In this study, building on the 1D topological Su–Schrieffer–Heeger (SSH) model, we propose a model of quantum dot arrays with odd and even parity and variable on-site local potentials to examine topological edge states and a possible quantum information encoding, using these states. We first investigate the SSH model with alternating tunneling amplitudes $t1$ and $t2$. We study the model in a ring-like structure and then proceed to minimal open-end chains with even ($N=4$) and odd ($N=5$) number of dots. Furthermore, we depart from the basic SSH model by introducing local potentials $\mu i$, which offer additional control at the cost of breaking the chiral symmetry of the Hamiltonian and study the implications. Then, we propose an idealized “static” charge qubit design, based on encoding the topological invariant $\nu $ as qubit states, that exploits the topological nature of the edge states and their collective character. We introduce perturbing noise $\delta tij(t)$ into the system and demonstrate the robustness of the states for some range of the ratio $\xi =t1/t2$. Moreover, we show a possible way to detect the presence of topological order in the system using equilibrium dynamics for both even and odd chains. We utilize the quantum informatic measure of bipartite mutual information $I{b:e}(2)(t)$ as a measure of bulk-edge quantum correlations and a quantitative indicator for the manifestation of bulk-edge correspondence; here, we also propose a dynamical qubit encoding with $\nu $ for specific quantum chain parity. Finally, we offer a few remarks on potential future explorations.

## I. INTRODUCTION

Quantum computers^{1,2} are a promising upcoming technology that will revolutionize computation and, as a consequence, many other fields such as quantum chemistry,^{3,4} pharmaceutical drug design,^{5,6} finance,^{7,8} quantum machine learning, and AI.^{9,10} They are a new generation of computers, which take advantage of “spooky” laws that reside in the quantum realm, such as superposition and entanglement, in order to encode, decode, and process information. These logical processes are done in a parallel manner,^{11,12} which results in an exponential speedup in computational time and reduction of computational resources needed. Also, different processes that occur purely in the quantum world like atomic, molecular, and nuclear dynamics become intractable for large systems to simulate on a classical computer since the resources needed to simulate a $p$-level system (e.g., a molecule with $p$ orbitals) with $N$ components scale as $pN$; this is in sharp contrast to a quantum computer with $p$-level qudits, which would only need $N$ qudits (a qudit is a generalization of a qubit from a $2$- to a $p$-level bit of information; instead of a binary, we have $p$-nary logic). Thus, from a $O(t)$, we get to a $O(logp\u2061(t))$ computational time. The physical implementation of qubits and qudits, as well as the techniques to encode information through them, has been an immensely studied research topic over the recent decade.

Since quantum computers exploit the laws of quantum mechanics, there are many different kinds; the most popular technologies are those of superconducting qubits,^{13,14} ion-traps,^{15,16} optical,^{17,18} Majorana anyons^{19,20} albeit purely theoretical at the moment, spin,^{21,22} charge,^{23–25} and hybrid qubits,^{26,27} which are a combination of the last two. To implement different technologies, many models are borrowed from condensed matter physics, since one can think of a quantum dot as an artificial atom and a chain of dots as an artificial molecule.

Moreover, since semiconductor quantum dots and qubits have been becoming widespread,^{28,29} computation using quantum dot arrays (QDAs) turns into a realistic prospect. With this development, we would like to rethink and have a look into QDA models from the perspective of information encoding in the context of quantum computing.

Here, we focus on the (semiconductor) charge-based qubit technology. As with any other technology, there are some advantages as well as drawbacks over others. On the one hand, the benefits include the relatively low cost of production, potential for massive scalability, and fast gate operation speed.^{30} On the other hand, some major disadvantages are that they constitute low-quality qubits, are easily corrupted due to electronic and/or thermal noise, and are trickier to control.^{30}

One possible way to attempt to circumvent the drawbacks is to somehow make the electron qubit states to be more tolerant to noise. This is one reason why we chose to study the most simple system that manipulates electrons but that also has some properties that manifest some kind of noise tolerance in 1D, namely, the so-called Su–Schrieffer–Heeger (SSH) model.^{31,32} This model exhibits interesting properties in its spectrum, which may be exploited in order to have more stable electron configurations and, hence, possibly more stable qubit states.

In reverse, we can also apply electronic measurements to specific architectures, which are modeled by the SSH model, in order to extract some useful properties of the system. For example, as we also show later on, one can determine if the system has topologically nontrivial or trivial states present in its eigenenergy spectrum (band structure).

