## Abstract

Simulations using highly tunable quantum systems may enable investigations of condensed matter systems beyond the capabilities of classical computers. Quantum dots and donors in semiconductor technology define a natural approach to implement quantum simulation. Several material platforms have been used to study interacting charge states, while gallium arsenide has also been used to investigate spin evolution. However, decoherence remains a key challenge in simulating coherent quantum dynamics. Here, we introduce quantum simulation using hole spins in germanium quantum dots. We demonstrate extensive and coherent control enabling the tuning of multi-spin states in isolated, paired, and fully coupled quantum dots. We then focus on the simulation of resonating valence bonds and measure the evolution between singlet product states which remains coherent over many periods. Finally, we realize four-spin states with *s*-wave and *d*-wave symmetry. These results provide means to perform non-trivial and coherent simulations of correlated electron systems.

## Introduction

Quantum computers have the potential of simulating physics beyond the capacity of classical computers^{1,2,3,4}. Gate-defined quantum dots are extensively studied for quantum computation^{5,6}, but are also a natural platform for implementing quantum simulations^{7,8,9,10,11}. The control over the electrical charge degree of freedom has facilitated the exploration of novel configurations such as effective attractive electron–electron interactions^{12}, collective Coulomb blockade^{13} and topological states^{14}. Coherent systems may be simulated when using the spin states of electrons in quantum dots, though experiments thus far have relied on gallium arsenide heterostructures^{15,16,17}, where the hyperfine interaction limits the spin coherence and therefore the complexity of simulations that can be performed. This bottleneck can be tackled by using group IV materials with nuclear spin-free isotopes. A natural candidate would be silicon, but this material comes with additional challenges due to the presence of valley states and a large effective electron mass^{18}.

Hole quantum dots in planar Ge/SiGe heterostructures exhibit many favorable properties found in different quantum dot platforms^{19}. Natural germanium has a high abundance of nuclear spin-free isotopes and can be isotopically purified^{20}. Holes in germanium benefit from a low effective mass^{21,22}, absence of valley degeneracies, ohmic contacts to metals^{23}, and strong spin-orbit coupling for all-electrical control^{24,25}. Recent advances in heterostructure growth have resulted in stable, low-noise germanium devices^{26}. This has sparked rapid progress, with demonstrations of hole quantum dots^{23}, single hole qubits^{25}, singlet-triplet (ST) qubits^{27}, two-qubit logic^{28} and a four qubit quantum processor^{29}.

Here, we explore the prospects of hole quantum dots in Ge/SiGe for quantum simulation. We focus on the simulation of resonating valence bond (RVB) states, which are of fundamental relevance in chemistry^{30} and solid state physics^{31,32,33,34} and have been used in other platforms as a feasibility test for quantum simulation^{35,36,37,38}. In our simulation, we probe RVB states in a square 2 × 2 configuration. First, we realize ST qubits for all nearest-neighbor configurations. We then study the coherent evolution of four-spin states and demonstrate exchange control spanning an order of magnitude. Furthermore, we tune the system to probe valence bond resonances whose observed characteristics comply with predictions derived from the Heisenberg model. We finally demonstrate the preparation of *s*-wave and *d*-wave RVB states from spin-singlet states via adiabatic initialization and tailored pulse sequences.

