**2.1 Generation of Matryoshka states using spin systems in condensed matter physics**

In this chapter, the generation of Matryoshka states will be explored in spin systems in condensed matter physics. Unlike in the case of the aforementioned algorithm, instead of composite operators, in this case we have localised generation and minimal interactions between different GHZ and GHZ-like states to create the Matryoshka states. In this case, we consider *N* spin-<sup>1</sup> <sup>2</sup> particles, with each spin coupled to its nearest neighbours by the XY Hamiltonian

$$H = \sum\_{i=1}^{N-1} \left( J\_{X,i} \hat{X}\_i \hat{X}\_{i+1} + J\_{Y,i} \hat{Y}\_i \hat{Y}\_{i+1} \right) \tag{10}$$

where *<sup>J</sup>σ*,*<sup>i</sup>* is the pairwise coupling constant with *<sup>σ</sup>* <sup>¼</sup> *<sup>X</sup>*^, *<sup>Y</sup>*^, *<sup>Z</sup>*^ being the Pauli operators. For the purposes of this chapter, we take *N* to be odd. *Franco et al* [58] showed that it is sufficient to state that the information flux between the *<sup>X</sup>*^ (*Y*^<sup>Þ</sup> operators of the first and last qubits in the spin-chain depends on an alternating set of coupling strengths. For example, the information flux from *X*^<sup>1</sup> to *X*^ *<sup>N</sup>* depends only on the set *JY*,1, *JX*,2, … , *JY*,*N*�<sup>1</sup> � � and is independent of any other coupling rate in the spin-chain. *Christandl et al* [60, 61] showed that after a time *<sup>t</sup>* <sup>∗</sup> <sup>¼</sup> *<sup>π</sup>=<sup>λ</sup>* with *<sup>λ</sup>* being a scaling constant (as mentioned in the definition of the case of a perfect state transfer in a linear spin-chain given by weighted coupling strengths: *J<sup>σ</sup>*,*<sup>i</sup>* ¼ *λ* ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *i N*ð Þ � *<sup>i</sup>* <sup>p</sup> ), the state of the first qubit in the spin-chain can be perfectly transferred to the last qubit. We see that by preparing the initial state of this spin-chain in an completely separable eigenstate of the tensorial product of *Zi* operators, say ∣Ψð Þi ¼ 0 ∣000*:::*0i<sup>12</sup> … *<sup>N</sup>*, we obtain an information flux towards symmetric two-site spin operators, and a final state of the form [58].

$$|\psi\_0\rangle = |0\rangle\_\varepsilon \otimes\_{i=0}^M |\psi\_+\rangle\_{2i+1,N-2i} \otimes\_{i=1}^M |\psi\_-\rangle\_{2i,N-2i+1} \tag{11}$$

$$|\psi\_1\rangle = |1\rangle\_\varepsilon \otimes\_{i=0}^M |\psi\_-\rangle\_{2i+1,N-2i} \otimes\_{i=1}^M |\psi\_+\rangle\_{2i,N-2i+1} \tag{12}$$

where *<sup>c</sup>* labels the central site of the spin-chain, *<sup>M</sup>* <sup>¼</sup> *<sup>N</sup>*�<sup>3</sup> <sup>4</sup> and ∣*ψ*�i ¼ 1ffiffi 2 <sup>p</sup> ð Þ j00i�j11i . An illustration of the setup has been shown in **Figure 1**.

The critical step in the creation of the Matryoshka GHZ-Bell state is the evolution of the central and two neighbouring qubits to the GHZ state, without disturbing the rest of the spin-chain. This is a key result around the generation of *Matryoshka* GHZ-Bell states in this chapter, which can be extended to other classes of *Matryoshka* states.

#### **Figure 1.**

*Scheme for the generation of Matryoshka GHZ-Bell resource-states, where the effective spin–spin XY Hamiltonianan is obtained as an effective adiabatic Hamiltonian for a linear chain of optical cavities with each interacting with a three-level atomic system. The ground states of each atomic unit provide the computational space of each spin, and the dipole-forbidden transition between these states is realised as an (adiabatic) Raman transition through the excited state:* ∣*e*i*<sup>i</sup> with i* ¼ 1, 2, … , *N. The cavity field drives off-resonantly the dipoleallowed channel* ∣*j*i*<sup>i</sup>* \$ ∣*e*i*<sup>i</sup> with the Rabi frequency g <sup>j</sup> , j* ¼ 0, 1*. Two lasers are also coupled to these atomic transitions with strength* Ω *<sup>j</sup> and detuning* Λ*j.*

