Currently, the most important semiconductor qubits are Quantum Dot (QD) or donor based qubits. The Quantum Dot (QD) qubits are mostly based on either self-assembled or lateral semiconductor heterostructures (typically GaAs/AlGaAs). The self-assembled QDs are grown epitaxially on lattice-mismatched crystalline materials. These are for ex- ample InAs or GaAs on Ge or Si, with a lattice mismatch between around 5-10% [96]. The self-assembled QDs form zero dimensional semiconductor nanostructures and their formation is steered by mechanical strain, or more generally by the growth conditions like the temperature and substrate properties. Typical sizes of these self-assembled QDs are around approximately 10 nm, thus the dots have a discrete level structure and electrons and holes are strongly confined in the QDs. Therefore, these systems can also be termed artificial atoms or even molecules [96–99]. The self-assembled QDs are manipulated with electromagnetic radiation in the optical domain and many studies regarding their prop- erties and manipulation have been pursued [100–107]. Applications for the QDs are for example the realization of lasers or qubits. The qubits can be defined via charge or spin ex- citations in the QDs. From the quantum computation point of view, these devices possess long coherence times and are only weakly coupled to the evironmental degrees of freedom.
2.2 Semiconductor qubits 25
Figure 2.14: Lateral double quantum dot structure that was used for the realiza- tion of a charge qubit, the readout of the charge qubit is done by a bias volt- age pulse, which transforms the system to a molecular state. Finally, a finite cur- rent depending on the state of the qubit is measured [94]. Note that the QD struc- tures presented in Figs. 2.14 and 2.15 can in principle be used to realize both types of QD qubits.
Figure 2.15: Lateral quantum dot struc- ture used for the experimental realization of a QD spin qubit, the readout of the spin qubit is done via spin to charge con- version using the second dot of the dou- ble dot system. The spin state in the left dot (qubit spin) is transferred in a spin- dependent charge state in the right dot, which is then detected by the Quantum Point Contact (QPC) [95].
However, on the other hand, the fabrication of the QD structures is not yet fully deter- ministic and the coupling of several qubits is hard to control. Two qubits in self-assembled QDs can be coupled via their Coulomb (dipol-dipol) interaction or via a single mode of the electromagnetic field inside a cavity.
In contrast to self-assembled QDs, the system parameters of QDs formed in a two- dimensional electron gas (2DEG) in lateral semiconductor heterostructures are easily tun- able. In these structures the 2DEG forms approximately a few hundered nanometers below the surface, near the interface of the two semiconducting materials. The 2DEG is confined by the gates on top of the substrate and the QD is defined. The gate voltages allow to precisely control the tunnel barriers between the QD and the reservoirs, i.e., the source and drain leads, the coupling between the QDs in a double-dot system, and the shape of the QD (the electro-chemical potential inside the dot). For transport measurements with lateral QDs, a finite voltage is applied across the QD via leads connected to the quantum dot and thus a current is flowing through the QD.
In these QD structures, all electrons are bound and there is only a small number of additional free electrons (ranging from 0 to around 1000) [108]. Therefore, qubits based both on the charge degree of freedom [109–111] and the spin of individual electrons in a
26 2. Solid-state quantum computation
Figure 2.16: Sketch of an array of QD spin qubits, taken from Ref. [64]. The qubits are defined by the spin states of a single electron in a QD.
QD [64, 112] have been envisioned.
Up to now, in QD charge qubits, where the state of the qubit is defined by the position of an additional charge in a double-QD system, coherent oscillations have been demonstrated [94]. QD charge qubits are supposed to be hindered by strong decoherence due to phonons, 1/f-noise due to background charges, electronic baths and co-tunneling, for a quantitative analysis see Ref. [113]. However, recently substantial experimental progress regarding the coherence times has been made [114].
