Quantum walks of interacting fermions on a cycle graph

Quantum walks have been employed widely to develop new tools for quantum information processing recently. A natural quantum walk dynamics of interacting particles can be used to implement efficiently the universal quantum computation. In this work quantum walks of electrons on a graph are studied. The graph is composed of semiconductor quantum dots arranged in a circle. Electrons can tunnel between adjacent dots and interact via Coulomb repulsion, which leads to entanglement. Fermionic entanglement dynamics is obtained and evaluated.

where |m, k〉 is the state with the first and second electron being in the m-th and k-th vertex, respectively; |ψ (m,k) 〉 is the state of electrons occupying vertices m and k, which corresponds to a product state of two uncorrelated systems. These electrons have to be treated as indistinguishable because their wave functions overlap spatially in the quantum dots. The state in Eq. (1) is a superposition of electrons being in different vertices with time-dependent amplitudes ω mk (t), which form a matrix ω(t) 33 . The matrix ω(t) is antisymmetric, i.e. ω T (t) = − ω(t), and takes into account the antisymmetric nature of the fermionic wave function. The normalization of the |ψ(t)〉 state gives an additional condition on ω(t): ω ω = . † t t Tr( ( ) ( )) 1 ( 2) The wave function specified by ω matrix fully characterizes a fermionic state, however a correct definition of its subsystems is required for studying properties of the system. The problem of reduced fermionic density operators was addressed recently in refs 34 and 35, where it is shown that parity superselection rule should be applied to the fermionic state and the unique definitio of the reduced density operator is provided. In our case, the total number of fermions is constant, which leads to the standard procedure of obtaining the reduced state ρ 1 (t): This definition is used later to study the entanglement properties of the system.

Non-interacting indistinguishable electrons.
The dynamics of the two-electron fermionic state depends on an arrangement of quantum dots: if quantum dots are close enough -electrons will interact through Coulomb repulsion, otherwise electrons do not interact. First, we consider the case without interaction and move to the case with interaction afterwards. The evolution of an electron in an array of tunnel-coupled semiconductor quantum dots can be modelled by a continuous-time quantum walk, which is defined by a Hamiltonian with nearest-neighbour interactions 23 . By analogy, the evolution of two electrons can be modelled by a continuous-time quantum walk of two particles that is governed by the Hamiltonian where Ω is the tunneling frequency, which corresponds to the potential barrier height between neighbouring quantum dots. This Hamiltonian is defined for N > 2 (K > 1), for the smallest graph (K = 1) the Hamiltonian is equal to the half of the one in Eq. 4, i.e. H O /2, since in the circle of two dots clockwise and counterclockwise jumps correspond to the same transition. One can see, that the Hamiltonian H O only changes the spatial part of the total fermionic wave function |Ψ (t)〉 , leaving the spin part unchanged. In other words, the walk is performed in the space of coordinates of quantum dots, and the spins remain parallel as they were initially prepared. Therefore, the spin part of the wave function factors out from the evolution and will not be taken into account below. The remaining part of the total wave function, the antisymmetric spatial part, evolves according to the Schrödinger equation , where the unitary operator − e iH t/ O  can be shown to map any antisymmetric fermionic wave function to antisymmetric one. An exact matrix representation of this unitary operator can be obtained analytically for small K, but in general can only be computed numerically.
In Methods we provide exact solutions of the Schrödinger equation for K = 2, 3, 4. Exact solutions let us observe the periodic dynamics for K = 2 and 3 with periods T = π/2Ω and 2π/3Ω, respectively, and aperiodic dynamics for K = 4 (see Methods for details). From these results we conclude that in general the dynamics is aperiodic, as it was also shown in the case of discrete-time quantum walks on cycles 36,37 . Although the dynamics is aperiodic, it is known that by waiting enough time, an arbitrary precision of returning to the initial state can be achieved, as shown in Methods for K = 4. The possibility to achieve an arbitrary precision of the state revival holds for all K and is known from the Poincaré recurrence theorem 38,39 , although in general for different K it might take different time to achieve the same level of precision.
