#### 2.1 Quantum Dots: Localized charge and spin

Quantum dots are man-made structures designed to trap electrons on length scales varying from a few tens to a few hundreds of nanometers. These dots can be capacitively coupled to electrostatic gates that are used to modify their equilibrium charge state, and can be tunnel coupled to electron reservoirs or to neighbouring quantum dots [18, 19]. In this review, we will focus on gate-defined dots, as their intrinsic electrical tunability makes them particularly attractive for quantum simulations.

Gate-defined quantum dots are usually based on semiconductor heterostructures presenting a quantum well in the vicinity ( nm) of the surface. Different materials can be used, such as GaAs/AlGaAs [19] or Si/SiGe [27]. Figure 1a shows a typical heterostructure on which gate-defined quantum dots are built: it consists of an AlGaAs/GaAs heterojunction, grown by molecular beam epitaxy. The difference in bandgap energy for the two materials results in a step in the conduction band minimum at the interface. During the growth process, a thin Si-doped layer is placed above the heterojunction. As shown in the conduction band diagram of Fig. 1b, the ionized dopants locally bend the conduction band minimum. This effect creates a triangular quantum well at the interface in which electrons are bound, forming a two-dimensional electron gas (2DEG).

Using electron-beam lithography, one defines thin metallic gates on the surface of the semiconductor [20, 19]. Figure 1c shows a scanning electron microscope (SEM) image of the gates used to define a triple dot system (bottom half) capacitively coupled to a single dot (upper half). Applying a negative voltage on the gates creates potential wells surrounded by barriers tunable by the “B” gates, in which a certain number of electrons is trapped. Applying a negative voltage on the plunger gates (“LP”, “MP”, “RP” and “TP”) lowers the number of trapped electrons in each well or quantum dot down to the single-electron regime. The crossed squares in Figure 1c indicate electron reservoirs, which are connected to external electronics through ohmic contacts.

Gate-defined quantum dots can also be created in nanowires [28] or carbon nanotubes [29]. We will focus in this review on 2DEG-based devices, as the versatility of the 2D design is especially interesting for quantum simulations.

The 3D confinement of electrons in the quantum dot creates a discrete series of quantum states. Due to the huge difference in the amplitude of the in-plane confining potential (defined by the gates) and of the confinement in the growth direction (due to the heterostructure), the low-energy states depend only on the 2D lateral potential, and display therefore a 2D shell structure. Quantum dots are therefore often termed “artificial 2D atoms”. As the filling of these energy levels is determined by Hund's rule, one can build a periodic table for 2D elements [18].

If a single quantum dot behaves like an artificial atom, quantum dots placed next to each other can form “artificial molecules” [30, 19, 31]. For the device shown in Figure 1c a triple dot array is formed in the lower half of the image. The voltage on the “B” gates controls the interdot tunnel barriers, and the voltage on the plunger gates (“LP”, “MP” and “RP”), allows to shift the electrochemical potential ladders of the different dots individually. The interdot tunnel coupling hybridizes the charge states localized on each of the respective dots, thereby forming “bonding” and “antibonding” orbitals. This effect can be used to perform quantum simulations of chemical reactions [32].

The current through a quantum dot system as a function of the various gate voltages and the bias voltage can be used to infer the number of electrons occupying each dot, the orbital level spacing, spin splittings, and so forth. A development of the last ten years is that all this information can also be obtained via a charge sensor placed in the vicinity of the quantum dots. Such a sensor can be easily integrated in the form of a narrow channel (quantum point contact or QPC) [33] or another dot [34]. The sensor must be operated at a working point where its conductance is sensitive to the electrostatic environment (i.e. on a flank of a Coulomb peak). Changes in the occupation of any of the dots are easily detected using such sensors (see Figure 3). We also see in Figure 3 that by adjusting the gate voltages, any desired number of electrons can be placed on any of the dots.

As we have seen, gate defined quantum dots offer a quantum platform with very high controllability and very efficient charge readout schemes. We will see in the next section how this can be used for digital and analog quantum simulations.

#### 2.2 Towards digital quantum simulation

The principle of digital quantum simulations [1, 10, 37] is to program a quantum computer for simulating the dynamics of a certain Hamiltonian *H*. The simulated initial state is represented by the quantum state ψ_{0} of a qubit register, and the time-evolution operator is approximated by a succession of elementary quantum gate operations acting on this qubit register.

Quantum dots can be used as qubits in various ways. The most promising approaches rely on spin manipulation in quantum dots in the few-electron regime. Qubit implementations based on charge degrees of freedom suffer from very short coherence and relaxation times ( ns, ns) [39-41].