## Results

### RVB simulation in a quantum dot array with a square geometry

The experiments are based on a quantum dot array defined in a high-quality Ge/SiGe quantum well, as shown in Fig. 1a^{29,39}. The array comprises four quantum dots and we obtain good control over the system, enabling to confine zero, one, or two holes in each quantum dot as required for the quantum simulation. The dynamics of resonating valence bonds is governed by Heisenberg interactions. The spin states in germanium quantum dots, however, also experience Zeeman, spin–orbit and hyperfine interactions (see Supplementary Note 6). We therefore operate in small magnetic fields and acquire a detailed understanding of the system dynamics to apply tailored pulses. In the regime where Heisenberg interactions are dominating, the total spin is conserved. We can therefore study the subspaces of different total spin separately. The relevant subspace for the RVB physics is the zero total spin space spanned by the basis formed by the four-spin states \(\vert {S}_{x}\rangle =\vert {S}_{12}{S}_{34}\rangle\) and \(\frac{1}{\sqrt{3}}(\vert {T}_{12}^{+}{T}_{34}^{-}\rangle +\vert {T}_{12}^{-}{T}_{34}^{+}\rangle -\vert {T}_{12}^{0}{T}_{34}^{0}\rangle )\), where \(\vert {S}_{ij}\rangle =\frac{1}{\sqrt{2}}(\vert {\uparrow }_{i}{\downarrow }_{j}\rangle -\vert {\downarrow }_{i}{\uparrow }_{j}\rangle )\) and \(\vert {T}_{ij}^{0}\rangle =\frac{1}{\sqrt{2}}(\vert {\uparrow }_{i}{\downarrow }_{{{{\rm{j}}}}}\rangle +\vert {\downarrow }_{i}{\uparrow }_{j}\rangle )\), \(\vert {T}_{ij}^{+}\rangle =\vert {\uparrow }_{i}{\uparrow }_{j}\rangle\), \(\vert {T}_{ij}^{-}\rangle =\vert {\downarrow }_{i}{\downarrow }_{j}\rangle\) denote the singlet and triplet states formed by the spins in the quantum dots *i* and *j*. In this basis, the Heisenberg Hamiltonian *H*_{J} reads:

where *J*_{x} = *J*_{12} + *J*_{34} and *J*_{y} = *J*_{14} + *J*_{23}. Figure 1b, c shows the eigen energies and eigenstates of *H*_{S} for different regimes of exchange interaction. When the exchange interaction is turned on in only one direction, *J*_{x} ≫ *J*_{y} or *J*_{x} ≪ *J*_{y}, the system is equivalent to two uncoupled double quantum dots. The ground state is then a product of singlet states \(\left\vert {S}_{x}\right\rangle\) or \(\left\vert {S}_{y}\right\rangle =\left\vert {S}_{14}{S}_{23}\right\rangle\). However, when all exchanges are on and in particular when they are equal, *J*_{x} = *J*_{y}, the eigenstates are coherent superpositions of \(\left\vert {S}_{x}\right\rangle\) and \(\left\vert {S}_{y}\right\rangle\), which simulate the RVB state. In this regime, the ground state is the *s*-wave superposition state \(\left\vert s\right\rangle =\frac{1}{\sqrt{3}}(\left\vert {S}_{x}\right\rangle -\left\vert {S}_{y}\right\rangle )\) and the excited state is the *d*-wave superposition state \(\left\vert d\right\rangle =\left\vert {S}_{x}\right\rangle +\left\vert {S}_{y}\right\rangle\).

Figure 1b shows that RVB states can be generated from uncoupled spin singlets by adiabatically equalizing the exchange couplings. Alternatively, if the exchange couplings are pulsed diabatically to equal values, valence bond resonances between \(\left\vert {S}_{x}\right\rangle\) and \(\left\vert {S}_{y}\right\rangle\) states occur.

### Singlet-Triplet oscillations in the four double quantum dots

Probing the RVB physics relies on measuring the singlet probabilities in the (1,1) charge state^{17,36}. We thus investigate ST oscillations within all nearest-neighbor pairs.

To generate ST oscillations, we operate in a virtual gate landscape and apply pulses on the virtual plunger gates v*P*_{i} of each quantum dot pair according to the pulse sequence depicted in Fig. 2a ^{27,40,41,42,43}. The double quantum dot system is initialized in a singlet (0,2) state. Then, the detuning between the quantum dots is varied by changing the virtual plunger gate voltages. The system is diabatically brought to a manipulation point in the (1,1) sector creating a coherent superposition of \(\left\vert S\right\rangle\), \(\left\vert {T}^{-}\right\rangle\) and \(\left\vert {T}^{0}\right\rangle\)^{27,40,41,42,43}. After a dwell time *t*_{D}, the system is diabatically pulsed back to the (0,2) sector where the ST probabilities are determined via single-shot readout using (latched) Pauli-spin-blockade^{44,45,46}.

Results of such experiments performed at *B* = 3 mT with Q_{3}Q_{4} pair are presented in Fig. 2c. Clear oscillations between the \(\left\vert S\right\rangle\) and \(\left\vert {T}^{-}\right\rangle\) state are observed over a large range of gate voltage. Importantly, using this method we find the *S*-*T*^{−} anticrossing, which is the position where the frequency has a minimum. The observation of such oscillations, predominating over oscillations between \(\left\vert S\right\rangle\) and \(\left\vert {T}^{0}\right\rangle\) states, agrees with recent investigations suggesting that *S*-*T*^{−} oscillations dominate in germanium ST qubits placed in an in-plane *B* field^{43}.