Our work is structured as follows: in Sec. II, we make a short review on the SSH model and some of its interesting properties, extending Ref. 32. We analyze its band structure and topological characteristics in a ring-like structure, as an introduction to the basic characteristics of the model. In Sec. III, we probe the open chain case for an odd and even parity QDA and highlight the very different characteristics exhibited by the electron wavefunction; here, the topological character becomes more apparent. In Sec. IV, we propose a naive and idealistic way for a possible use of the topological protection of the states to construct noise-tolerant charge qubits for charge-based quantum computers. In Sec. V, we solve the time-dependent problem in equilibrium and propose a way to measure and identify the existence of topological states in the system from dynamical measurements and again offer a possible topological qubit encoding state based on the topological invariant number $\nu $. Finally, in Sec. VI, we make some further remarks, conclusions, and possible future outlooks of the current work.

## II. THE SU–SCHRIEFFER–HEEGER (SSH) MODEL

For the convenience of a reader, we outline some fundamental information on the SSH model. For more details, please refer to Refs. 33 and 34. The SSH model is a small divergence from the well-known and simplistic tight-binding model. It is a single-electron model (it is worth noting that similar dynamics at low energies can occur in a half-filled chain,^{35} with very strong on-site Coulomb interactions where there is an extra electron that hops in the chain with opposite spin), with nearest-neighbor allowing tunneling but with two alternating tunneling amplitudes. It has been proved very useful to study since it is the simplest topologically nontrivial 1D model, which captures some of the odd properties of the class of topological insulators,^{36} a particular case of the wider class of topological quantum matter. More specifically, it exhibits the existence of topologically protected states, whose topological character is quantified by a nontrivial topological invariant, namely the winding number $\nu $. This corresponds to the Zak phase,^{32,37} which is a geometric phase picked up by the electron during its motion in the periodic structure; we define it later on.

To construct this model, we need to have a lattice with alternating hopping coefficients and a unit cell composed of two lattice sites. That is, we have a crystal lattice with a presence of a Peierls-like instability (if we want to replicate a similar structure in a QDA, we can adjust the tunneling amplitudes through voltage manipulation of imposers in-between the dots).^{38} This induces distortion of the lattice and, hence, the alternating hopping coefficients in the Hamiltonian of the model; the geometrical representation is shown in Fig. 1.

To begin with, we will consider the Hamiltonian that captures the hopping (tunneling) of the electron(s) along the chain (we consider a single-electron non-interacting spinless model as the original SSH model, but given the shape of the SSH Hamiltonian, there is no *a priori* limitation to the number of electrons in the chain). The Hamiltonian for an $N$-site system is

where $cn,j\u2020$ creates an electron at unit cell $n$ and orbital $j=1,2$; $cn,j$ is the corresponding annihilation operator and $t1,t2$ are the intra- and inter-dot tunneling amplitudes that correspond to the tunneling integrals in first quantization formalism.

The model also exhibits chiral symmetry; it is more explicitly seen if we first define the sublattice projection operators $P1=\u2211ncn,1\u2020cn,1,P2=\u2211ncn,2\u2020cn,2$. Now, we can define the chiral operator $\Gamma =P1\u2212P2$ and see that

We require the chiral operator $\Gamma $ to be unitary and Hermitian $(\Gamma 2=I,\Gamma \u2020=\Gamma )$, local as it acts only inside the unit cell, and robust $(\Gamma H\Gamma \u2020=\u2212H,\u2200t1,t2\u2208R)$. The chiral symmetry of the Hamiltonian implies a reflective symmetry of the eigenenergy spectrum; that is, for each eigenstate $|\psi +\u27e9$ with energy $E+$, there is a corresponding eigenstate $|\psi \u2212\u27e9=\Gamma |\psi +\u27e9$ with energy $E\u2212=\u2212E+$.

Now, to be able to study the system more easily and make some properties more explicit, we impose periodic boundary conditions (PBCs) in the chain as $n=N+1\u21921$, or more concretely $n\u2208Z+\u2217/NZ+\u2217$. Since we have the translation symmetry in our Hamiltonian now, we can use Fourier transform on our creation and annihilation operators $cn,i\u2020,cn,i$ as (we can also express the creation/annihilation operators in Dirac’s bra-ket notation as)

We can see that the above Fourier transform acts on the whole unit cell and not just on one of the two components. The dimension of our SSH Hilbert space is $dim\u2061(HSSH\u2282Hfull)=N$.

