---
abstract: 'Quantum simulators are engineered devices controllably designed to emulate complex and classically intractable quantum systems. A key challenge lies in certifying whether the simulator is truly mimicking the Hamiltonian of interest. However, neither classical simulations nor quantum tomography are practical to address this task because of their exponential scaling with system size. Therefore, developing novel certification techniques, suitable for large systems, is highly desirable. Here, in the context of fermionic spin-based simulators, we propose a global many-body spin to charge conversion scheme, which crucially does not require local addressability. A limited number of charge configuration measurements performed at different detuning potentials along a spin chain allow to discriminate the low-energy eigenstates of the simulator. This method, robust to charge decoherence, opens the way to certify large spin array simulators as the number of measurements is independent of system size and only scales linearly with the number of eigenstates to be certified.'
author:
- Abolfazl Bayat
- Benoit Voisin
- Gilles Buchs
- Joe Salfi
- Sven Rogge
- Sougato Bose
title: 'Certification of spin-based quantum simulators '
---

*Introduction.–* Quantum simulators [@georgescu2014quantum; @cirac2012goals; @buluta2009quantum] are devices designed to emulate the behavior of quantum systems whose complexity generically increases exponentially with size. Their importance is multifold as they: (i) can provide new insights into complex quantum phenomena, e.g. high temperature superconductivity [@anderson1987resonating; @lee2006doping], non-Abelian gauge theories [@banerjee2013atomic], scattering effects [@zhang2018experimental], quantum criticality [@PhysRevA.94.063626], and long-term many-body dynamics [@daley2012measuring; @trotzky2012probing; @lv2018quantum]; and (ii) realize models that do not exist naturally, e.g. Kitaev Hamiltonian for Toric code [@kitaev2003fault]. One of the main challenges in quantum technology is to certify that an engineered quantum simulator, non-tractable classically, truly emulates the system of interest [@wiebe2014hamiltonian; @gao2017quantum; @hangleiter2017direct; @aolita2015reliable]. A necessary first step consists in matching the simulator low energy eigenstates with their expected counterparts in the emulated model. This task is highly challenging as it usually requires full quantum state tomography, because eigenstates may differ only by their global entanglement structure. However, this requires local addressability along with a number of measurements which scales exponentially with system size [@PhysRevLett.105.150401].

Recently, quantum simulators have been implemented in various setups, including cold atoms [@bernien2017probing], ion traps [@zhang2017observation; @dutta2012nonequilibrium], superconducting devices [@barends2014superconducting; @o2016scalable; @roushan2017spectroscopic; @barends2015digital], semiconductor quantum dots [@watson2018programmable; @hensgens2017quantum; @li2018crossbar; @zajac2016scalable; @nakajima2017robust] and dopant arrays [@salfi2016quantum]. Semiconductors offer a scalable platform with the natural presence of Fermi statistics (as opposed to simulating fermions with bosonic qubits via non-local interactions) and of Coulomb, electron-phonon and spin-orbit interactions. However, state-of-the-art spin readout techniques for single [@elzerman2004single; @pla2012single] and adjacent pairs [@petta2005coherent; @shulman2012demonstration; @broome2017high; @gray2016unravelling; @banchi2016entanglement; @gray2018machine] of electrons requires challenging single site accessibility. Therefore it is crucial to develop new certification schemes applicable to large spin systems.

![ **Schematic of the system.** (a) A chain of interacting electrons confined in the sites of a regular lattice. (b) Potential gradient (tilt) applied across the chain to change the charge configuration. (c) Low energy spectrum and corresponding charge configuration in tilted system.[]{data-label="Fig_Schematic"}](Fig_Schematic){width="7cm" height="4cm"}

Here, we propose a global spin to charge conversion readout scheme to discriminate between the low-energy entangled spin eigenstates of a spin chain. The basic principle is to measure the charge configuration of the simulator under different potential gradients (called tiltings) applied across the chain. Importantly the number of tilts is independent of the system size and only scales linearly with the number of eigenstates to discriminate. This readout scheme can be used to both certify (when the solution is known) and measure (when an unknown process is being simulated) the system evolution in the low energy regime. Our scheme can greatly facilitate the realization of a solid state spin-based quantum simulator as: (i) charge detections are easier to perform than direct spin measurements [@watson2018programmable; @hensgens2017quantum; @li2018crossbar; @zajac2016scalable; @nakajima2017robust; @salfi2016quantum]; (ii) a single capacitive detector is able to readout charge configurations of multiple sites [@nakajima2017robust]; (iii) global potential tilts are sufficient as opposed to local addressability; and, most importantly, (iv) the distinction of eigenstates sharing the same symmetries and the total spin (differing only in their entanglement structure) is possible without quantum tomography.