In experiment, the wave function |ψ(t)〉 cannot be directly observed, the measured data corresponds to a population in each quantum dot, i.e an average number of electrons in each dot. For this reason our function of interest is the population λ i in the vertex i of the cycle graph. The population λ i is equal to the probability to detect an electron in the vertex i and is related to the amplitudes ω mk of the wave function |ψ(t)〉 : Figure 2 shows population dynamics λ i (t) for the smallest K = 2, 3, 4 and 5. The solution in the case of K = 1 is trivial |ψ(t)〉 = |ψ(0)〉 and is not shown. From Fig. 2(a,b) one can immediately deduce that the charge dynamics is periodic, confirming the analytical results for the periods of quantum walks T = π/2Ω and 2π/3Ω in the case of K = 2 (a) and K = 3 (b), respectively. The dynamics in the case of K = 4 (c) is, however, aperiodic, which is proven in Methods. But as we discussed before, a nearly full state revival can be observed in this system, in particular in the case of K = 4 (c) and K = 5 (d).
A population distribution dynamics λ λ , similar to the one shown in Fig. 2, can be obtained by having only one electron initially prepared in a superposition of |0〉 and |K〉 coordinate states, where the scaling factor of 1/2 comes from the reduction of the total charge in the system. This can be seen from the right part of Eq. 5 -the position of the second particle k is irrelevant, the distribution λ i only depends on the position of the first particle. The probability to find this single electron in a certain node is half of the probability of finding one of two non-interacting electrons, which is also a consequence of Eq. 5. Hence a quantum walk of two non-interacting particles can be simulated by a one-particle walk, whose dynamics was studied in refs 23, 40 and 41. But because it is not straightforward to initialize an electron in a superposition of being in different nodes, two-particle walk can be used for studying one-particle walks with arbitrary initial conditions.

Interacting indistinguishable electrons.
Here we consider the case of two identical electrons that interact through Coulomb repulsion. The mutual repulsion between electrons becomes apparent when the distance between the quantum dots is such that the emerged Coulomb energy induced by one of the electrons prevents the second electron to tunnel to the adjacent dot. In order to model a fermionic quantum walk we approximate the Coulomb interaction by restricting the positions of electrons: electrons cannot be in the same or neighbouring vertices of the graph, and the effect of repulsion is negligible in all other situations, i.e. an electron does not "feel" an electric field of the distant electrons, if the distance between them is more than one empty quantum dot. This approximation is reasonable because neigbour dots are generally closer to each other than to metallic gates forming them so interaction can be strong, while interaction of electrons at distant dots is substantially suppressed not only by larger distance of interaction but also by screening due to presence of metallic gates between and nearby them. The Hamiltonian with the restriction of not being in the same and neighbouring vertices of the cycle graph is for small dimensions of the cycle graph K = 2, 3 and 4 (the case of K = 1 is unfeasible, because it is impossible to place two strongly repelling electrons in two quantum dots). The results, provided in Methods, demonstrate that there exists a period of quantum walks for K = 3, but not for K = 4. Hence, in general, the quantum walk of interacting particle on a cycle aperiodic. This fact can be seen in the population dynamics λ i (t) plotted in Fig. 3 for K = 3, 4, 5 and 6.

Fermionic entanglement by means of a quantum walk. It is known that interactions between
particles create quantum entanglement between these particles 42,43 . The qualification and quantification of an entanglement between several subsystems is one of the most important issues in quantum information theory. However, by describing an entanglement of two fermions we cannot use the standard definition of entanglement of distinguishable particles, because for identical particles the Hilbert space has no longer a tensor product structure. More specifically, the Hilbert space of two electrons is an antisymmetric product, not a direct product 44,45 .
To define entanglement of indistinguishable fermions one can use the Slater rank 33,46,47 . The Slater rank is the minimum number of Slater determinants, and this number is an analogue of the Schmidt rank for the distinguishable case. Fermions are called separable iff the Slater rank is equal to one. That is quantum entanglement arise in a pure state if there is no single-particle basis such that a given state of electrons can be represented as a single Slater determinant Fermionic quantum correlations defined above are the analogue of quantum entanglement between distinguishable systems and are essential for quantum information processing with indistinguishable systems. However these correlations should be quantified differently from the case of distinguishable systems by taking into account the definition of fermionic entanglement. Defining good measures of fermionic entanglement remains a field of active research 48,49 . In this paper we use three fermionic entanglement measures: von Neumann entropy 50,51 , linear entropy 50,51 and fermionic concurrence 49 .