Now, we can write our Hamiltonian in the more compact form as

where $ck\u2020=(ck,1\u2020ck,2\u2020)andck=(ck,1ck,2)$.

Now that we have expressed our Hamiltonian effectively as a $2\xd72$ matrix, we can easily obtain the eigenenergies, which in turn will give us the energy bands. We obtain the energy spectrum, which is visible in Fig. 2,

Looking at the energy bands of the system, we can observe that the two insulating phases for the different hopping coefficients are identical, and there is an intercepting metallic phase in between; that is, there is no possibility to distinguish between the insulating ones. The key lies in the eigenstates of the system, which hold some hidden extra clues. In order to find the eigenstates and visualize them more intuitively, we will first use the Pauli matrix vector $\sigma =(\sigma x,\sigma y,\sigma z)$ to rewrite the Hamiltonian as

Solving the eigenvectors (wavefunctions), we get

Now, there are two ways to distinguish between the two insulating phases. One is to plot the $d(k)$ vector in the plane for the two phases, with $k$ taking values over the Brillouin zone and observe the trajectories. The other one, which is more straightforward mathematically, is to calculate a topological invariant called the winding number $\nu $. In Figs. 3(a)–3(e), we have the projection of the $d(k)$ on the 2D plane, which adds a geometrical intuition in the definition of $\nu $. Hence, it is defined as

where “BZ” is the Brillouin zone, “$\xd7$” is the cross-product, and $n(k)=d(k)/|d(k)|$.

Now, in order to distinguish the two insulating phases, we inspect the ratios of the hopping integrals of the two insulating phases $t1/t2<1$ and $t1/t2>1$. We obtain

Thus, we can see that although the dispersion plot of the two insulating phases is identical, the corresponding eigenstates are not. The topological phase is protected and in order to go from the nontrivial insulating topological phase $(\nu =1)$ to the trivial one $(\nu =0)$, the system needs to undergo a topological phase transition.^{31,36,39} That is, it goes first through a metallic phase $(\nu =?)$ or breaks somehow the chiral symmetry (e.g., if we include a non-uniform local chemical potential term) and the winding number $\nu $ “jumps” from 0 to 1 and vice versa; this is depicted in Fig. 3(f) as a function of both $t1$ and $t2$. In the energy dispersion relation, this manifests as a closing and reopening of the bandgap, shown in Fig. 2, as the system undergoes the transition.

## III. ODD AND EVEN OPEN*N*-CHAIN

The characteristics of the topological insulating phase become more explicit in the case where we do not have PBC, but open boundary conditions (OBCs) at the ends of the chain. The different wavefunctions for both the odd and even dot cases are plotted in Fig. 4. In order to make the visualization easier, we have included visualizations of the ground state functions of a harmonic oscillator. The form of these functions for finite-dimensional dots would be Gaussian,

where $C(\alpha ),D(\alpha )$ constants that depend on the physical characteristics of the system and $Ri$ is the site vector). We believe this is also a realistic approximation since we are operating in non-thermal regimes $T=0$ and the electrons are spinless and have no angular momentum, so they could resemble the $s$-orbitals of a hydrogen atom in each dot. In what follows from this point onward, we will not make an explicit distinction between the different sub-cells in our notation for the states. For example, the state $|0100\u27e9$ corresponds to having an electron on the first unit cell, on the second sub-cell; another way to write it would be $|01;00\u27e9$ to make the bipartition of the unit cells more apparent. This makes it easier to write in a more compact notation the states for a general number of quantum dots/cells.

We can see that in the case of OBC, if the system is in the topologically insulating phase, the wavefunction becomes localized at the edges and insulated in the bulk; these are also called edge states, which are a subset of the more general surface states,^{33} a proper manifestation of the bulk-boundary correspondence. (This is a kind of duality, in which we have an emergence of features on the one side of the duality that is the boundary of a material, which is dependent on but, at the same time, distinct from the properties of the other side, that is, the bulk of the material.^{40}) Again, they are characterized by a nontrivial topological invariant^{37} (in this case the winding number $\nu $) and remain stable through variation of the hopping coefficients for some range of the ratio $\xi =t1/t2$.