*Model.–* The Heisenberg spin chain is a key model in condensed matter physics [@sachdev2011quantum; @amico2008entanglement], spintronics [@vzutic2004spintronics] and quantum technologies [@farooq2015adiabatic; @yang2010spin]. To simulate this model we consider $N$ interacting electrons hopping among $N$ sites (i.e. half filling) in a regular 1D lattice. The Hamiltonian is characterized by the Fermi-Hubbard model $$\begin{aligned}
 \label{Hubbard_Hamiltonian}
H &=& t \sum_{k=1}^{N-1}  \sum_{\sigma=\uparrow,\downarrow}  \left( c_{k,\sigma}^\dagger c_{k+1,\sigma}+ c_{k+1,\sigma}^\dagger c_{k,\sigma} \right) + \sum_{k=1}^{N} \tilde{\epsilon}_k n_k \cr 
&+& V \sum_{k=1}^{N-1} n_k n_{k+1} + \frac{U}{2} \sum_{k=1}^{N} n_k (n_k-1),\end{aligned}$$ where $c_{k,\sigma}$ ($c_{k,\sigma}^\dagger$) is the annihilation (creation) fermionic operator for an electron at site $k$ with spin $\sigma$, number operator $n_k=\sum_{\sigma=\uparrow,\downarrow}c_{k,\sigma}^\dagger c_{k,\sigma}$ counts the number of electrons at site $k$, $t$ is the tunnel coupling between neighboring sites, $\tilde{\epsilon}_k$ is the local potential at site $k$, $V$ is the Coulomb interaction between adjacent sites and $U$ is the on-site energy. In the case of a homogeneous 1D array, i.e. $\tilde{\epsilon}_k{=}0$, the Hamiltonian (\[Hubbard\_Hamiltonian\]) is solvable [@Elliott1968]. Throughout this letter we consider a chain made of an even number of sites $N$, with on-site energy $U/t{=}40$, Coulomb interaction $V/t{=}10$ and local potential of the form $\tilde{\epsilon}_k=(k-1)\epsilon$ where $\epsilon$ is the potential difference between two adjacent sites. A schematic picture of the system is shown in Fig. \[Fig\_Schematic\](a). In a homogeneous lattice ($\tilde{\epsilon}_k{=}0$), whenever $U{\gg} t$, the low energy eigenstates take the charge configuration $(1,1,\cdots,1)$ and the system effectively becomes a Heisenberg spin chain with exchange coupling $J{\sim} t^2/U$ (with possible corrections due to $V$) [@pica2014exchange]. These eigenstates form a low energy manifold separated by units of $U$ from the eigenstates with double charge occupancies for which the map to the Heisenberg model fails. For even $N$ the ground state $|S_1\rangle$ is always a global singlet with total spin $S_{tot}{=}0$. The first two excited states $|T_1\rangle$ and $|T_2\rangle$ are triplets with the total spin $S_{tot}{=}1$. The fourth eigenstate is again another global singlet $|S_2\rangle$. In a chain of length $N=4$ these four eigenstates form the low energy manifold.

*Charge configurations.–* Many-body spin eigenstate measurement is challenging. For example, $|S_1\rangle$ and $|S_2\rangle$ have the same total spin $S_{tot}{=}0$ and share various symmetries (e.g. SU(2) invariance) making them difficult to be distinguished locally. To achieve spin eigenstate readout, we apply a potential tilt across the chain, i.e. a finite $\epsilon$, to provide enough energy for electrons to overcome $U$, as shown in Fig. \[Fig\_Schematic\](b), the charge configuration is measured. Since the eigenstates are always orthogonal, their charge configuration, which are experimentally measurable, depend on their spin state. This is the core of our certification method.