In the QD spin qubit, the qubit is defined by the spin state of an electron in a single QD [112]. It is considered to be more stable with respect to decoherence due to the stability of the electron spin in the QD. On the other hand, the electron spin is also much harder to manipulate and to read out. The most important decoherence sources in QD spin qubits are expected to be hyperfine interactions between the electron spin and the nuclear spins in the bulk material [115] and spin-phonon or spin-orbit phonon coupling [116]. It has been shown that it is possible to fabricate QDs with just one electron in the dot [117] and also the readout of QD spin qubits has been demonstrated via spin-charge conversion and readout with a quantum point contact (QPC) [95]. In Fig. 2.16 an array of several coupled QD spin qubits for quantum computation is sketched.
A schematic view of a single lateral QD with several top gates for manipulation of the QD is shown in Fig. 2.17. The source-drain voltage corresponds to the difference in the electrochemical potentials µS and µD, cf. Fig. 2.18. In figure 2.18 a more detailed picture of the energy levels of a single lateral quantum dot is depicted. The chemical potentials of source and drain are denoted by µS and µD, respectively. The charge states inside the quantum dot are indicated by black lines, the grey line is an orbital excited state. The electrochemical potential of the quantum dot with N electrons is µ(N). It is defined as the energy that is required to add an electron to the dot. Without loss of generality, the local electron level (or electrochemical potential) is given for the difference betweenN−1
2.2 Semiconductor qubits 27 V SD Γ ΓL R V G
Figure 2.17: Schematic drawing of a sin-
gle lateral quantum dot. The 2DEG
that forms approximately 100 nm below the surface, constricted by the gate elec- trodes, is indicated by the grey circle. The gate voltage can be used to displace the charge levels inside the quantum dot, cf. Fig. 2.19. Moreover, it is also possible to manipulate the barrier heights ΓLand ΓR externally.
eV
SDµ
Sµ
DΓ
LΓ
R∆E
(N)
µ
(N+1)
µ
(N−1)
µ
Figure 2.18: Energy levels inside a sin- gle lateral quantum dot. Depending on the source-drain voltage VSD, transport through the quantum dot is possible. The charge levels are given by black lines and the orbital excited levels are drawn in
grey. A gate voltage can be used to
shift the energy levels upwards through the transport window given by VSD. The Coulomb peaks in the transport current appear when the charge levels are inside the transport window, cf. Fig. 2.19. Pic- ture taken from [113].
and N electrons [108, 118] µ(N) ≡ U(N)−U(N −1) = N −N0− 1 2 EC− EC |e| (CSVSD+CgVg) +EN, (2.22)
where EC = e2/C is the charging energy of the dot and C is the capacitance given by the network of capacitances connected to the dot. The source-drain voltage VSD and the gate voltage Vg are used to manipulate the local electron level according to Eqn. (2.22). Moreover, N0 is the number of electrons in the QD without applying a gate voltage and the Ci are the corresponding capacitances. The discrete levels in the QD are separated by the addition energy Eadd(N), which is given by [108, 118]
Eadd(N) = µ(N + 1)−µ(N) =EC + ∆E, (2.23)
where ∆E is the level spacing between two discrete quantum states, indicated in Fig. 2.18. Note that this energy ∆E is zero when two electrons are added to the same spin-degenerate
28 2. Solid-state quantum computation
Figure 2.19: Transport through a QD. The left plot depicts the Coulomb peaks in the linear-response regime. The right plot shows the Coulomb diamonds in the differential conductance dI/dVSD, where VSD is the source-drain or bias voltage. The edges of the diamond-shaped regions (marked black) correspond to the onset of current. The grey lines indicate the onset of transport through excited states. Pictures are taken from Ref. [118].
level and the first term in the expressions for the addition energy, the charging energyEC, is purely electrostatic.
In transport experiments with QDs, a finite source-drain voltage is applied across the quantum dot and when one of the charge levels is tuned by the gate voltage into the transport window, the so-called Coulomb peaks appear. Thus, in a single quantum dot there are charge ground states, however, there also exist elementary excitations just like in real atoms, which lead to orbital excited states.
