1.4 Efforts toward experimental quantum computing
1.4.2 Superconducting qubits
Electrons are great for classical computing and people are now very good at manufacturing small electronic devices. With the nano-fabrication tech- niques available, one can routinely make conductors sufficiently small for the increase in potential caused by adding one electron to be significant. This leads to the classical effect known as Coulomb-blockade [34] in which move- ment of an electron onto a conducting region can be blocked by the excess of one electronic charge. Single electron transistors (SET) [35, 36] that are based on this effect can measure with good fidelity the movements of sin- gle charges. What stops single electrons being useful in implementing single 6It should be mentioned that there are actually schemes for state labelling that don’t have a exponential signal decrease [32]. However such schemes are not practical with the very small polarisations that are present in liquid state NMR.
1.4 Efforts toward experimental quantum computing 21
Gate voltage
Energy
Figure 1.7: Energy level diagram for stationary states of the Cooper pair box as a function of control gate voltage. The left minima corresponds toN Cooper pairs in the box and the right to N + 1 Cooper pairs. The dotted line represents the absence of Josephson tunnelling. Only theN andN+ 1 states are shown here. In reality the figure should be periodic along thex-axis with minima corresponding to {...N−1, N, N+ 1, N+ 2...} in the Cooper-pair box.
qubits is their very short coherence times. These are hundreds of picosec- onds at best [31]. Because of their charge, electrons strongly interact with everything, especially the ions of the host lattice. One way of improving this situation dramatically is using superconducting materials. In these materi- als, at temperatures below their transition temperature, the electrons pair up intoCooper pairs. These electrons that make up these Cooper pairs are bound by their interaction. The net interactions of a pair and the lattice are much smaller than those of single electrons. As a result the coherence times for qubits based on Cooper pairs can be much larger — typically nanosec- onds.
A simplified description of such a qubit is shown in Fig. 1.6. The diago- nal terms of the Hamiltonian for the qubit are determined by the potential difference between the box and the reservoir. This depends both on the self capacitive charging energy for the box and on the effect of the control gate, which is capacitively coupled to the box. The off-diagonal terms of the Hamiltonian are constant and are given by the tunnelling rates through the Josephson junctions. In experiments, the readout is effected by having the Cooper pair box weakly coupled to a “probe” via an incoherent tunnel junc- tion. The presence of an excess Cooper pair on the box will cause a current through this probe when it jumps off.
Nakamura and his coworkers at NEC in 1997 [37] showed an effect similar to Rabi flopping with such a system. They set the control gate so that they were operating at, for example, the left minima of Fig. 1.7. After sufficient time for the system to fall into the ground state the voltage was ramped
quickly to the anti-crossing. Ideally this would be done much quicker than any other of the system dynamics and the sudden approximation [38] could be made.
The eigenstates for the system at the anti-crossing are linear combinations of those states with definite numbers of pairs. An initial state with a definite pair number will undergo a nutation between N and N + 1 Cooper pairs on the box. After waiting for a chosen length (∆t) of time, the voltage is quickly ramped back away from the anti-crossing. By looking for an excess Cooper-pair tunnelling through the probe junction, the probability of finding the qubit in the excited state can be measured. The probe current shows a sinusoidal variation as the length of time at the anti-crossing (∆t) is varied. Very recently, the same group at NEC has shown similar oscillations for two Cooper pair boxes capacitively coupled together [39]. They infer from their measurements that the two qubits were at some part of the process entangled, although no direct measurements were made.
Many of the problems encountered while carrying out these experiments came from the extreme timescales involved. While the coupling can be made large enough for the coherence time to be large compared to the interaction, other factors are also important. The coherence time of ≈ 10 ns [39] was estimated. The Josephson coupling terms were of the order of 10 GHz. This means that rise times much quicker than 100 ps would be required to make the sudden approximation strictly valid. This pushes the limits of available technology and rise times of approximately 35 ps were estimated for the experiment.
In these experiments the coupling between the Cooper-pair box and the probe was constant. In order that the decoherence caused by this coupling be kept to a reasonable level the coupling must be weak. Given this situation it would be difficult to imagine that the near perfect readout that can be achieved with ion traps could be attained.