Single spins can be manipulated by electron spin resonance [43] (ESR): an oscillatory current applied to a strip line placed next to a dot, induces an oscillatory magnetic field in a direction perpendicular to the magnetic field, which rotates the electron spin when it is on resonance with the Zeeman splitting. Alternatively, one can use electric-dipole spin resonance (EDSR) [35, 44], which allows all-electrical control of the spin: applying ac excitation on the electrostatic gates forces the quantum dot position to oscillate. Due to the spin-orbit coupling in the semiconductor, the motion of the electrons results in an effective oscillating magnetic field that rotates the spins with Rabi frequencies in the Megahertz range. EDSR can also be performed using a local micromagnet to create a magnetic field gradient. The motion of the electrons in the inhomogeneous magnetic field also allows controlled spin rotations [45].

Due to the combined effect of Pauli exclusion principle and Coulomb interactions, single electrons in two neighbouring dots feel an effective spin-spin interaction. This interaction takes the form of an Heisenberg exchange coupling [42]:

- (4)

where is the spin of the electron in dot *i*. The coupling constant J is negative (antiferromagnetic coupling) and depends on the spatial overlap of the electron wave functions, which is controlled by gate voltage. This coupling is the basis of a two-qubit gate for Zeeman qubits [42]. The effect of the exchange interaction is literally to swap or exchange the state of the two spins [46]. When exchange acts for half the duration of a SWAP gate, the gate is obtained, which is universal when combined with single-spin rotations. Exchange gates can be very fast, and are in practice limited only by the achievable rise times of gate voltage pulses (sub-ns).

An alternative qubit representation based on spins in quantum dots uses two-electron singlet and triplet spin states in a double quantum dot to define a single qubit [47]. The two-spin exchange interaction now acts as a single qubit gate [36]. For such singlet-triplet (ST) qubits a two-qubit gate requires to couple two double dots. Capacitive coupling between the dots can be used here: the difference in charge configuration between singlet and triplet state in one double dot slightly modifies the singlet-triplet time evolution in the other, which allows to entangle the spin states of the two double dots [48]. More recently, also a triple quantum dot containing three spins was operated as a single qubit [49-52], using the spin exchange mechanism to drive single qubit operations. Each of the possible qubit representations has its own advantages and disadvantages, and the improvement of the spin qubit performances is an actively pursued research topic.

To read out spin qubit states, two methods have been successfully demonstrated (see Figure 5). In both cases, tunneling of electrons is made to depend on their spin states. Simultaneous detection of the charge occupation of the dots, then allows one to infer the spin states. For single spins, the energy difference between spin-up and spin-down electrons is exploited to induce spin-selective tunneling (see Figure 5a) [53]. For two-spin states, the so-called Pauli spin blockade technique is used (see Figure 5b) [54, 55].

Relaxation of spins in quantum dots typically occurs via phonon-emission [20, 56]. Coupling of spins to phonons is usually dominated by the spin-orbit interaction (SOI), which admixes spin and orbital states, making spins sensitive to electric field fluctuations. Spin relaxation times strongly depend on the spin splitting, as this affects the relevant phonon density of states and electron-phonon coupling [57]. For low magnetic fields, relaxation times as high as s have been reached, both in GaAs [58] and in Si quantum dots [59].

The spin-orbit coupling does not induce pure dephasing of the spin-state, so it only influences spin coherence through relaxation and sets a limit [60]. Unfortunately, other perturbations strongly influence the phase of the spin states. The main source of spin decoherence is the hyperfine coupling between the electron spin and the spin of the Ga and As nuclei forming the quantum dots [20, 61, 62]. Interestingly, this hyperfine-induced decoherence in quantum dots is a physical realization of the well-known Central Spin problem, a heavily studied problem in condensed matter physics that in general cannot be solved exactly by analytical or numerical methods. The basic question is how the coherence evolves for one central spin that is coupled to a bath of other spins [63]:

- (5)

where is the spin of the nuclei and the coupling strength, proportional to the value of the electron wavefunction at the position of the nuclei. For large numbers of nuclei (for GaAs quantum dots, [20]), the influence of the nuclei can be modelled semiclassically as an Overhauser magnetic field , and the hyperfine interaction can be described by

- (6)

In strongly polarized systems, can be as high as a few Tesla, while its statistical fluctuations are of the order of a few mT [20, 61, 62]. This effective magnetic field adds up to the applied magnetic field and dephases the spins with characteristic time ns [36, 43]. Using spin-echo pulses, this dephasing can be compensated for. The resulting decoherence time is limited by nuclear spin dynamics: s [36, 64, 65].

Intense efforts focus nowadays on improved performance of coherence and control of spins. As the most important limitation comes from the hyperfine coupling, nuclear spin-free systems would be highly desirable. Si/SiGe heterostructures offer very interesting possibilities in this domain: they have a very low concentration of nuclear spins [27], and can be isotopically purified [66]. The improvements due to the weak hyperfine coupling in Si/SiGe systems are seen in a measured singlet-triplet dephasing time of ns [67], which is up to 40 times longer than the typical case in GaAs [36]. Furthermore, singlet-triplet relaxation times as high as 3s have been measured [68], compared to at most 10 ms in GaAs systems [69].