In addition, it is very interesting to note that if we have an even SSH chain, the behavior of both the eigenenergy spectra and the wavefunctions is very different compared to having an odd chain; this can be seen explicitly in Figs. 5–7. Also, in Fig. 7, one can see that the topology of the band structure and the wavefunctions is independent of the number of dots and manifests in a similar way for any number $N$.

Observing now the band structure for the two cases, i.e., even and odd chains, we instantly see some qualitative differences. That is, for the even chain, we have zero-energy edge states for a specific ratio of $\xi <1/4$ and for $\xi \u22731/4$ there is a level splitting, while for the odd chain we have a zero-energy state that persists throughout the whole spectrum of $\xi $.

Moreover, for the odd chain, by taking a quick look at the wavefunction probability distributions in Fig. 5, we see that the zero-energy state is the one in which the electron is completely localized in one edge dot and it is unique ($q=0$ degeneracy). It is worth noting that the insulating states, both topological and trivial, presented here for both the even and odd chains, can be generalized easily for any $N$, e.g., the topological. That is,

On the contrary, in the even chain Fig. 6, the zero-energy states have a $q=2$ degeneracy with a phase difference of $\Delta \phi =\pi $ in its superposition states and the electron’s wavefunction becomes “split” between the two edges with equal probability as

Finally, it is worth noting that we can reproduce the completely localized electron at the edge for the even chain as well if we add on-site potential terms as

where $\mu n,j$ is an on-site tunable chemical potential of the $n$th unit cell on the $j$th site.

The states obtained are shown in Figs. 6(h) and 6(i). We can see that they are identical to the trivial and nontrivial insulating states of the $N=5$ chain. It is also worth noting that, although in our plot we have the completely localized states with a small on-site potential $\mu 1,\mu 4\u224810\u22121$, as we increase the hoppings $t1,t2$ in order to achieve the most localized states and have minimum leakage in the bulk, we would need them to be larger by an order of magnitude.

Physically, these local potential terms could correspond to, e.g., impurities in the crystal or on-site voltage differentiation. Nevertheless, introducing these local terms comes at the cost of breaking the chiral symmetry of the system, since ${H,\Gamma}\u22600$, and the eigenenergy spectrum will no longer have reflection symmetry along the $tj$-axis (the same effect would be also manifested if we added more electrons in the chain and allowed them to interact via, e.g., electrostatic Coulomb interaction $U$ in the Hamiltonian); the band structure becomes vertically skewed even for the slightest $\mu j\u22600$, which is the case in Fig. 6(g).

Therefore, we can see that we have a relative freedom to fine-tune the architecture of the dots, depending on what kind of electronic detectors we have at our disposal and the states we want to probe. Of course, the experimental task of controlling simultaneously all the on-site potentials and tunneling amplitudes through voltage control is a very nontrivial task but feasible nevertheless.

## IV. QUBIT ENCODING WITH TOPOLOGICAL INVARIANT *ν*

One way to exploit this topological phase stability is to use it in quantum computing. More specifically, we can use localized measurements at the edges of our quantum dots configuration in order to detect the presence or lack of the topological edge states. After measuring, we can use the topological invariant $\nu $ to encode our qubits. That is, if we detect a topological edge state, we label it as qubit $|\nu =1\u27e9$, otherwise for the trivial state, as $|\nu =0\u27e9$. Focusing on the odd chain, the two states correspond to the occupational basis states as

Naturally, these can be generalized to any number of dots $N$ as $|\nu =1\u27e9=|1\u27e9\u2297|0\u27e9\u2297N\u22121,|\nu =0\u27e9=|0\u27e9\u2297N\u22121\u2297|1\u27e9$. Of course, in order to keep the system at $|E|=0$, we assume, as a first approximation, that we inject electrons of fixed energy into the system (e.g., by adjusting the Fermi level of an external injector or lead) and there are no losses. Thus, as long as the system remains in the topologically trivial insulating phase, the state will be labeled as $|\nu =0\u27e9\u2261|0\u27e9$, and for the nontrivial phase as $|\nu =1\u27e9\u2261|1\u27e9$. We first advocate here for an architecture with an odd number of dots since the two phases are completely distinguishable and localized in measurements.

Now, if we allow the tunneling coefficients to vary totally independently, i.e., we have for each dot connection a different hopping $tij$, then we can reproduce the edge states of the even chain; thus, we have to depart from the SSH model. Our Hamiltonian now would take a similar form

where $i,j\u2208{1,2,\u2026,N}$ and $\u27e8\cdots \u27e9$ is index notation for nearest neighbors.