![ **Singlet charge configurations.** Charge occupancies of a chain of length $N=4$ for the state: (a) $|S_1\rangle$; and (b) $|S_2\rangle$. (c) Energy spectrum of the first three singlet eigenstates. []{data-label="Fig_Charge_S_N4"}](Fig_Charge_S_N4){width="7.5cm" height="5.5cm"}

We now develop the evolution of the charge configurations versus the tilt for a chain of $N{=}4$. Longer chains are discussed in the Supplementary Material (SM). The charge configuration of the two singlet eigenstates $|S_1\rangle$ and $|S_2\rangle$ as a function of $\epsilon/t$ are plotted in Figs. \[Fig\_Charge\_S\_N4\](a)-(b). The charge configuration changes for both eigenstates around $\epsilon/t {\sim} 13.4$ and one electron moves from either site $4$ (in the case of $|S_1\rangle$) or site $3$ (in the case of $|S_2\rangle$) to site $1$, creating two different charge configurations for $|S_1\rangle$ and $|S_2\rangle$. At around $\epsilon/t{\sim} 30$ in the eigenstate $|S_2\rangle$ an electron moves from site $4$ to site $2$ resulting in the charge configuration $(2,2,0,0)$. Finally, at $\epsilon/t {\sim} 50$ the charge configuration of $|S_2\rangle$ evolves to $(2,1,1,0)$ while $|S_1\rangle$ rearranges to $(2,2,0,0)$. All these charge configurations are summarized in Fig. \[Fig\_Schematic\](c). To understand this charge dynamics we plot the energies of the first three singlet eigenstates in Fig. \[Fig\_Charge\_S\_N4\](c). Any charge movement in the eigenstates corresponds to an anti-crossing between two eigenstates with the same $S_{tot}$. This is evident at $\epsilon/t {\sim} 13.4$, $\epsilon/t {\sim} 30$ and $\epsilon/t {\sim} 50$ where $E_{S_1}$ and $E_{S_2}$, $E_{S_2}$ and $E_{S_3}$ and $E_{S_1}$ and $E_{S_2}$ again, anti-cross.

![ **Triplet charge configurations.** Charge occupancies of a chain of length $N=4$ for the state: (a) $|T_1\rangle$; and (b) $|T_2\rangle$. (c) Energy spectrum of the first two triplet eigenstates. []{data-label="Fig_Charge_T_N4"}](Fig_Charge_T_N4){width="7.5cm" height="6cm"}

A similar analysis can be performed for the triplet states. The charge configurations of the two triplets $|T_1\rangle$ and $|T_2\rangle$ are depicted in Figs. \[Fig\_Charge\_T\_N4\](a)-(b), respectively. The charge configuration of both eigenstates changes around $\epsilon/t {\sim} 13.4$ and one electron moves from either site $4$ (in the case of $|T_1\rangle$) or site $3$ (in the case of $|T_2\rangle$) to site $1$. In Fig. \[Fig\_Charge\_T\_N4\](c) we plot the energy eigenvalues of both $|T_1\rangle$ and $|T_2\rangle$ as functions of $\epsilon/t$ which show an anti-crossing at the charge transition point $\epsilon/t {\sim} 13.4$. For larger systems (see the SM), the final charge configurations are $(2,\cdots,2,0,\cdots,0)$ for $|S_1\rangle$ and $(2,\cdots,2,1,1,0,\cdots,0)$ for $|T_1\rangle$. This important feature will be used for certification later in the letter.