Another option is to control the nuclei and stabilize the Overhauser field by electron spin-nuclear spin feedback [61, 62]. Using these methods, a increase of a factor of 10 has been demonstrated in GaAs [70].

These ideas could lead to important improvements of the performance of quantum dot spin qubits and could open the way towards digital quantum simulations with quantum dots.

#### 2.3 Analog Quantum Simulations

Whereas using quantum computers should lead to a universal quantum simulator, analog quantum simulations can be performed by constructing dedicated quantum systems designed to simulate a specific Hamiltonian. It is of course not sufficient for a quantum system to evolve according to a Hamiltonian to qualify as a quantum simulator. A system becomes a useful simulator when it has control knobs that allow to freely tune the relevant parameters of the Hamiltonian, or when it permits measurement of important quantities that are not accessible in natural systems. In this way, the Hamiltonian can be taken into regimes that were not accessible before, and that are too complex to be studied via classical computations, giving new insight in the underlying physics.

Quantum simulators are especially interesting to study the emergence of strongly correlated phases in many-body systems, as these regimes are the most difficult to study theoretically or simulate classically. For this purpose, quantum dots show very interesting properties: as we have seen in the previous section, both charge and spin correlations are present in quantum dots, and can present a complex interplay already in double or triple dot systems.

An array of tunnel coupled quantum dots is naturally described by the Hubbard hamiltonian [71] and all the parameters of the hamiltonian (interdot detuning, tunnel couplings, ...) can be widely varied by controlling the gate voltages. Quantum dot systems can therefore be used to simulate Hubbard physics. The form of this hamiltonian is

- (7)

The first term describes the tunnel coupling between dot *i* and dot *j*, is the on-site interaction energy in dot *i*, and the is the chemical potential of electrons in dot *i*, which depends linearly on gate voltages. Note that this hamiltonian is a single band hamiltonian that takes into account only one orbital state per dot. As explained above, in typical structures, *U* is of the order of a few meV, *t* is tunable by gate voltages from to eV and all can be controlled individually over a range of several meV. In a dilution refrigerator, such a system can be cooled down to a few tens of mK, ensuring that the thermal energy is significantly lower than the other energy scales in the system (). This regime is particularly important for condensed matter, yet is hard to access using cold atoms in optical lattices. Therefore quantum dots offer a very interesting platform for simulation of the Hubbard hamiltonian.

The few-quantum dot systems recently studied experimentally [72, 26, 73, 48, 74] show an increasing complexity. They include triple dots in either a linear or triangular geometry, and quadruple dots arranged along a line or a square. In long chains (⩾4 dots), 1D correlated electron physics [75] could be simulated. Two-dimensional arrays of dots () could lead to the simulation of Hubbard ladder physics [76], and the observation of d-wave hole pairing and possibly superconductivity [77].

The currently available systems already present very interesting possibilities: for instance, using a fully tunnel-coupled quadruple dot with three electrons, one could observe Nagaoka's ferromagnetism [78-80], which takes its origin in a two-path interference process (see Figure 6). Let us first assume infinite interactions (), which totally prevent double occupancies. The four dots are denoted by the index , and neighbouring dots are tunnel coupled. Without double occupancies, the charge state of the three-electron system is completely characterized by the position of the hole (). The origin of Nagaoka's ferromagnetism is found in the matrix elements and , describing the coupling between next-nearest neighbour sites on the quadruple dot plaquette. For aligned spins (ferromagnetic phase), the two paths and are indistinguishable and their contributions to interfere constructively. In contrast, the two paths are distinguishable for non-aligned spins, and therefore their contributions do not interfere (see Fig. 6a-b). Due to this effect, is higher in the ferromagnetic case, which means the hole can lower its kinetic energy by delocalizing. This makes the ferromagnetic state the ground state for large *U*. For negligible *U*, double occupancy of any site is allowed. Now the ground state has two electrons in the lowest energy (delocalized) orbital (forming a singlet) and one electron in the next orbital. The ferromagnetic (spin-3/2) state can have only one electron in each orbital, giving a larger overall energy. This destroys the ferromagnetic phase for (see Figure 6d). The gap between the two spin configurations for meV is shown in Fig. 6e. We see that for realistic values of *t* in a quantum dot array, this gap should be observable, as well as the quantum phase transition between the ferromagnetic and unpolarized states.