Von Neumann entropy of the pure state ρ = |ψ〉 〈 ψ| is Quantum walk dynamics for K = 2, initial state is fully recovered after the time π/2Ω. (b) Quantum walk dynamics for K = 3, initial state is fully recovered after the time 2π/3Ω. (c) Quantum walk dynamics for K = 4, initial state is partially recovered after the time π π Ω = ≈ .
where ρ 1 is the single-particle reduced density matrix defined in Eq. 3, and ξ j are the nonzero eigenvalues of the ρ 1 matrix. It was shown, that a pure state ρ has the Slater rank equal to one iff S vN (ρ) = 0 33,52,53 . An entanglement criterion for states of two fermions can also be formulated in terms of the linear entropy L 1 2 which is the approximation of the von Neumann entropy. A pure state ρ has the Slater rank equal to one iff S L (ρ) = 0. We also use the fermionic concurrence 49 which by analogy with the linear entropy gives 0 for separable states and nonzero values for entangled fermionic states. In addition, the fermionic concurrence in Eq. 10 is normalized between 0 and 1. We calculate S vN (ρ(t)), S L (ρ(t)) and C f (ρ(t)) functions using Eqs 8-10, respectively, for K = 3, 4, 5 and 6. These entanglement measures are shown in Fig. 4. It can be seen that two electrons are initially separable, but after a time, which increases with K, they become entangled. In the case of K = 3, shown in Fig. 4(a), the entanglement dynamics is periodic, as expected due to the periodicity of the wave function. The maximum entanglement is achieved at times , for the state The minimum entanglement corresponds to the separable state |ψ (0,3) 〉 , which is present at times π = Ω t n 2 / 6 ,  ∈ n . The evolution of entanglement for K = 4 (N = 8 vertices) is shown in Fig. 4(b). One can see that the particles entangle slower (initial slope in Fig. 4) than in case of K = 3, because electrons are initially further away from each other and it takes more time for particles to meet each other. The evolution is aperiodic and the entanglement never disappears, but because of a partial revival of the initial separable state, the entanglement of electrons drops suddenly at times of the largest overlap with the initial state Ωt ≈ 3.1π and Ωt ≈ 7.5π (see also Fig. 3). The maximum entanglement is achieved for multiple states. For instance, at times π π Ω = + ≈ . t 7 / 6 3 2 2 2 , π π Ω = + ≈ . t 17 / 6 3 2 5 3 and π π Ω = + ≈ . t 27 / 6 3 2 8 4 the following fermionic state is generated  Figure 4(c,d) show the entanglement dynamics for higher cycle graph dimensions K = 5 and 6, respectively. Similar to the case of K = 4, the dynamics is aperiodic and maximal entanglement is achieved for many states. There are also occasional drops of entanglement caused by a partial return to the initial state |ψ (0,K) 〉 . The fermionic entanglement initiation described here is due to the Coulomb interaction. This repulsive interaction can be interpreted as a condition that restricts the positions of electrons -a quantum walk of one electron is conditioned on the state of the second electron and vice versa. In contrast to the interacting case, non-interacting electrons do not have this conditioned dynamics; dynamics of electrons is independent from each other. It can easily be shown that in absence of this Coulomb repulsion condition, entanglement is not initiated and all mentioned fermionic entanglement measures are equal to zero throughout the entire quantum walk evolution. Indeed, the Hamiltonian in Eq. 4 that governs the evolution of non-interacting electrons, leads to an independent unitary dynamics of two electrons   where by U O (t) we denote a time-dependent unitary matrix that acts locally on a subspace of one particle. Because the initial state of two electrons is separable, local operations clearly cannot create an entangled state. To verify this we compute the reduced density state from Eq. 3: Using the explicit form of the reduced density matrix ρ 1 (t) that we obtained, we compute ρ = t Tr ( ) 1/2 1 2 and ρ ρ = − t t Tr( ( )ln ( )) ln 2 1 1 , which leads us to the observation that all entanglement measures, S vN (ρ, t), S L (ρ, t) and C f (ρ, t) are equal to zero for all times t. Because these measures are zero iff the fermionic state ρ is separable, we conclude that, as expected, only by allowing electrons to interact, one can introduce fermionic entanglement in the system of coupled quantum dots that we consider.