*Adiabatic tilting.–* In order to readout the many-body spin eigenstate, we tilt the system, initially prepared in one of the low energy eigenstates, adiabatically such that it remains in the local eigenvector of the Hamiltonian at any time $\tau$. The eigenstates can be discriminated by measuring the charge configuration at different potentials $\epsilon$. The tilt potential varies as $$\label{epsilon_tau}
\epsilon (\tau)=
\left\{ {\begin{array}{c}
\frac{\tau}{T_{max}}  \epsilon_{max},  \quad \text{ for: } \tau \le T_{max}  \\
\epsilon_{max}, \quad \qquad \text{for: } \tau > T_{max}   \\    
    \end{array}  } \right.$$ where $\epsilon_{max}$ is the maximum tilt potential considered here to be $\epsilon_{max}/t=70$. For any initial state $|\Psi(0)\rangle$ the system evolves to the state $|\Psi(\tau)\rangle$ according to the Schrödinger equation under the action of the time dependent Fermi-Hubbard Hamiltonian described in Eq. (\[Hubbard\_Hamiltonian\]). The choice of $T_{max}$ is important as it results in different system dynamics. Adiabaticity, which notably protects the evolution against Landau-Zener transitions while sweeping through anticrossings, is achieved for slow dynamics and large $T_{max}$. However, faster dynamics minimizes charge decoherence effects at these transitions. In Fig. \[Fig\_Adiabatic\_ST\_N4\](a) we plot the charge occupancies for the quantum state $|\Psi(\tau)\rangle$, taking $T_{max}=2\times 10^4/t$, as a function of time when the system is initially prepared in the state $|S_1\rangle$. The charge configurations are very similar to the real eigenstates displayed in Fig. \[Fig\_Charge\_S\_N4\](a), with the fidelity of the evolution $F=\left| \langle \Psi(\tau)|S_1(\tau)\rangle \right|^2$ remaining above 0.98 throughout the evolution, which demonstrates that the adiabatic condition is well satisfied. In Fig. \[Fig\_Adiabatic\_ST\_N4\](b) we depict the charge occupancies when the system is initialized in the state $|T_1\rangle$. Again the charge configurations are very similar to the ones for the real eigenstate shown in Fig. \[Fig\_Charge\_T\_N4\](a) with the fidelity above $0.97$ throughout the evolution. In Figs. \[Fig\_Adiabatic\_ST\_N4\](c) and (d) we plot the charge occupancies of the state $|\Psi(\tau)\rangle$ when the system is initially in the state $|T_2\rangle$ and $|S_2\rangle$, respectively. In these two cases, the evolution is very different from the charge configurations of the local eigenstates given in Fig. \[Fig\_Charge\_T\_N4\](b) and Fig. \[Fig\_Charge\_S\_N4\](b), respectively. Here $T_{max}$ is not large enough to keep an adiabatic evolution for these two eigenstates and their fidelity reaches levels as low as ${\sim} 0.2$. In the SM, we show that $T_{max}$ values in the order of $(10^7-10^8)/t$ would be required to ensure and adiabatic evolution of $|S_2\rangle$ and $|T_2\rangle$, due to smaller gaps between higher energy eigenstates. Nonetheless, as we will show below, only an adiabatic evolution of $|S_1\rangle$ and $|T_1\rangle$ is enough to distinguish all four eigenstates, enabling complete certification.

![ **Adiabatic evolution.** Charge occupancies in the evolution of a system of length $N=4$ when $T_{max}{=}2\times 10^4/t$ and the system is initialized in the state: (a) $|S_1\rangle$; (b) $|T_1\rangle$; (c) $|T_2\rangle$; and (d) $|S_2\rangle$. This choice of $T_{max}$ results in an adiabatic evolution only for $|S_1\rangle$ and $|T_1\rangle$. []{data-label="Fig_Adiabatic_ST_N4"}](Fig_Adiabatic_ST_N4){width="7.5cm" height="6cm"}

*State discrimination.–* First we consider the ideal case in which the potential tilting is performed adiabatically for all eigenstates. The number of required tilts depends on the number of eigenstates to be discriminated. For instance, if one wants to distinguish between $|S_1\rangle$ and $|T_1\rangle$ then only one tilt, namely $\epsilon/t\simeq 50-60$, is enough as $|S_1\rangle$ takes the configuration $(2,2,0,0)$ and $|T_1\rangle$ goes to $(2,1,1,0)$. Only two tilts are then required to fully distinguish the four lowest eigenstates. For instance, by tilting to $\epsilon/t{=}35$ we can fully distinguish $|S_2\rangle$, with configuration $(2,2,0,0)$, and $|T_2\rangle$, with configuration $(2,1,0,1)$. However, both $|S_1\rangle$ and $|T_1\rangle$ share the same configuration $(2,1,1,0)$ and cannot be distinguished. Therefore, another charge configuration measurement must be performed at a larger detuning $\epsilon/t {\sim} 50-60$ when the charge configuration for $|S_1\rangle$ changes to $(2,2,0,0)$ while $|T_1\rangle$ remains in the $(2,1,1,0)$ configuration. The key feature of our proposal lies in its scalability: only two tilts are needed to fully distinguish the four lowest eigenstates, irrespective of the system size (see SM). In fact, for distinguishing $n$ low-energy eigenstates only $n/2$ tilts are required.

Now, we consider an evolution which is only adiabatic for $|S_1\rangle$ and $|T_1\rangle$, like depicted in Fig. \[Fig\_Adiabatic\_ST\_N4\]. For $|S_2\rangle$, the outcome of the charge measurement will be time averaged over the charge occupancies due to rapid charge oscillations. Therefore, by using the same procedure, at $\epsilon/t{=}35$ the states $|S_2\rangle$ and $|T_2\rangle$ take the configurations $(1.5,1,1.5,0)$ and $(1.2,1.6,1,0.2)$, respectively, which are very distinct from each other as well as from the configuration of $|S_1\rangle$ and $|T_1\rangle$. Note that the partial charges mean that the quantum states are in a superposition of multiple charge states. This means that even when the evolution for $|S_2\rangle$ and $|T_2\rangle$ is non-adiabatic the proposed discrimination procedure still holds.