In this case, the SSH model could be retained in the special case of $t12=t34$ and $t23=t45$, assuming $N=5$. As is apparent in Fig. 5(e), the wavefunction for the configuration $t12=t45$ and $t23=t34$ with $t12<t23$, splits equally between the two edge dots, where if we express it in the edge-local basis, we can create a superposition of the trivial and nontrivial SSH edge states as

We see that we can reproduce the even-dot-number edge state in an odd dot configuration, if we fine-tune our tunneling amplitudes away from the SSH configuration space. The process that leads to this state can be considered as an analog to a Hadamard-like gate; one key difference though is that here it is not yet clear how to be able to introduce a phase detuning between the two qubit states. This is also the reason this kind of qubit encoding structure is not enough to build a universal quantum computer.

The benefit of the proposed system is that it is robust to electronic noise, usually present in electronic systems and controls; we can also see this by adding a perturbing time-dependent term in the Hamiltonian as

where $\delta t1(t),\delta t2(t)$ can be random functions in time representing electronic (or other) noise as a perturbation in the system.

We can see in Fig. 5(f) that as long as the amplitude of the noise is preserving the ratio $\xi \u226b1$ (trivial phase), then the system preserves its topological characteristics, hence rendering the qubits tolerant to noise. Although we have a relatively high amplitude noise with $\delta t1/t1\u22480.69$, we can see that the wavefunction remains localized at a high degree on the last dot with probability $Pq=5>0.7$. This is also true for the case $\xi \u226a1$. Now, the system being in the topological phase, its wavefunction still preserves its key characteristics with localization at the first edge $Pq=1>0.8$, despite the high amplitude noise. Last, it is worth noting that the superposition state is more vulnerable to noise but we can still obtain it with relatively high fidelity by measuring the wavefunction at the edges with $PHedges\u22480.5$.

Something similar could also be done for the even chain, as shown in Figs. 6(e) and 6(f). The key difference, and also disadvantage, would be that in order to distinguish between the two states $|\nu =1\u27e9$ and $|\nu =0\u27e9$ by measurement, one would need to make simultaneous measurements of both the edge dots, which is more difficult to implement. Also, the difference in conditional probabilities $\Delta P=P(\nu =1|edge)\u2212P(\nu =0|edge)$ given the edge measurements is very small when noise is included, so there is a high probability of error in the readout $\u03f5\u221dO(\delta tij)$; this is why the odd chain is preferred here for the “static” encoding. On the other hand, one advantage is that the doubly degenerate edge state has a phase difference $\Delta \phi $ which, with the help of on-site potentials $\mu j$, could be used for $z$-control of the qubit (phase detuning). In both cases, a supervised learning classification algorithm could be used in order to implicitly set a type of “detection probability threshold” to be able to classify the two topological states with better precision and hence exploit the robustness of the different topological states to encode the qubit; we mention this in the end as well and leave it for future explorations in order to improve the scheme. Furthermore, as we briefly mention in Sec. V, a useful dynamical qubit encoding could potentially be utilized in the even chain.

## V. DETECTING TOPOLOGICAL EDGE STATES VIA MEASUREMENTS

We can detect the existence of a topological state with $\nu \u22600$ by doing measurements of the first and last dots in our structure. As mentioned above, the existence of an edge state requires that the relation $\xi \u226a1$ be preserved. Now, we need to let the system evolve in time and see what that would imply for our measurement. In order to see the evolution of the wavefunction $|\Psi \u27e9$ of our system, first we need to solve the time-dependent Schrödinger equation (TDSE),

where $\u210f$ is the reduced Planck’s constant.

The wavefunction can then be written as a linear superposition of all the possible single occupation basis states,

where $aj(t)$ are the wavefunction amplitudes.

where $a(t)\u2208RN$ is the $N$-dimensional vector with entries $an(t)$ and $a(0)$ is the initial condition of the wavefunction.