Figure 2c also suggests that a (1,1)-singlet can be initialized from a (0,2)-singlet, by changing the energy detuning between the quantum dots while avoiding to pass the *S*-*T*^{−} anticrossing. We achieve this by shifting the anticrossing towards the center of the (1,1) charge sector by decreasing the magnetic field to *B* = 1 mT and increasing the tunnel couplings (Supplementary Fig. 1). Figure 2d demonstrates clear *S*-*T*^{−} oscillations observed in this regime for all nearest-neighbor configurations (see also Supplementary Fig. 2). Importantly, these oscillations also enable to determine the singlet/triplet states on two parallel quantum dot pairs by using sequential readout^{47}.

### Tuning of individual exchanges using coherent oscillations

The overlap of the *H*_{S} eigenstates with \(\left\vert {S}_{x}\right\rangle\) and \(\left\vert {S}_{y}\right\rangle\) depends on *J*_{x} and *J*_{y} (see Supplementary Note 4). A quantitative comparison between experiments and theoretical expectations thus requires fine control over the exchange couplings.

In this purpose, we focus on the evolution of coherent four-spin ST oscillations. These oscillations are induced using the experimental sequence depicted in Fig. 3a (see also Supplementary Fig. 3). We turn off two parallel exchange couplings and initialize a \(\left\vert {S}_{x}\right\rangle\) or a \(\left\vert {S}_{y}\right\rangle\) state in parallel double quantum dots. We then rotate one of the singlet pairs to a triplet \(\left\vert {T}^{-}\right\rangle\) state through coherent time evolution after pulsing to the *S*-*T*^{−} anticrossing, creating a four-spin singlet-triplet product state (e.g., \(\left\vert {T}_{34}^{-}{S}_{12}\right\rangle\) or \(\left\vert {T}_{23}^{-}{S}_{14}\right\rangle\)). All barrier gate voltages are then diabatically pulsed to turn on all the exchange couplings leading to coherent evolution of the four-spin system. After a dwell time *t*_{D}, two pairs are isolated (not necessarily the initial ones) and their spin-states are readout sequentially, which allows to deduce spin-correlations of opposite pairs, as was realized in linear arrays in GaAs^{17}.

The observation of resonating valence bond requires equal couplings between all four quantum dots. In navigating to this point, we carefully develop a virtual landscape, keep control over all the individual exchange interactions. First, we separately equalize the horizontal (*J*_{12} = *J*_{34}) and vertical (*J*_{14} = *J*_{23}) exchange couplings. Then, we tune the vertical and horizontal exchanges to the same coupling strength. The Chevron patterns displayed in Fig. 3c, d are consistent with a Heisenberg Hamiltonian (see Supplementary Figs. 4–6) and the minima in the oscillation frequency mark the location of equal exchange couplings for horizontal (*J*_{12} ≃ *J*_{34} ≃ *J*_{x}/2 for Fig. 3c) or vertical pairs (*J*_{14} ≃ *J*_{23} ≃ *J*_{y}/2 for Fig. 3d). Through an iterative process, we can find ranges of virtual gate voltages where *J*_{12} ≃ *J*_{34} and *J*_{23} ≃ *J*_{14}.

We can now control the spin pairs simultaneously, while maintaining the exchange couplings in both the horizontal and vertical directions equal (see Supplementary Note 5), with a priori *J*_{x} ≠ *J*_{y}. Through the readout of both pairs, we can obtain the frequency of four-spin ST oscillations observed in this regime (Fig. 3e), which is given by *f*_{ST} = *J*_{y}/2*h* or *J*_{x}/2*h* depending on the initial state, and with that determine the exchange interaction. As highlighted in Fig. 3f, the virtual control enables to tune *J*_{x} from 15 MHz to 109 MHz with *J*_{y} remaining between 46 and 56 MHz. Clearly, the exchange interaction can be controlled and measured over a significant range and tuned to a regime where all couplings are equal (we obtain a precision of ≈ 3 MHz, as discussed in Supplementary Note 5, mostly determined by drifts between experiments).