![ **Decoherence.** The time evolution in the presence of decoherence in a system of length $N=4$ when $T_{max}{=}2\times 10^4/t$. The charge occupancies are given for $\gamma/t=0.001$ when the system is initialized in the eigenstate: (a) $|S_1\rangle$; and (b) $|T_1\rangle$. (c) The distance between the charge probability distributions of $|S_1\rangle$ and $|T_1\rangle$ as a function of $\gamma$ when $\epsilon/t=70$. []{data-label="Fig_Decoherence_ST_N4"}](Fig_Decoherence_ST_N4){width="7.5cm" height="5.5cm"}

*Decoherence.–* Interaction with the environment results in non-unitary dynamics and decoherence. For itinerant particles, the most common source of decoherence is charge fluctuations [@wardrop2014exchange; @nakajima2018coherent] which destroys the superposition of different charge configurations. Therefore, if $\{L_n\}$ represent the projection operators on $n$-th charge configuration then the non-unitary dynamics can be modeled using the Lindblad master equation $$\nonumber
\frac{\partial \rho}{\partial \tau}=-i[H(\tau),\rho]+\gamma \sum_n \left( L_n \rho L_n^\dagger -\frac{1}{2} L_n^\dagger L_n \rho
-\frac{1}{2} \rho L_n^\dagger L_n  \right)
$$ where $\gamma$ represents the decoherence strength, $\rho$ is the density matrix of the system and $L_n$’s are the Lindblad operators which depend on the decoherence source. In Fig. \[Fig\_Decoherence\_ST\_N4\](a) we plot the charge occupancies for the evolution of $|S_1\rangle$ in a chain of $N=4$ when $\gamma/t{=} 10^{-3}$ and $T_{max}=2\times 10^4/t$. Decoherence leads to partial charge transitions for the quantum states to become mixtures of charge configurations. The same evolution for the triplet state $|T_1\rangle$ is depicted in Fig. \[Fig\_Decoherence\_ST\_N4\](b). The evolution for $|T_1\rangle$ is less affected than $|S_1\rangle$ as there are less charge transitions. In the SM we discuss the fidelities and entropy production resulting from this evolution.

As decoherence affects charge transitions, it is important to address its impact on on our protocol to distinguish quantum states. Each measurement outcome is associated to a charge projection operator $L_n$ with respective probability $p_n=\operatorname{Tr}{(\rho L_n)}$. Distinguishing the two eigenstates, e.g. $|S_1\rangle$ and $|T_1\rangle$, equivalent to distinguishing two probability distributions $\{p_n: p_n=\operatorname{Tr}{(\rho_{S_1} L_n)}\}$ and $\{q_n: q_n=\operatorname{Tr}{(\rho_{T_1} L_n)}\}$, where $\rho_{S_1}$ ($\rho_{T_1}$) is the solution of the above Lindblad master equation with the initial state $|S_1\rangle$ ($|T_1\rangle$). Experimentally the true probability distribution can be obtained by averaging over $M$ charge measurements at each tilt. The distance (or relative entropy) defined as $d(S_1,T_1){=}\sum_n p_n \log_2 \frac{p_n}{q_n}$ can be used to quantify the distinguishability between the two distributions. The error in discriminating the two probability distributions after $M$ samples scales as ${\sim} 2^{-Md}$ [@vedral1997statistical], for $M$ large. Therefore, by repeating the experiments at each tilt for $M{\sim} 10^2{-}10^3$ one can reconstruct the probability distributions and discriminate between the eigenstates when $d{>}1$. In Fig. \[Fig\_Decoherence\_ST\_N4\](c) we plot $d(S_1,T_1)$ versus $\gamma$ for a tilt set to $\epsilon/t{=}70$. The distance drops as $\gamma$ increases, however it remains above $10$ even for $\gamma/t{=}0.01$, and discrimination is still achievable.