Now, considering the unperturbed time-independent SSH Hamiltonian $H(t)=HSSH$, we end up with the time evolution of the system wavefunction,

Starting by injecting our electron in the left edge dot (labeled as $q=1$), the initial state of the system is the one which is (ideally) completely localized in the first dot $|\Psi (0)\u27e9=|\nu =1\u27e9$. When the configuration of tunneling coefficients obeys $\xi \u226a1$, which means operating in the “topologically insulating” regime, we see that for the $N=5$ case $|a1(t)|2=|\u27e8\nu =1||\Psi (t)\u27e9|2\u22481$, and $|a\u02d91(t)|2\u2261d|a1(t)|2dt$ is small [Fig. 8(a)]; in other words, $|a1(t)|2\u226b|ai(t)|2,\u2200i\u22601,t\u2208R+$. This is significantly different from the even $N=4$ case, where $|a1(t)|2$ and $|a4(t)|2$ are the dominating amplitudes for the most part, as seen in Figs. 8(d) and 8(e). Moreover, we can see that in the even chain, the electron spends a lot of its time mostly localized in one of the two edge dots. This can be also utilized to encode a qubit state that occupies the two edge dots dynamically. Since there is a very clear oscillation between the edge dots, the qubit state would be easily readable; hence, we would label this time-evolving state $|\nu =1\u27e9\u2261|1\u27e9$.

Consequently, although in both cases we can dynamically detect the existence of topological states, they manifest differently with unilateral localization for the odd-case and bilateral edge localization for the even case. It is also worth noting that, as stated above, by introducing a local chemical potential $\mu i$ in the Hamiltonian on the $N=even$ chain, we can reproduce the topological states of the $N=odd$ one. This is visually manifested in Figs. 8(h) and 8(i); both for $\mu 1>0$ and $\mu 4>0$, we can produce the completely localized states (since chiral symmetry is broken from on-site chemical potential $\mu j$, these two states will not have the same energy) but with $\Delta \phi =(2k+1)\pi $ and $\Delta \phi =2k\pi ,k\u2208Z$ between the two edge dot states, respectively.

On the other hand, for the case of a topological phase transition and the metallic configuration, we can see that although $|\Psi \u27e9$ becomes delocalized throughout the whole structure, we have fairly intense alternating localization on the first and last dots for both chain parities. That is, $\u27e8|a1(t)|2\u27e9+\u27e8|aN(t)|2\u27e9>0.5$ and their periodicity is high for both even and odd chains, as is obvious from Figs. 8(b) and 8(f).

Finally, in the topologically trivial insulating regime of the parameters, we have very different behaviors. For the odd chain, we have large $\u27e8|a\u02d91(t)|2\u27e9$, similarly to the metallic case but $|a\u02d95(t)|2\u22480$. In addition, for the even chain, we have an almost identical behavior of $|a\u02d91(t)|2$ with the odd chain, but in sharp contrast, we have $|a\u02d95(t)|2=|a\u02d91(t)|2$. Here, the localization alternates between the first and last two adjacent dots, with no intermediate cases. Also, note that $\u27e8|a\u02d9i(t)|2\u27e9=\u27e8|a\u02d9j(t)|2\u27e9,\u2200i,j\u2208{1,\u2026,4}$. In this case, for the even chain, we could use this oscillating state to encode dynamically another qubit state, namely, the $|\nu =0\u27e9\u2261|0\u27e9$. The angular frequency dependence of the wavefunction probability distribution function oscillations on the number of dots $\omega (N)$ in the chain for even and odd dots, is shown in Figs. 9(a) and 9(b), respectively.

We can examine further the presence of topological order dynamically by means of entanglement (von Neumann) entropy $Sent$ and bipartite mutual information $I{a:b}(2)$. Since we have a single electron present in the system, there can be no entanglement but only quantum correlations of the wavefunction between the bulk and the edge dots; this is why we deemed the measure of mutual information to be more fit for the purpose. Another perspective here, if we view the dots as logical qubits, is that since our states are expressed in the $N\u2212$site basis, we can consider them as qubits with one of them always having value 1 and the rest 0. No matter the interpretation, we use it to measure the degree of quantum correlation between the edge and bulk quantum dots. For this, we will first define our dynamical density matrix $\rho (t)$,

In order to make a proper bipartition of our system, we will have to consider the density matrix corresponding to the full Hilbert space $Hfull$, which, if we want to stay faithful to our $N$-site state representation of the SSH model, would be constructed as

where $HSSH(k)$ is the $N$-site Hilbert subspace with $k$ particles and $HSSH(0)\u2261C$ is the SSH Hilbert subspace with zero particles, which corresponds to complex scalars and $(Nj)=N!j!(N\u2212j)!$.