Electrons for quantum information processing. We showed that a variety of entangled fermionic states can be created by means of quantum walks. However, it is not apparent how useful are these fermionic states for quantum information processing, because of inconsistency between fermions and qubits 34 . Here we show how one can define qudits by using the freedom of dividing the graph into two subgraphs. We divide the cycle graph into two equal parts: the first subgraph contains the vertices … and + + ⌊ ⌋ K K /2 1 dots. As we demonstrate below, in our framework, due to the symmetry of the initial state, it is possible to see that electrons are confined in different subgraphs with the unit probability. In this case, we say that an electron in the upper subgraph (vertices … and an electron in the lower subgraph (vertices represent two distinguishable qudits. Below we show that this definition of two qudits in terms of the upper and the lower subspaces allows obtaining highly entangled states of two qudits. The described "cuts" of the circle are depicted in Fig. 5(a-d) for K = 3 (a), K = 4 (b), K = 5 (c) and K = 6 (d). Figure 5(a) schematically shows two qutrits (three-level systems with basis states |0〉 , |1〉 and |2〉 ) defined on a cycle graph. At time π = Ω t / 6 , as we have shown before, the quantum dynamics on a cycle graph with 6 vertices leads to the state in Eq. 11 with two-particle correlation matrix shown in the right part of Fig. 5(a). One can see, that if one raises a potential barrier between quantum dots 1 and 2, 4 and 5, as shown in the left part of Fig. 5(a), one traps electrons in separate subgraphs, because at this time the particles can only be detected in the strictly opposite sites at the circle. After defining qudits at this time step we obtain an entangled state ψ = − + + .
, the subspace of the second qubit is shown in violet (vertices The brown curves in Fig. 5(a) represent the type of a superposition in Eq. 15, which corresponds to the Bell-type entanglement. Figure 5(b) depicts a larger cycle graph with 6 vertices. Similar to the case of K = 3, two qudits are defined on the circle at a certain time π = + Ω t 17 / 6 3 2 . At this time one can separate two halves of the circle and obtain the following state of two ququarts (2 02 2 20 11 ) (16) 4 Although this state is similar to the one in Eq. 15 up to a local phase, the type of entanglement in space of the graph is different and shown in the left part of Fig. 5(b). From the right part of the Fig. 5(b) one can see that electrons are distributed in different subgraphs. The cases of K = 5 and K = 6 are shown in Fig. 5(c,d), respectively. As one can see from correlation matrices, the same types of quantum correlations can be achieved with high probability.
The fermionic entanglement dynamics studied in this paper is obtained for a pure state of a quantum system. However, in experiment the quantum state is subjected to decoherence. In particular, a change in the state of the qudits can be caused by the white noise from the quantum point contact detectors, which was shown to be one of the major concerns facing experimental realization of quantum walks in quantum dots structures 23 . This noise can be modelled by a depolarizing channel that acts on a density matrix of two fermions ρ as follows which is the solution of the differential equation dρ(t)/dt = − Γ (ρ(t) − ρ M ) with the initial state ρ(0) = |ψ (0,K) 〉 〈 ψ (0,K) |, where Γ is the relaxation rate that corresponds to the coupling between the quantum dot and the quantum point contact. The density matrix ρ M is the maximally mixed state of the coordinate part of two electrons. Because of the antisymmetric fermionic state the maximally mixed state is not the normalized identity matrix, but is defined as follows Combining the dissipative dynamics from Eq. 17 with the coherent evolution with Hamiltonian H C from Eq. 6 we write the general expression for the fermionic density matrix We are able to analyze the combination of the two different processes and to do the simplification of the expression due to the relation [H C , ρ M ] = 0, which implies that the operator  − e iH t/ C commutes with ρ M . The decoherence described by Eq. (17) leads to errors in quantum information stored in the system of electrons. In order to quantify this error we use the measure of decoherence 22,54 to quantify the amount of errors, where the operator norm of the matrix X is given by = ∈ X x max x X spec( ) with spec(X) being the spectrum of the operator X. The measure of decoherence D can be thought of as a probability of obtaining an error. In our scenario getting an error would correspond to getting a completely classical state of two particles, which are uniformly distributed over the circle, instead of getting entangled states shown in Fig. 5. As shown in Fig. 6, where we plot an error for each state in the set from Fig. 5 (entangled states with K = 3, 4, 5 and 6), this error can become large for large circles and strong couplings Γ between the quantum dot and the quantum point contact.