*Experimental realization.–* The most relevant platforms to realize our proposal are fermionic optical lattices [@schreiber2015observation] and dopant arrays [@fuechsle2012single; @salfi2016quantum]. We specifically consider the latter. The atomic precision of scanning tunneling microscopy lithography [@fuechsle2012single] provides the required versatility to fabricate 1D or 2D phosphorus donor-bound spin arrays in silicon with charge sensors in their proximity calibrated to accurately deduce charge configurations [@mahapatra2011charge]. The dopant charging energy is $U{\sim} 47$ meV for bulk donors and both $t$ and $V$ can be engineered via the physical separation between sites. For dopants placed $10$ nm apart, $t$ is about $1$ meV [@gamble2015multivalley] and $V$ around $10$ meV as considered in this letter. From these values, an adiabatic evolution is achieved for $T_{max} {\ge} 13$ ns. Experimental charge dephasing values can be converted to $\gamma {\sim} 0.02-1$ $\mu$eV [@pashkin2003quantum; @dupont2013coherent; @van2018microwave; @van2018time]. More precisely the ratio $\gamma/t$ is relevant, which is found to be ${\sim} 10^{-5}{-}10^{-3}$ as strong tunneling interactions are considered here. As shown in Fig. \[Fig\_Decoherence\_ST\_N4\](c), this results in $d{>}20$ (and fidelities above $0.8$, see SM), and hence precise certification to be achievable in dopant systems. The hyperfine interactions, coupling electron and nuclear spins, are another possible source of errors in dopant systems as they mix the singlet and triplet subspaces. For the hyperfine coupling of $A{\sim} 0.4$ $\mu$eV this mixing rate is ${\sim} A^2/(E_{T_1}-E_{S_1})$ due to the energy difference between $|S_1\rangle$ and $|T_1\rangle$. As the minimum $E_{T_1}{-}E_{S_1} {\sim} 100$ $\mu$eV is found for $N{=}4$ the role of hyperfine interaction can be neglected. However, as the energy gap scales as $1/N^2$, we predict that hyperfine interactions will be relevant for $N{>}20$ making the nuclear spin initialization essential.

*Conclusion.–* We have proposed an efficient procedure for certifying the performance of spin-based quantum simulators via discriminating between the low energy eigenstates without using quantum tomography. This is a nontrivial task as the eigenstates cannot be distinguished locally because of: (i) being many-body entangled; and (ii) having the same symmetries and total spin. Given the fact that our scheme can be implemented without local addressability, it opens up the possibility to scale up the simulators to large sizes. The proposed mechanism can potentially be exploited to detect low energy phenomena such as quantum phase transitions, electronic thermometry and emergent Kondo screening clouds. After certification of the spin Hamiltonian in the low energy regime, the same simulator can be used to reveal classically inaccessible features such as high energy long-time dynamics and complex two-dimensional structures. Relevant platofrms to implement our certification method include dopant arrays [@fuechsle2012single; @salfi2016quantum] and fermionic optical lattices [@schreiber2015observation].

*Acknowledgment.–* The authors would like to thank Didier ST Medar for helpful discussions. We acknowledge support from the ARC Centre of Excellence for Quantum Computation and Communication Technology (CE170100012) and an ARC Discovery Project (DP180102620). JS acknowledges support from an ARC DECRA fellowship (DE160101490). AB thanks the National Key R&D Program of China, Grant No. 2018YFA0306703. SB acknowledges support from the UK Engineering and Physical Sciences Research Council (EPSRC) grant EP/R029075/1.

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Supplementary Material
======================

In the supplementary material we provide further investigations on some of the subjects which were not discussed in details in the main text of the paper.

1. Charge Configurations and State Discrimination for Large Chains
------------------------------------------------------------------

The proposed mechanism can also be applied to large chains with $N>4$. The charge configurations become more diverse as the size of the system increases. Using the same parameters as in the main text, namely $U/t=40$ and $V/t=10$, one can plot the charge occupancies of each site as a function of the tilting potential $\epsilon$. As a typical example we plot the data for $N=8$ in Fig. \[Fig\_Supp\_Charge\_ST\_N8\] for the first four eigenstates, namely $|S_{1,2}\rangle$ and $|T_{1,2}\rangle$. As the figure shows the overall picture is similar to the case of $N=4$ except that there are more charge movements. The eigenstate charge configurations as a function of $\epsilon/t$ for chains of length $N=6$ and $N=8$ are represented schematically in Figs. \[Fig\_Supp\_Schematic\](a)-(b). It can be shown that the final configuration of the eigenstate $|S_1\rangle$ is always $(2,\cdots,2,0,\cdots,0)$ and for the eigenstate $|T_1\rangle$ it is $(2,\cdots,2,1,1,0,\cdots,0)$. An important feature which arises in large chains is that the final charge configuration of $|T_2\rangle$ shows partial charge occupancies. This is due to a superposition of charges.

