the two-dimensional calculation. The second column defines the dynamic working points for the gates.
Gate I σz σx T H
log10² −3.202 −3.197 −4.157 −3.453 −2.529 E(eV ) 8.548 8.548 6.896 8.548 9.476
φ1 3.282 7.280 −0.528 2.268 7.261 φ2 −3.282 2.718 0.528 −4.294 4.605
one-dimensional calculation, we expect other quantum gates could reach the threshold fidelity as well.