### Valence bond resonances

Valence band resonances can occur when all *J*_{ij} are equal. To experimentally assess this, we prepare \(\left\vert {S}_{x}\right\rangle\) or \(\left\vert {S}_{y}\right\rangle\), which are superposition states of *H*_{S}. We then pulse the exchanges such that *J*_{x} ≈ *J*_{y}. Figure 4a shows the result of the time evolution in this regime of equal exchange couplings. Since we start from a superposition state of *H*_{S}, the time evolution leads to coherent oscillations between \(\left\vert {S}_{x}\right\rangle\) and \(\left\vert {S}_{y}\right\rangle\), which results in periodic swaps between the singlet states as depicted in Fig. 4b. In addition, we readout both in the horizontal and vertical configuration, and observe an anti-correlated signal, consistent with signatures of valence bond resonances^{32,36}. The observation of more than ten oscillations shows the relatively high level of coherence achieved during these experiments further confirmed by the characteristic dephasing time *T*_{φ} ≈ 150 ns.

Figure 4c, d shows a more detailed measurement, which we can fit using \(\frac{{{{{\mathcal{V}}}}}_{x,y}}{2}\cos (2\pi {f}_{SS}{t}_{{{{\rm{D}}}}}+\phi )\exp (-{({t}_{{{{\rm{D}}}}}/{T}_{\varphi })}^{2})+{A}_{0}\) to extract the evolution of the frequencies *f*_{SS} and of the visibilities \({{{{\mathcal{V}}}}}_{x,y}\), plotted on Fig. 4e and Fig. 4f. We find a quantitative agreement between the measured frequencies and the theoretical expectation \({f}_{SS}=\sqrt{{J}_{x}^{2}+{J}_{y}^{2}-{J}_{x}{J}_{y}}/h\) despite deviations for the lowest values of \(\delta {V}_{x}^{{\prime} }\) that could result from the uncertainties in the exchange couplings. We also find a qualitative agreement for the visibilities though the measured \({{{{\mathcal{V}}}}}_{x,y}\) remain lower, in particular when the exchange is larger. Fermi-Hubbard simulations and further analysis (see Supplementary Notes 7 and 8) reveal that part of the visibility loss can be attributed to leakage and to the insufficient diabaticity of the voltages pulses. We speculate that the rest of the visibility loss is mainly due to the decoherence induced by the voltage pulses at the manipulation stage, or by pulse distortion arising from the non-ideal electrical response of the wiring. The underlying mechanism affects similarly the results of the measurements in the both readout directions over most of the voltage range spanned (see Supplementary Note 8). Consequently, a more quantitative agreement is reached when comparing the ratio \({{{{\mathcal{V}}}}}_{y}/({{{{\mathcal{V}}}}}_{x}+{{{{\mathcal{V}}}}}_{y})\) (Fig. 4g) of the visibilities measured over the visibilities predicted, similarly as is done in ref. ^{36}. Overall, the good agreement observed confirms that the dynamics is governed by *H*_{S}.

### Preparation of resonating valence bond eigenstates

Having observed valence bond resonances, we now focus on the preparation of eigenstates of *H*_{S} which are the \(\left\vert s\right\rangle\) and \(\left\vert d\right\rangle\) RVB states. \(\left\vert s\right\rangle\) is the ground state of *H*_{S} when *J*_{x} = *J*_{y}, whereas \(\left\vert {S}_{x}\right\rangle\) and \(\left\vert {S}_{y}\right\rangle\) are the ground states when *J*_{x} ≫ *J*_{y} and *J*_{x} ≪ *J*_{y}. Experimentally we therefore prepare \(\left\vert s\right\rangle\) from \(\left\vert {S}_{x}\right\rangle\) or \(\left\vert {S}_{y}\right\rangle\) by adiabatically tuning the exchange interactions to equal values. Figure 5a shows experiments where we control the ramp time *t*_{ramp} to tune to this regime and we observe a progressive vanishing of phase oscillations. For large *t*_{ramp} ≳ 140 ns, the oscillations nearly disappear and the measured probability saturates to \({P}_{{S}_{12}{S}_{34}}\simeq 0.78\). Performing similar experiments starting from a \(\left\vert {S}_{y}\right\rangle\) state or measuring \({P}_{{S}_{23}{S}_{14}}\) leads to identical features with singlet-singlet probabilities saturating between 0.66 and 0.72 (see Supplementary Fig. 22). These values are close to the probabilities \({| \langle {S}_{x,y}| s \rangle |}^{2}=3/4\) expected when the *s*-wave state is prepared.