Remarkably, independently of the system size we can discriminate between the four eigenstates using only two potential tilts. For instance, in the case of $N=6$, with $\epsilon/t{=}35$ we can fully discriminate the eigenstate $|S_2\rangle$ and $|T_2\rangle$ from the rest but we cannot distinguish $|S_1\rangle$ from $|T_1\rangle$. Note that, at this value of the potential tilt the charge measurement outcome for $|T_2\rangle$ is not unique as that eigenstate is a superposition of different charge configurations but due to orthogonality it does not share any charge configuration with $|T_1\rangle$ (which has the same charge configuration as $|S_1\rangle$) and $|S_2\rangle$. If the charge measurement shows the configuration $(2,2,1,1,0,0)$ this means that the quantum state is either $|S_1\rangle$ or $|T_1\rangle$ and to discriminate between them one has to tilt the system further to $\epsilon/t{=}70$ for which the two eigenstates take different charge configurations. The same argument is valid for $N=8$ and $N=10$ (data not shown) in which the two potential tilts should still be performed at $\epsilon/t{=}35$ and $\epsilon/t{=}70$ for full discrimination between the four eigenstates.

![ **Charge configuration for a chain of size $N=8$.** The charge configuration of a spin simulator with $N=8$, $U/t=40$ and $V/t=10$. The charge configrations are given for: (a) the ground state $|S_1\rangle$; (b) the first singlet excited state $|S_2\rangle$; (c) the first triplet excited state $|T_1\rangle$; and (d) the second triplet eigenstate $|T_2\rangle$. []{data-label="Fig_Supp_Charge_ST_N8"}](Fig_Supp_Charge_ST_N8){width="8cm" height="7cm"}

2. Adiabatic Evolution
----------------------

![image](Fig_Supp_Schematic){width="12cm" height="6cm"}

[ | m[1cm]{} | m[1cm]{}| m[1cm]{} | m[1cm]{}| m[1cm]{} | ]{} N & 4 & 6 & 8 & 10\
$\Delta E_{S}/t$ & 0.2231 & 0.1255 & 0.1011 & 0.0710\
$\Delta E_{T}/t$ & 0.0913 & 0.0757 & 0.0684 & 0.0575\

As mentioned in the main text, the time $T_{max}$ needed to keep the evolution of $|S_2\rangle$ and $|T_2\rangle$ adiabatic must lie in the range $10^7-10^8$. An example is given for $T_{max} {\sim} 8\times 10^7/t$. In Fig. \[Fig\_Supp\_Adiabatic\_ST\_N4\](a) we plot the charge occupancies as functions of time when the initial state is $|S_2\rangle$ for $N=4$. As the figure shows the charge occupancies are almost identical to the charge configuration shown in Fig. \[Fig\_Charge\_S\_N4\](b) for the local eigenstate. Similarly, in Fig. \[Fig\_Supp\_Adiabatic\_ST\_N4\](b) we plot the charge occupancies for the initial state $|T_2\rangle$ which is also close to the charge configuration of the local eigenstate given in Fig. \[Fig\_Charge\_T\_N4\](b). To asses the adiabatic evolution we plot the fidelities for $|S_2\rangle$ and $|T_2\rangle$ in Fig. \[Fig\_Supp\_Adiabatic\_ST\_N4\](c). The fidelity for $|S_2\rangle$ is always above $0.99$ and for $|T_2\rangle$ it always remains above $0.9$. The better fidelity for $|S_2\rangle$ is due to a larger energy gap for higher singlet eigenstates. It is worth emphasizing that, as discussed in the main text, a large value of $T_{max}$ to keep the evolution of $|S_2\rangle$ and $|T_2\rangle$ adiabatic is not needed for state discrimination between the first four eigenstates.