Our full density matrix $\rho Hfull(t)$ will be a sparse $2N\xd72N$ matrix with the only non-zero entries corresponding to the single-electron Hilbert subspace $HSSH(1)$. Now if we make a bipartition of the edge and bulk dots, we can trace out one of the two parts, in order to construct the reduced density matrix. Writing our states as $|i\u2026jk\u2026l\u27e9\u2261|il\u27e9e\u2297|\u2026jk\u2026\u27e9b$ with “e” and “b” subscripts corresponding to edge and bulk dots respectively, the bulk density matrix is constructed as

where $1\u22611e\u22971b:HSSH(k)\u2192HSSH(k)$ is the identity projection operator acting on our SSH Hilbert subspace and the bulk part acts as

with $i,j,\u2026,k\u2208{0,1}$. The full density matrix $\rho Hfull(t)$ can be constructed as

Entropy has been calculated numerically by diagonalizing the density matrix as

where $\lambda k$ are the eigenvalues of the reduced density matrix $\rho bulk(t)$.

Now, in order to make more clear the quantum correlations between the edge and the bulk dots, we introduce the quantum informatic measure of bipartite mutual information $I{e:b}(2)$ as

where $e,b$ correspond to edge and bulk dots and $Se,Sb$ to their associated von Neumann entropies, respectively.

We can see how quantum correlations evolve in our system between the bulk and edges in Fig. 10. As expected, the maxima and minima of $I{e:b}(2)(t)$ evolve in tandem with the evolution of the wavefunctions $|\Psi (t)\u27e9$. For the $N=5$ case, when the electron becomes completely localized at the edge(s), $I{e:b}(2)$ becomes zero; same goes for the $N=4$ case. We see that when states start to overlap between the edge and bulk, $I\u02d9{e:b}(2)(t)>0$ and the other way around when they do not. However, there are some key qualitative differences distinguishable between the two cases.

In the $N=5$ chain, the time evolution has a very peculiar periodicity and, in most of the cases, we cannot see where the sequence repeats itself. We can understand this asymmetry in the odd chain if we consider the tunneling structure. The chain ends with a different tunneling coefficient than it begins with; hence, since the chain is asymmetrical hopping-wise, this becomes manifested here. This is in contrast to the $N=4$ chain where it is more symmetrical and we can see clear periodicity $T$. This happens because in the $N=4$ case, the wavefunction utilizes both edge dots most of the time, while in $N=5$, one dot dominates. This is another consequence of the difference between the topological state characteristics between the two cases. Finally, in both cases, we can see that when the tunneling configuration allows the existence of topological states in the system, $max(I{e:b}(2))<0.5$ is smaller overall than when we have a metallic or non-topological state, where $max(I{e:b}(2))=0.7$, which is the maximum possible value and also $\u27e8I{e:b}(2)\u27e9topological<\u27e8I{e:b}(2)\u27e9trivial,metallic$. Moreover, the lower values occur when we turn on our local potential $\mu i$ to create the completely localized states in the system for the even case as seen in Figs. 10(g) and 10(h). This stems both from our initialization of the electron and from the isolationist character of the topological states at the edges.

From the above, we see that if we can inject a single electron into a device with a QDA, we can figure out by measuring the edges alone if there are topological states present in the system. Inversely, given the tunneling amplitudes of the inter-dot barriers, we can probe to see if there are present odd or even number of dots in the system, no matter how large it is. Moreover, we can see that we can dynamically encode qubit states labeled by topological order (not to be confused with superconducting Majorana anyon topological qubits^{19,20}), which are robust to some degree to electronic noise present in electronic circuits and could be defined even more narrowly using mutual information to separate them.

## VI. CONCLUSIONS, REMARKS, AND FUTURE OUTLOOKS

Charge-based quantum computation has immense potential to be applied in the NISQ era and beyond. Although there are a few drawbacks like electronic noise rendering the qubits of low quality, it has the potential to be scaled to a large number of qubits that can compensate for that.

In this work, we have first of all investigated thoroughly a very simple and interesting quantum condensed matter model, which uses a single electron in an equipotential chain and captures “bizarre” topological characteristics. That is, we examined how in the SSH model, the winding number $\nu $, a manifestation of the Zak phase,^{33,37,39} can be defined as a topological invariant which, in turn, points to the existence of a topological phase in the band structure of the system. Moreover, we analyzed a ring structure with PBC and even and odd chains for $N=4$ and $N=5$ quantum dots with OBC, respectively. We also reviewed how there are some important qualitative and quantitative differences between the two latter cases, with the topological protected phase eigenstates manifesting in very different ways.