In addition to the described noise, the system of electrons is subjected to the inevitable phase noise caused by the deformation interaction of electrons with acoustic phonons 18 . As a result, the energy levels in quantum dots where electrons reside become not fully determined, which effectively distorts the nondiagonal elements of the density matrix ρ  as follows where I is the identity matrix, = − γ − E e i i 1 i for i < K and E K = e −γ/2 I with γ = Ξ 2 /2ħπ 2 ρs 3 a 2 . Note that the |ψ K 〉 state is written in the basis of separated electrons, which corresponds to the states in Eqs 15 and 16. The following parameters are taken for electrons in silicon: effective deformation potential Ξ = 3.3 eV, speed of sound s = 9.0 × 10 3 m/s, density ρ = 2.33 g/cm 3 , and quantum dot size a = 10 nm 18 . For this set of parameters and K = 4 the obtained phase error is D ≈ 1.4 × 10 −5 , which suggests that this additional phase error is negligible and the error is mostly determined by the depolarizing noise in the range of relaxation rates we consider in Fig. 6.

Discussion
In the paper we considered the dynamics of two-particle fermionic system. We analyzed quantum walks in two possible setups, which lead to a walk with and without interaction between electrons. We preferred quantum walks approach to quantum information processing for a number of reasons. Quantum walks dynamics is a natural process for many quantum systems compared to more artificial gate implementation. It is therefore easier to build and implement in experiment. One may also hope that relatively complex gates sequences could be replaced by simpler quantum walks processes. It was shown before that one can do arbitrary quantum operations using only particles free propagation 12 . One way to realize quantum walks algorithms is to use silicon quantum dots that form a cycle graph. We showed the electrons entanglement dynamics in this structure. The value of fermionic entanglement was calculated using measures in Eqs (8)- (10), which were proven to correctly quantify entanglement 49,50,51 . We showed that fermionic entanglement can be used to prepare quantum states for quantum information processing. These highly entangled states of qudits can be obtained by only using the free quantum evolution of identical particles, without relying on any additional manipulations with electrons. In addition, we supplemented our protocol of obtaining entangled states with analytical solutions for certain sizes of a graph and proved a general aperiodic nature of the continuous-time quantum walk of identical particles on a cycle graph.

Methods
In this section we explain the fermionic quantum walks dynamics in details by obtaining explicit analytical solutions of the Schrödinger equation. Throughout the section we use the series expansion of the quantum walk unitary operator Period of quantum walks of non-interacting particles. We first start our analysis with the case of quantum walks of non-interacting indistinguishable electrons, whose dynamics is described by the Hamiltonian H O in Eq. 4. We first consider the smallest sizes of the cycle graph with K = 1 (2 vertices), K = 2 (4 vertices), K = 3 (6 vertices) and show the periodicity of the underlying dynamics. Next we show that, in general, the dynamics is aperiodic, i.e. there is no time T ≠ 0 s.t. |ψ(T)〉 = |ψ(0)〉 , by obtaining the solution for K = 4 (8 vertices).
A cycle graph with 2 vertices. For K = 1 the evolution of the state is trivial:   Fig. 5(a-d).
Scientific RepoRts | 6:34226 | DOI: 10.1038/srep34226 An overlap of the state |ψ(t)〉 with the initial state is equal to ψ ψ = Ω t t (0) ( ) c os (2 ) , therefore the dynamics of the wave function (as well as the population λ i and the fermionic entanglement functions) is periodic. The period of the dynamics is T = π/2Ω, after this time the initial state is fully revived (we neglect a global phase). It is worth noting, that at time t = π/Ω the unitary matrix − e iH t/ O  is equal to the identity matrix I, so any initial state is recovered after this time. The specific choice of the symmetric initial state we use recovers twice more frequent.