For this, we need to switch off all the interactions except for those connecting the central qubit to the neighbouring ones. A point to note here is that had we started with ∣Ψð Þi ¼ 0 ∣111*:::*1i<sup>12</sup> … *<sup>N</sup>*, we would have obtained a final state of the form

$$|\boldsymbol{\Psi}\_{0}\rangle = |\mathbf{0}\rangle\_{\varepsilon} \otimes\_{i=0}^{M} |\boldsymbol{\Psi}\_{-}\rangle\_{2i+1,N-2i} \otimes\_{i=1}^{M} |\boldsymbol{\Psi}\_{+}\rangle\_{2i,N-2i+1} \tag{13}$$

$$|\psi\_1\rangle = |1\rangle\_\varepsilon \otimes\_{i=0}^M |\psi\_+\rangle\_{2i+1,N-2i} \otimes\_{i=1}^M |\psi\_-\rangle\_{2i,N-2i+1} \tag{14}$$

We use this principle and the idea that after evolution over time *t* <sup>∗</sup> , the states in Eqs. (2) and (3) transform back to ∣000*:::*000i<sup>12</sup> … *<sup>N</sup>* and states in Eqs. (4) and (5) transform back to ∣111*:::*11i<sup>12</sup> … *<sup>N</sup>*. We can utilise this concept, by taking the state in Eq. (2) and evolving it, for the truncated subsystem comprising of the central qubit and the adjoining qubits. A point to note here is that due to only coupling that connects to the central qubits, the coupling strength (*J* 0 *<sup>σ</sup>*,*<sup>i</sup>* <sup>¼</sup> *<sup>λ</sup>*<sup>0</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *<sup>i</sup>*ð Þ <sup>3</sup> � *<sup>i</sup>* <sup>p</sup> ) and time of evolution (*t* <sup>00</sup> ¼ *π=λ*<sup>0</sup> ) vary accordingly. Before carrying out this evolution, we perform a Hadamard operation on the central qubit to give

$$|\psi\_0\rangle = \frac{1}{\sqrt{2}} (|0\rangle\_\epsilon + |1\rangle\_\epsilon) \otimes\_{i=0}^M |\nu\_+\rangle\_{2i+1, N-2i} \tag{15}$$
 
$$\otimes\_{i=1}^M |\nu\_-\rangle\_{2i, N-2i+1}$$

We now perform the truncated subsystem time-evolution with the parameters *J* 0 , *t* <sup>00</sup> ð Þ to give us the state

$$|\psi\_0\rangle = \frac{1}{\sqrt{2}} (|000\rangle + |111\rangle)\_{c-1,c+1} \otimes\_{i=0}^{M-1} |\psi\_+\rangle\_{2i+1,N-2i} \tag{16}$$
 
$$\otimes\_{i=1}^M |\psi\_-\rangle\_{2i,N-2i+1}$$

Therefore, we can obtain a Matryoshka GHZ-Bell state using nearest spin–spin interactions in a spin-chain. A similar generation protocol can be defined for the other two classes of Matryoshka states. The teleportation of an arbitrary n-qubit state can be performed using Matryoshka GHZ-Bell States [62].

Given the triangular three-qubit configuration, we can also consider the anisotropic Heisenberg Hamiltonian, which describes the interaction between three spins that are located at the corners of an equilateral triangle lying in the xy-plane, as shown in **Figure 2**.

$$H = -J\_{xy} \sum\_{i=1}^{3} \left( \mathbf{S}\_i^x \mathbf{S}\_{i+1}^x + \mathbf{S}\_i^y \mathbf{S}\_{i+1}^y \right) - J\_x \sum\_{i=1}^{3} \mathbf{S}\_i^x \mathbf{S}\_{i+1}^x + H\_x \tag{17}$$

here the three spins *Si*, with S = 1/2, are located at the corners i = 1, 2, 3, and *S*<sup>1</sup> ¼ *S*4. *Jxy* and *Jz* are the in-plane and out-of-plane exchange coupling constants respectively, and *HZ* <sup>¼</sup> <sup>P</sup><sup>3</sup> *<sup>i</sup>*¼<sup>1</sup>*bi:Si* denotes the Zeeman coupling of the spins *Si* to the externally applied magnetic fields *bi* at the sites *i*. If we consider isotropic exchange couplings: *Jxy* ¼ *Jz* ¼ *J* > 0 (ferromagnetic coupling) and *bi* ¼ 0∀*i*, we have a ground-state qudruplet that is spanned by the GHZ states: <sup>1</sup>ffiffi 2 <sup>p</sup> ð Þ j000iþj111i and 1ffiffi 2 <sup>p</sup> ð Þ j000i�j111i , along with the W- and spin-flipped W-states. A set of appropriately chosen magnetics fields will allow us to split off an approximate GHZ state from this degenerate eigenspace. If we find a set of magnetic fields that, in classical spin systems, shall result in exactly two degenerate minima for the configurations ∣000i, representing the ↓↓↓ spin configuration, and ∣111i, representing the ↑↑↑ spin configuration, with an energy barrier in between, quantum mechanical tunnelling shall yield the desired states. The magnetic fields must be of the same strength, in-plane and sum to zero, with a convenient additional choice being that of the field pointing radially outward. Therefore, the successive directions of the magnetic fields have to differ by an angle of 2*π=*3 with respect to each other. Going by the schematic in **Figure 2**, we can write the hamiltonian