We can now also prepare the ground state *H*_{S} for arbitrary exchange values, by carefully tuning the ramp time (*t*_{ramp} = 160 ns in our experiments). Figure 5b shows the evolution of \({P}_{{S}_{12}{S}_{34}}\) for different \(\delta {V}_{x}^{{\prime} }\). Since we prepare the ground state, coherent phase evolution results in a \({P}_{{S}_{12}{S}_{34}}\) that is virtually constant for any \(\delta {V}_{x}^{{\prime} }\) and only faint oscillations are observed. \({P}_{{S}_{12}{S}_{34}}\), however, is strongly dependent on \(\delta {V}_{x}^{{\prime} }\), as increasing *J*_{x} changes the ground state to \(\left\vert {S}_{x}\right\rangle\).

The measured \({P}_{{S}_{12}{S}_{34}}\) values can be compared with predictions using *J*_{x,y} values extracted from four-spin singlet-triplet oscillations (see Supplementary Note 4). Figure 5c shows that a good agreement exists between the theory and the experiments. The raw experimental probabilities \({P}_{{S}_{12}{S}_{34}}\) remains smaller than the theoretical predictions due to systematic errors during the experiments, which are most likely state initialization and readout errors (see Supplementary Note 8). Measuring \({P}_{{S}_{23}{S}_{14}}\) leads to a similar agreement, although the imperfections have a larger impact in this experiment. Rescaling the data by constant factors, that compensate for systematic errors, allows to reach a quantitative agreement, as shown in Fig. 5c. From this we conclude that the ground state of *H*_{S} is adiabatically initialized in these experiments.

We prepare the *d*-wave state by including an additional operation where we exchange two neighboring spins^{36}. This results in a transformation of neighboring spin-spin correlations to diagonal correlations. We experimentally implement this step by adding, before the free evolution step, an exchange pulse of duration *t*_{J} during which only one exchange coupling is turned on (see Fig. 5d).

Figure 5e, f shows \({P}_{{S}_{12}{S}_{34}}\) and \({P}_{{S}_{23}{S}_{14}}\) measured as functions of *t*_{D} and *t*_{J} in experiments where the system is initialized in \(\left\vert {S}_{x}\right\rangle\) and the exchange *J*_{23} is pulsed. As a function of the exchange pulse duration, we observe a periodic vanishing of RVB oscillations (linecuts provided in Fig. 5g, h, imperfections in exchange control cause residual oscillations). Due to the exchange pulse, a periodic swapping of neighboring spins occurs, and thus a periodic evolution between neighboring spin-spin correlations and diagonal correlations. Thus the regime where the *d*-wave eigenstate is prepared is marked by the vanishing of RVB states. The mean of the probabilities, \({P}_{{S}_{23}{S}_{14}}\simeq 0.21\) and \({P}_{{S}_{12}{S}_{34}}\simeq 0.13\), measured for *t*_{J} = 25ns are in the direction of theoretical expectations \({| \langle {S}_{x,y}| d \rangle }|^{2}=1/4\).

## Discussion

In this work we demonstrated a coherent quantum simulation using germanium quantum dots. Clear evolution of resonating valence bond states appeared after tuning to a regime where all nearest neigbours have equal exchange coupling. We furthermore established the preparation of the *s*-wave and *d*-wave eigenstates. In addition, we have shown that we can control the exchange interaction over a significant range in a multi-spin setting.

The low-disorder and quantum coherence make germanium a compelling candidate for more advanced quantum simulations. Improving the initialization and readout fidelities will enable to observe a stronger correspondence between ideal predictions and experimental results. Additionally, advanced voltage pulsing may facilitate to reduce errors occurring when controlling the spin states. Furthermore, a significant improvement in the quantum coherence may be obtained by exploring sweet spots^{48} and by using purified germanium.

Controlling multi-spin states is also highly relevant in the context of quantum computation. The realization of exchange-coupled singlet-triplet qubits enables to implement fast two-qubit gates^{49,50,51,52}. Leakage may then be reduced by exploiting the large out-of-plane *g*-factor for holes in germanium ^{27,43}. Also, operation with four-spin manifolds provides means for decoherence-free subspaces^{53}.