There also exist other proposals for qubits in semiconductors. The most important ones are based on phosphorus donors in silicon. Again, a distinction between donor charge and spin qubits is made, the charge qubit states are given by lowest states of an electron which is localized in the potential generated by two P+ ions. The donor spin qubit is based on the nuclear spin of a positively charged donor in a semiconductor [119], the qubit can be manipulated by controlling the resonance frequency of the nuclear spins and the coupling between spins with external gates. Then ac electromagnetic fields are used at resonance to alter the state of the nuclear spin. The measurement of this type of qubit could be performed by spin-charge conversion [120, 121].
Chapter 3
Quantum gate operations and
readout
A quantum gate operation in solid-state devices can be performed using electromagnetic fields, i.e., by means of gate voltages and magnetic fields. The simplest experiment, yet the most important one to begin with, is to demonstrate coherent Rabi oscillations in a driven qubit circuit. These have been shown first in superconducting charge qubits [122] and later also for superconducting flux qubits [85].
Typically, in superconducting charge qubits quantum gate operations are performed via application of DC pulses, i.e., short rectangular pulses. The individual pulses to complete a more complex quantum gate, for example acnotgate can be viewed as different single- qubit or two-qubit gates given by quantum mechanical propagatorsU = exp(−(i/~)H∆t), where the length of the pulse is denoted by ∆t and the time evolution is generated by a static HamiltonianH. Recently, also manipulation schemes that are inspired by protocols from NMR quantum computation [123] have been presented [124]. In chapter 10, another method to manipulate superconducting charge qubits will be presented. There, a short pulse-trajectory of rather soft (small Fourier bandwith compared to the time and frequency scale that is set by the pulse duration) pulses will be presented that drastically improves the gate performance of a charge qubit system and it is also feasible from the point of view of experiments. Superconducting flux qubits can also be manipulated with DC pulses. This is done by rapidly switching the two fields Bz and Bx that determine the energy bias and the tunnel matrix element of the qubit, respectively. Usually, the tunnel matrix element
Bx is set fixed. However, it is possible to replace one of the junctions in a rf-SQUID or 3jj flux qubit by another SQUID loop with two Josephson junctions. Then, the tunnel matrix element Bx(Φx) of the qubit can be tuned by adjusting the flux through the addtional SQUID loop. Note that the requirement to steer two fluxes that act spatially very close is hard to achieve and flux crosstalk then will certainly degrade the performance of these types of qubits. Moreover, also an additional noise source is introduced to the system. With this control over Bx and Bz the elementary single-qubit gates could be performed.
However, flux qubits are most conveniently addressed by AC fields and resonant pulses. This is done in the following way. In these systems first a coherent flux driving of the type
30 3. Quantum gate operations and readout
Figure 3.1: SEM image of a flux qubit with a readout DC-SQUID, which is surrounding the qubit loop. The qubit state can be measured using the current biased DC-SQUID, which will switch into the finite voltage state for dif- ferent values of the bias currentIB depending on the state of the qubit. Image courtesy of TU Delft.
Figure 3.2: SEM image of a flux qubit and attached readout SQUID. In experiments performed with this kind of qubit coherent Rabi oscilla- tions have been demonstrated. Be- low the image of the qubit circuit, an equivalent circuit diagram and the pulse sequence used for demonstra- tion of Rabi oscillations are shown. Image courtesy of TU Delft.
Hd = δcos(ωt) is applied. The oscillatory driving field can be applied at the resonance frequency of the qubit like in NMR experiments [76]. Then the manipulation pulses are convoluted with the driving [125]. The effective manipulation of the qubit is then described best in the reference frame that rotates with the driving frequency. Two-qubit operations can be performed with the same technique, for example in a two-qubit system the level splitting of one qubit depends on the state of the other qubit. Therefore, the transition of the first qubit, will be conditionally shifted depending on the state of the first qubit, e.g., when the second qubit is in the|1istate. A pulse can only be resonant with the transition of the first qubit when the state of the second qubit (the control qubit) is correct. Obviously, the timescale of the irreducible two-qubit operations is determined by the coupling strength between the qubits.