*<sup>H</sup>* ¼ �*Jxy*<sup>X</sup> 3 *i*¼1 *Sx i Sx <sup>i</sup>*þ<sup>1</sup> þ *S y i S y i*þ1 � � � *Jz* X 3 *i*¼1 *Sz i Sz <sup>i</sup>*þ<sup>1</sup> þ *Hz* þ X *Nl il*¼1 �*J* ð Þ*il xy* <sup>X</sup> 3 *i*¼1 *Sx i Sx <sup>i</sup>*þ<sup>1</sup> þ *S y i S y i*þ1 � � � *<sup>J</sup>* ð Þ*il z* X 3 *i*¼1 *Sz i Sz <sup>i</sup>*þ<sup>1</sup> <sup>þ</sup> *<sup>H</sup>*ð Þ *il z* " # X *Nr ir*¼1 �*J* ð Þ *ir xy* <sup>X</sup> 3 *i*¼1 *Sx i Sx <sup>i</sup>*þ<sup>1</sup> þ *S y i S y i*þ1 � � � *<sup>J</sup>* ð Þ *ir z* X 3 *i*¼1 *Sz i Sz <sup>i</sup>*þ<sup>1</sup> <sup>þ</sup> *<sup>H</sup>*ð Þ *ir z* " # þ X *Nl il*¼1 *λl* ð Þ *il*,*il*þ<sup>1</sup> *<sup>S</sup>nr <sup>i</sup> :Snl <sup>i</sup>*þ<sup>1</sup> þ *N* X*<sup>r</sup>*�<sup>1</sup> *ir*¼0 *λr* ð Þ *ir*,*ir*þ<sup>1</sup> *<sup>S</sup>nr <sup>i</sup> :Snl i*þ1 (18)

#### **Figure 2.**

*Schematic for all (three) classes of Matryoshka states for d* ¼ 2 *levels of the quantum system, explored in this chapter. The triangular formations encapsulate the logical units of two/three qubits mediated by CNOT gates. Each of these triangular units are weakly coupled to each other (shown with light blue patches). In the case of the* Matryoshka GHZ-Bell *states, we only have the black links, while for the* Matryoshka Generalised GHZ *states and* Matryoshka Q-GHZ *states, we also have the blue links.*

where the superscripts *il* and *ir* denote the left and right branches respectively of the schematic arounnd a central triangular unit. For *il* ¼ 1, we have the leftmost triangular unit and for *ir* ¼ *Nr*, we have the rightmost triangular unit. *Nl* and *Nr* denote the number of units on the left and right side of the central triangular unit. In principle, we can have an asymmetric case where *Nl* 6¼ *Nr*. In the fourth line, the term *SNl*þ<sup>1</sup> and *S*<sup>0</sup> refer to the spins in the central triangular unit connected to the adjacent left and right triangular units respectively. Moreover, both *λ<sup>l</sup>* ð Þ *il*,*il*þ<sup>1</sup> and *λl* ð Þ *ir*,*ir*þ<sup>1</sup> are coupling constants between adjacent triangular units that are numerically negligible with respect to *J* but are non-zero, to account for inter-unit coupling. *Snr <sup>i</sup>* and *Snl <sup>i</sup>* are right and left connecting nodes of the *i th* triangular unit.

An important point here is the condition: *GHZak*,*di* ,� *di* <sup>j</sup>*GHZak*<sup>0</sup>,*di* ,� *di* D E <sup>¼</sup>

*δkk*<sup>0</sup>∀*i*, *A<sup>k</sup>* <sup>1</sup> <sup>j</sup>*Ak*<sup>0</sup> 1 D E <sup>¼</sup> *<sup>δ</sup>kk*<sup>0</sup> in Eqs. (4), (6) and (9). This is ensured by the additional application of single qubit gates on the nodes of the triangular units. For instance, 1ffiffi 2 <sup>p</sup> ð Þ! <sup>j</sup>000iþj111<sup>i</sup> *<sup>σ</sup>*<sup>2</sup> *<sup>x</sup>* <sup>1</sup>ffiffi 2 <sup>p</sup> ð Þ j010iþj101 . Using combination of such single qubit operations, we can span the entire space of GHZ and GHZ-like states. The important point here is the synchronised timing of these operations, with the inter-unit coupling, so as to give us a superposition over orthogonal GHZ and GHZ-like states for all triangular units, as shown in **Figure 2**.