Extensions of this work leveraging the full tunability of germanium quantum dots could provide new insights for extensive studies of strongly-correlated magnetic phases and associated quantum phase transitions. In particular, the implementation of similar simulations in triangular lattices offer new possibilities to investigate the emergence of non-trivial phases arising from frustration^{33,34}. Likewise, the preparation of RVB states and the investigation of their dynamics in larger devices may help to probe their properties experimentally and explore how they relate to superconductivity in doped cuprates^{31}.

## Methods

### Materials and device fabrication

The device is fabricated on a strained Ge/SiGe heterostructure grown by chemical vapor deposition. Starting from a natural Si wafer, a 1.6 μm thick relaxed Ge layer is grown, followed by a 1 μm reverse graded Si_{1−x}Ge_{x} (*x* going from 1 to 0.8) layer, a 500 nm relaxed Si_{0.2}Ge_{0.8} layer, a 16 nm Ge quantum well under compressive stress, a 55 nm Si_{0.2}Ge_{0.8} spacer layer and a < 1 nm thick Si cap. The quantum well is contacted by aluminum ohmic contacts after a buffered oxide etch of the oxidized Si cap. The ohmics are isolated from the gates by a 10 nm ALD grown alumina layer. Two sets of Ti/Pd gates, separated by 7 nm of alumina, are deposited on top of the heterostructure to define the quantum dots. The potential of the quantum dots is tuned using the plunger gates (blue in Fig. 1a) while barrier gates are used to tune the tunnel couplings between the quantum dots (green).

### Experimental set-up

Experiments are performed in the 2 × 2 array of quantum dots showed in Fig. 1a and the changes of the charge states in the array are measured using two single hole transistors (yellow in Fig. 1a) via rf-reflectometry. Further information regarding the experimental set-up used are provided in ref. ^{29}.

### Measurement techniques

Virtual barrier and plunger gate voltages, defined as linear combinations of real gate voltages, are used to tune independently the potentials/couplings and compensate effects of cross-capacitances (see Supplementary Note 1). For four-spin coherent oscillations, the spin–spin probabilities (or equivalently the spin-spin correlations) are determined by reading out sequentially the states of two parallel quantum dot pairs, either first Q_{3}Q_{4} and then Q_{1}Q_{2} or first Q_{2}Q_{3} and then Q_{1}Q_{4}. While reading one pair, the second is stored deep in the (1,1) charge sector to prevent cross-talk between the measurements^{17,47}. The state of each pairs is determined for each single shot-measurements by comparing the sensor signal to a predetermined threshold.

## Data availability

Data underlying this study are available on a Zenodo repository at https://zenodo.org/record/7998145.

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## Acknowledgements

We thank T.-K. Hsiao, M. Rimbach-Russ, L.M.K. Vandersypen for their valuable advices and feedback. We also thank the other members of the Veldhorst and Vandersypen groups for stimulating discussions. We acknowledge O. Benningshof and R. Schouten for their technical support, and S.G.J. Philips and S.L. de Snoo for their help on software development. We acknowledge support through an ERC Starting Grant and through an NWO projectruimte. Research was sponsored by the Army Research Office (ARO) and was accomplished under Grant No. W911NF-17-1-0274. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the Army Research Office (ARO), or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation herein. This publication is part of the ‘Quantum Inspire—the Dutch Quantum Computer in the Cloud’ project (with project number [NWA.1292.19.194]) of the NWA research program ‘Research on Routes by Consortia (ORC)’, which is funded by the Netherlands Organization for Scientific Research (NWO).

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C.-A.W., C.D. and H.T. performed the experiments. C.-A.W. and C.D. analyzed the data with inputs from all authors. C.-A.W. performed the numerical simulations. C.D. and M.V. wrote the paper with the inputs of all coauthors. W.I.L.L. fabricated the device with inputs from NWH. A.S. and G.S. provided the heterostructure. M.V. supervised the project.

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Wang, CA., Déprez, C., Tidjani, H. *et al.* Probing resonating valence bonds on a programmable germanium quantum simulator.
*npj Quantum Inf* **9**, 58 (2023). https://doi.org/10.1038/s41534-023-00727-3

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DOI: https://doi.org/10.1038/s41534-023-00727-3