Another recent development could have interesting applications for quantum simulations: in a triple-dot system, long-distance charge coupling (due to cotunneling) has been recently demonstrated [26]. This could lead to new schemes of spin-spin coupling between non-nearest neighbours and could allow simulating frustrated spin systems combining exchange coupling between nearest neighbours with non-local coupling as the frustrating link.

The flexibility of the design of quantum dots constitutes one of their major advantages: indeed, complex geometries can be built, allowing the study of certain Hamiltonians in very different situations, bringing new insight into interesting quantum effects. In that respect, the study of the Kondo effect in quantum dots gives a very interesting illustration of the versatility of quantum dots systems.

The Kondo effect [81] was first observed in metals doped with magnetic impurities. These systems show a resistivity that, below a certain critical temperature, increased upon decreasing temperature. This effect is the consequence of the coupling between the spin of the free conduction electrons with the localized magnetic moments on the impurities. This spin coupling forces the localized and delocalized electrons to form a spin singlet, which increases the density of states at the Fermi energy. This increased density of state favors scattering and increases the resistance in metallic system.

The Kondo hamiltonian can be derived from the Anderson impurity model [82] describing electrons bound to localized states inside a metal host, which explains the origin of the spin interaction as cotunneling (second-order tunneling) of conduction-band electrons on the impurity. A quantum dot with an odd number of electrons coupled to electron reservoirs is a powerful quantum simulator for the Anderson impurity model. Indeed, conduction can take place in a Coulomb-blockade valley due to cotunneling [83], which couples the dot electron spin to the density of spins in the leads, giving rise to an increased density of states at the Fermi energy. This peak in the density of states can be accessed through differential conductance measurements [84, 85]: curves show a zero-bias peak whose height increases upon decreasing temperature (see Figure 7a). Note that the coupling between localized and conduction spins manifests itself through an increasing resistance in metals and through an increasing conductance in quantum dots.

The tunability of the parameters of the quantum dot also allowed to study spin-1 Kondo physics in an evenly-occupied quantum dot [86]. Furthermore, as we discuss next, the versatility of the design of quantum dot systems allowed simulating more complex hamiltonians, giving insight into multichannel Kondo coupling or multi-impurity Kondo effects.

For impurities coupled to two reservoirs independent from each other (electrons cannot be exchanged between them), a two-channel Kondo effect can occur if the coupling to both reservoirs has the same strength: instead of building a singlet with one of the reservoirs, the spin of the electron in the dot induces correlations between the two reservoirs. This effect is relevant for understanding heavy fermions [87, 88] and glassy metal [89] physics. To study it, Potok *et al*. [90] built a quantum dot coupled to both a finite (Fig. 7e, red colored) and an infinite reservoir (Fig. 7e, blue colored). Due to its finite size, the red reservoir has a finite charging energy, and electrons cannot be exchanged between red and blue reservoirs. Varying the voltage on a plunger gate capacitively coupled to the finite reservoirs, the authors could tune the strength of its Kondo coupling with the quantum dot, leaving the coupling to the infinite reservoir unchanged. Measuring the conductance between the two (blue colored) leads of the infinite reservoir, they observed a Kondo peak when the localized electron couples most strongly with the infinite reservoir, a Kondo dip when the localized electron couples most strongly with the finite reservoir, both resulting from single-channel Kondo effects. Between these two situations, the scaling of the conductance with bias and temperature (see Fig. 7f-g) showed a two-channel Kondo effect, the signature of dot-mediated spin correlations in the two reservoirs.

Future experiments could study Kondo physics with three dots, either in a in-line geometry [96], or in a triangular geometry [97-100]. The triangular geometry is especially interesting as it adds Aharonov-Bohm physics, and allows studying its interplay with the Kondo and Heisenberg coupling. Experimental developments are very encouraging. Indeed, a triangular triple quantum dot with tunnel coupling between every pair of dots has been realized recently [73]. In this paper, the authors showed that the capacitive coupling between different dot pairs could simulate a Ising hamiltonian acting on the charge degrees of freedom. Taking advantage of the triangular geometry and the associated frustration of the Ising couplings, they created a sixfold degenerate ground state, similar to the ground state of a frustrated spin system [7]). It is interesting to note that while spin-spin coupling in quantum dots follows the Heisenberg hamiltonian (see equation (4), charge-charge interaction allow simulating the Ising model. Other proposals exit for capacitively coupled quadruple dots, similar to those used for double singlet-triplet qubits [48], to be used to study in details the competition between Kondo, Heisenberg, and Ising coupling [101].

As we have seen, quantum dot systems offer a very interesting platform for quantum simulations. Not only can they be operated as qubits, but the versatility of their design offers the possibility to perform analog quantum simulations for Hubbard, Ising, Heisenberg or Kondo hamiltonians in different situations. Moreover, the efficient control of the quantum dot properties through gate voltages offer a very practical way to tune the parameters of the Hamiltonian and explore various physical regimes.