Then, building on the above, we speculated about a possible construction of a qubit, which is based on this topological order; this has the advantage that the qubit is encoded in a collective way throughout the whole chain and not just on a single dot. Of course, the electron injected should occupy a particular energy level, $|E|=0$, in order to be able to exhibit the edge state behavior. We also showed how a Hadamard-like state $|\Psi H\u27e9$ can be constructed for the $N=5$ case, if we allow our tunneling coefficients to depart from the SSH model to a more general hopping model; this state is identical to the $N=4$ edge states. Unfortunately, as far as our investigation goes, there might not be a universal quantum computer utilizing this qubit. Furthermore, we have considered a symmetry-preserving time-dependent perturbation as the noise of the Hamiltonian $\delta H(t)$ and demonstrated the robustness of the edge states, even for a considerable amount of noise, at specific time instances $t=t0$; we showed that as long as the noise introduced in the model preserves the ratio $\xi $, the topological order is preserved as well and the edge states keep their identity up to some point, even for relatively high noise amplitude. Additionally, by introducing local potential terms $\mu i$ in the Hamiltonian, we have reproduced the odd chain edge wavefunctions in the even chain structure; this comes at the cost of breaking the symmetry of the original SSH model.

In addition, we have shown how it is possible to detect the presence of topological order in the system considering the time dynamics of the single-electron wavefunction. We expressed our total wavefunction $|\Psi (t)\u27e9$ as a superposition in the single occupation basis. By measuring the two edge dots, after initializing our wavefunction to $|\Psi (0)\u27e9=|\nu =1\u27e9$ for the odd chain and $|a1(0)|2=1$ for the even chain, from the frequency of the oscillation of probability $|\Psi (t)|2$ and localization of the electron at the edges, we showed that deduction of the winding number $\nu =1$ or $\nu \u22601$ is possible; in other words, we deduct the presence of topological order in the system. To show this claim more clearly, we have utilized bipartite mutual information $I{a:b}(2)$ as a measure of the quantum correlation of bulk-edge dots. Moreover, we showed that from a measurement of the two edge dots, a deduction of which particular phase is present in the system (topological, metallic, or trivial) is possible as well. In addition, we provided a way to probe for the number parity of the quantum dots present in the structure, given that the tunneling amplitudes $tij$ are derived from the SSH model. Finally, we showed that even dynamically, we can use the winding number to encode qubit states for the even chain, albeit not so robust as the “static encoding.”

Now, we believe a few future explorations on this subject are in order. First, it would be interesting to investigate the time dynamics of the entanglement between the edge dots and the bulk (in a similar manner to Ref. 35) in the multiple electron case for variable length chains, which measures entropy $Sent(t)$ for the full Hilbert space of $N$ electrons $HFull$. This would allow for a variation in the number of electrons present in the chain and could potentially lead to some interesting encoding of a qubit constructed by the collective behavior of many electrons in the chain. Moreover, a more thorough investigation of the possible dynamic encoding of qubits with the winding number $\nu $, as proposed above, and the possible quantum gates construction to transition between different states. Moreover, in order to make the qubit design more realistic, one could include open leads at the structure with some Fermi energy, from which energy or electrons are pumped into the system dynamically. Of course, all of the above could also be put to test experimentally, given the physical parameters of the system.

Finally, an interesting interdisciplinary application could be to use machine learning to train a network to recognize the presence of topological order (as in Ref. 41) through time dynamics at the edges for different initial conditions. The sweep in the initial condition configuration space could offer a more general overview of the model and a more precise image of the different phases when dynamics are included. This could be also another way to encode a robust qubit through, e.g., a classification algorithm on the readout.

## ACKNOWLEDGMENTS

We would like to thank Andrii Sokolov for his contribution in creating some of the visualizations. This research was funded by the University College Dublin.

## AUTHOR DECLARATIONS

### Conflict of Interest

There are no conflicts of interest to be disclosed.

## DATA AVAILABILITY

The data that support the findings of this study are available within the article.

## REFERENCES

*Quantum Computing*, Lecture Notes (

*An Introduction to Quantum Computing*, Proof, Computation and Agency (International Journal of Mathematical Sciences and Computing (IJMSC), 2011), pp. 145–180.

*et al.*, “Qubits made by advanced semiconductor manufacturing,” arXiv:2101.12650 (2021).