Since the seminal proposal of Loss and DiVincenzo [18], many groups have tried to embed a spin qubit based on the single electron spin in a quantum dot [65–68]. The logical qubit states|0i and|1i, are the projection of the spin to the direction of the external magnetic field | ↑i and | ↓i. The ↑ and ↓ stand for the values of the spin projection ms= 1/2, ms=−1/2 respectively.
The initialization of single electron spin states can be achieved by coupling a quantum dot to a nearby lead [68]. Assuming that the width of the Fermi-Dirac distribution of the lead is smaller than the Zeeman splitting of the electron spin, the chemical potential of the lead is tuned between the electron spin eigenstates.
If this is the case the electron spin can only tunnel to the lower eigenstate and this is how the electron spin can be initialized [68].
Two single qubit axes of control are given by the external magnetic field (taken along z) and the inhomogeneous plane magnetic field. The inhomogeneous in-plane magnetic field can originate from the spin-orbit field [66,69], stray magnetic field of a micromagnet [70,71] or random in-plane nuclear magnetic fields [72].
When the in-plane magnetic field originates from the spin-orbit field, stray magnetic field of a micromagnet or random in-plane nuclear magnetic field, the wavefunction of the electron needs to be periodically displaced inside the quantum dot [17,69,73]. As the wavefunction of the electron is oscillating back and fourth, the electron experiences a periodic time-dependent magnetic field and coherently
changes the spin. The periodic oscillations of the electron wavefunction require a source of microwaves. This technique of controlling the electron spin is called Electric Dipole Spin Resonance (EDSR).
However, the efficiency of EDSR is often limited by un unknown static com-ponent of the magnetic field, parallel with thez-axis, originating in the unknown static magnetic field of the nuclear spins. This is often the case in InGaAs as this material does not have stable isotopes with a non-zero value of the nuclear spin.
On the other hand natural Si has≈95.3% of stable isotopes with a zero value of the projection of the nuclear spin. The remaining≈4.7% can be eliminated with isotopic purification techniques [74–76]. When the unknown magnetic field of the nuclear spins is present the electron spin is not rotating around the x-axis but around a sum axis of the nuclear spin magnetic field and the external magnetic field (parallel withz) and the x-axis of control.
However, the magnetic field of the nuclear spins is also varying in time. The nuclear spins are going to experience the magnetic field of the electron and are going to start to precess in this field at different rates. This precession of the nu-clear spins in the magnetic field of the electron is called Knight shift and happens onµstimescales [62]. As a consequence the electron spin is going to experience a time dependent magnetic field causing the electron spin to decohere.
Furthermore, the nuclear spins are also exchanging angular momentum with distant dipole-dipole interactions [62]. This can be understood on a simple exam-ple comprising of two nuclear spins, one close to the electron spin and one further
Figure 2.2. Quantum control of single electron qubit states on the Bloch sphere.
away from the electron spin. The initial system is a | ↑inuclear spin close to the electron spin and a nuclear spin further away from the electron spin initially in the| ↓istate. When the close and the distant nuclear spin exchange spin angular momentum the electron spin experiences a small change in the nuclear magnetic field, as the two nuclear spins are interacting with the electron spin with different magnitudes.
One of the methods for readout of the single electron spin qubit is spin-to-charge conversion [68]. Similarly to when the electron spin is initialized a chemical potential of a nearby lead is tuned so that only one of the spin eigenstates can tunnel out of the quantum dot. If a tunneling event occurs the current of a nearby quantum point contact is changed accordingly. An alternative method for spin readout is having an adjacent quantum dot near to the spin qubit, not occupied with an electron [77,78]. The adjacent quantum dot is required to have a different Zeeman splitting compared to the spin qubit quantum dot. This can be achieved by g-factor engineering [79] or by embedding a micromagnet on top of one of the quantum dots. The energy levels of one of the qubit states (for instance the spin down state) are aligned in the spin qubit quantum dot and the adjacent quantum dot. If the spin qubit was in the spin down state a tunneling event occurs and a nearby quantum point contact modifies the current. On the other hand, the spin up state is forbidden to tunnel to the adjacent quantum dot due to the large energy mismatch between the spin up states in the spin qubit quantum dot and the adjacent dot.
The exchange interaction produces a convenient platform for implementing two qubit gates between two single electron spin qubits [18]. However, the exchange interaction is highly susceptible to electric noise. The exchange interaction can mediate ∼ 10 coherent exchange oscillations before the quantum mechanical na-ture of the two qubit state is irreversibly lost [58]. A “sweet spot” is such a value of the detuning between the two quantum dots, in which the exchange interac-tions would be first order insensitive to the detuning noise. A double quantum dot loaded with two electrons only has a trivial “sweet spot”, one in which the exchange interaction is close to zero. Possible ways of reducing the sensitivity of exchange interaction to electric noise is to control the exchange by controlling the tunneling between the dots instead of the detuning [80,81]. As shown in Chapter 6, another way of reducing the sensitivity of the exchange interaction to electric noise is to perform two qubit operations with an adjacent quantum dot between the spin qubit quantum dots. A system like this has four “sweet spots”, giving rise to a possibility of charge-noise-insensitive exchange interaction.