Another possibility for qubit manipulation is the adiabatic quantum computation[126, 127]. This refers to a composite quantum system of qubits that is always kept in its ground state, where the Hamiltonian of the system is different for different quantum gates or algorithms and varied slowly. Here, it is important to note that the speed-up of the adiabatic quantum computer is due to an energy gap between ground state and excited states. Adiabatic quantum computation has been shown to be equivalent to standard quan- tum computation [128]. Qubit setups with flux qubits have been proposed for adiabatic quantum computation [129].
31 However, prior to the demonstration of single-qubit or two-qubit operations, the basic experiment is the aformentioned demonstration of coherent Rabi oscillations, where the frequency of the oscillations should vary linearly with the amplitude of the driving field, and Ramsey fringe experiments, where the coherent evolution in the ˆσx-component of the qubit is probed. The latter is done by two π/2 pulses, before the coherent evolution and in the end to readout the pseudo-spin. Both of these experiments indicate whether the qubit system can really be manipulated quantum coherently. From these experiments the so-called quality factor of quantum coherence Qϕ = πνqT2 [93] can be extracted, where νq is the transition frequency of the qubit and T2 is the coherence time of a superposition of qubit states. It follows from the threshold theorem for QECCs, cf. section 1.2, that a Qϕ > 104νqtop is required for quantum computation. Here, top is the duration of an elementary operation [130]. For superconducting phase qubits, the quality factor can be as large as Qϕ ≈ 25000. For comparison, the QD qubits exhibit a much smaller quality factor, namely Qϕ ≈ 7 [94], Qϕ ≈ 3 [131], and Qϕ ≈ 6 [114], as summarized in [113]. However, not only the number of coherent oscillations within the decoherence time is an important benchmark for the qubit system but also the amplitude (“visibility”) of the coherent oscillations [132, 133]. Many experiments have observed [71, 93, 122, 134] and analyzed [135, 136] this loss of oscillation amplitude of the coherent oscillations. Thus, additionally to the exponential decay of the coherent oscillations due to decoherence the oscillations do not extrapolate back to the full expected amplitude at timet = 0, which is an indication for non-Markovian effects due to the environment [132, 137].
The readout of a flux qubit can be done by fabricating the flux qubit loop inside a DC-SQUID, which is then used for readout of the qubit state. For readout, the DC-
SQUID is biased by a current IB, which is increased until the DC-SQUID switches to
the finite voltage state (where it also becomes dissipative, i.e., it is not superconducting anymore). The point where the DC-SQUID switches to a finite voltage depends on the flux that is threading the DC-SQUID loop and therefore on the state of the qubit. Due to a signal-to-noise ratio (SNR) that is smaller than unity and the statistical nature of this measurement, a huge number of experiments have to be performed. However, there are many more readout schemes for flux qubits, which in the best cases even provide single- shot resolution, or are quantum-non-demolition (QND) measurements, as in the case of a flux qubit inside a cavity, which will be presented later.
In general, the ideal readout process should work as follows [138]. As long as the detec- tor is switched off, the qubit should not experience any backaction from the detector,i.e., dephasing and relaxation due to the detector circuit need to be avoided during the initial- ization and manipulation of the qubit. But when the detector is switched on to detect the state of the qubit, the qubit has to be dephased quickly by the detector in order to project the qubit state onto one of the basis states. Thus, for a fast measurement, the detector has to be coupled strongly to the qubit. During the readout process energy relaxation, which leads to mixing, should be suppressed, though. The relaxation of the qubit is also due to the coupling between the meter and the qubit, as long as the measurement is not a quantum non-demolition measurement (QND) for which the qubit and qubit-detector coupling terms in the Hamiltonian commute [138]. Note that the coupling strength of the
32 3. Quantum gate operations and readout readout device, the meter, and the qubit during the measurement also determines the basis in which the qubit is read out. However, also measurements that do not lead to a complete collapse of the wavefunction can be performed repeatedly in order to obtain information about the system [139]. This is for example the case for the readout of superconducting