A crucial issue for the adiabatic evolution is the estimation of $T_{max}$ needed to evolve larger systems. As we discussed above, it is important to keep the evolution for both $|S_1\rangle$ and $T_1\rangle$ adiabatic, even if the higher energy eigenstates do not follow an adiabatic evolution. The criteria for the validity of the adiabatic theorem has been the subject of research over many years [@comparat2009general; @marzlin2004inconsistency; @tong2007sufficiency]. The standard criteria implies that one has to satisfy $|\frac{\langle \dot{S}_2(\tau)|S_1(\tau)\rangle }{E_{S_2}(\tau)-E_{S_1}(\tau)}|\ll 1$ where $|\dot{S}_2(\tau)\rangle$ is the time derivative of the eigenstate $|S_2\rangle$ with respect to $\tau$. Similar criteria can be written for triplets as well. Using perturbation theory one can show that in a pessimistic estimation $\langle \dot{S}_2(\tau)|S_1(\tau)\rangle \sim T_{max}^{-1} [ E_{S_2}(\tau)-E_{S_1}(\tau)]^{-1}$. This implies that for the validity of the adiabatic evolution one has to keep $T_{max}>1/\Delta E^2$ where $\Delta E$ is the the energy gap. To see how the energy gap scales with system size we present the minimum energy gap during the adiabatic evolution for both singlets (i.e. $\Delta E_S=E_{S_2}-E_{S_1}$) and triplets (i.e. $\Delta E_T=E_{T_2}-E_{T_1}$) in TABLE \[Table\_Energy\_Gap\]. As the data show, the energy gap decreases fairly linearly as the system size increases. This means that for a chain of size $N=10$ the time $T_{max}$ is almost $10$ times larger than the one needed for $N=4$.

![ **Adiabatic evolution of $|S_2\rangle$ and $|T_2\rangle$.** The adiabatic evolution for the excited states $|S_2\rangle$ and $|T_2\rangle$ in a system with $N=4$ when the ramping time is chosen to be $T_{max}=8\times 10^7/t$. (a) The charge occupancies as functions of time when the system is initialized in the eigenstate $|S_2\rangle$. (b) The charge occupancies as functions of time when the system is initialized in the eigenstate $|T_2\rangle$. (c) The fidelity of the adiabatic evolution for both $|S_2\rangle$ and $|T_2\rangle$. []{data-label="Fig_Supp_Adiabatic_ST_N4"}](Fig_Supp_Adiabatic_ST_N4){width="8cm" height="7cm"}

3. Decoherence
--------------

In order to understand the full effect of decoherence in the Lindblad master equation, we consider an adiabatic evolution of both $|S_1\rangle$ and $|T_1\rangle$ in a system of length $N=4$ with the total evolution time $T_{max}=2\times 10^4/t$. In Fig. \[Fig\_Supp\_Fidelity\_Entropy\_Decoherence\_ST\_N4\](a) we plot the fidelity of the evolution for the state $|S_1\rangle$ as a function of time $\tau$ for different values of noise strength $\gamma$. As the figure shows, by increasing $\gamma$ the fidelity decreases. To understand this, it is important to note that such dynamics is not unitary. This means that the quantum state of the system becomes mixed during the time evolution. To see this, one can compute the von Neumann entropy of the whole system which is defined as $$\label{von_Neumann_Entropy}
S(\rho)=-\operatorname{Tr}\left( \rho\log_2 \rho \right).$$ In Fig. \[Fig\_Supp\_Fidelity\_Entropy\_Decoherence\_ST\_N4\](b) we plot the von Neumann entropy of the system when the quantum state is initially $|S_1\rangle$ as a function of time $\tau$ for different values of noise strength $\gamma$. As the figure shows the entropy increases monotonically and sharp rises happen during the charge movements when the charge state is delocalized. In Fig. \[Fig\_Supp\_Fidelity\_Entropy\_Decoherence\_ST\_N4\](c) we also plot the fidelity for the quantum state $|T_1\rangle$ keeping all the parameters the same as for the singlet $|S_1\rangle$. Finally, in Fig. \[Fig\_Supp\_Fidelity\_Entropy\_Decoherence\_ST\_N4\](d) we plot the von Neumann entropy of the evolution of the triplet state $|T_1\rangle$ as a function of time. Figs. \[Fig\_Supp\_Fidelity\_Entropy\_Decoherence\_ST\_N4\](c)-(d) show that the fidelity of the triplet is slightly higher and its von Neumann entropy is smaller in comparison with the singlet. This is due to less charge movements for triplets, or equivalently fewer energy anti-crossings between the eigenstates, which makes the triplet evolution less prone to decoherence.

![ **Fidelity and Entropy.** The evolution of a system with length $N{=}4$ in the presence of decoherence. (a) The fidelity $F$ versus time for $|S_1\rangle$. (b) The von Neumann entropy $S$ versus time for $|S_1\rangle$. (c) The fidelity $F$ versus time for $|T_1\rangle$. (d) The von Neumann entropy $S$ versus time for $|T_1\rangle$. []{data-label="Fig_Supp_Fidelity_Entropy_Decoherence_ST_N4"}](Fig_Supp_Fidelity_Entropy_Decoherence_ST_N4){width="8cm" height="7cm"}
