# Pairing Hamiltonians of Nearest-Neighbor Interacting Superconducting Qubits on an IBM Quantum Computer

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Model

#### 2.1. Pairing Hamiltonians in Superconductivity

#### 2.2. Nearest-Neighbor Coupling Interactions in Superconductivity

## 3. Methods of Simulation for Hamiltonian Operators

_{12}U3

_{1}(CNOT)

_{12}. This is shown in Figure 9a.

_{12}Rz

_{2}(CNOT)

_{12}. This is shown in Figure 9c.

## 4. Results

#### 4.1. Initial State Preparation and Unitary Operation

- For the XX type of interaction: Comparing with the standard U3 gate (Equation (15)), the parameters for the gate are found to be $\mathsf{\theta}=2{J}_{l}t$, $\varphi =-\mathsf{\pi}/2$, and $\mathsf{\lambda}=\mathsf{\pi}/2$;
- For the YY type of interaction: The parameters obtained for the U3 gate are $\mathsf{\theta}=2{J}_{l}t$, $\varphi =-\mathsf{\pi}/2,$ and $\mathsf{\lambda}=\mathsf{\pi}/2$. The parameters obtained for the $U{3}^{\u2020}$ gate are $\mathsf{\theta}=2{J}_{l}t$, $\varphi =\mathsf{\pi}/2$, and $\mathsf{\lambda}=-\mathsf{\pi}/2$;
- For the ZZ type of interaction: Comparing with the standard U1 gate,$$U1=\left[\begin{array}{cc}1& 0\\ 0& {e}^{i\lambda}\end{array}\right]\phantom{\rule{0ex}{0ex}}\mathsf{\lambda}=2{J}_{l}t,$$

#### 4.2. Quantum State Tomography for Suzuki–Trotter Decomposition of Quantum Circuits

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 3.**Quantum circuits for the nearest-neighbor interacting (

**a**) Heisenberg, (

**b**) XY, (

**c**) transverse Ising, and (

**d**) longitudinal Ising Hamiltonian models.

**Figure 4.**(

**a**) Theoretical density matrix for the Heisenberg model, (

**b**) experimental density matrix for 2 iterations, (

**c**) experimental density matrix for 10 iterations, and (

**d**) fidelity as a function of iteration.

**Figure 5.**(

**a**) Theoretical density matrix for the XY model, (

**b**) experimental density matrix for 2 iterations, (

**c**) experimental density matrix for 10 iterations, and (

**d**) fidelity as a function of iteration.

**Figure 6.**(

**a**) Theoretical density matrix for the transverse Ising model, (

**b**) experimental density matrix for 2 iterations, (

**c**) experimental density matrix for 10 iterations, and (

**d**) fidelity as a function of iteration.

**Figure 7.**(

**a**) Theoretical density matrix for the longitudinal Ising model, (

**b**) experimental density matrix for 2 iterations, (

**c**) experimental density matrix for 10 iterations, and (

**d**) fidelity as a function of iteration.

**Figure 8.**(

**a**) Time evolution of the different models at $\frac{\theta}{2}$ for (

**a**) Heisenberg, (

**b**) XY, (

**c**) transverse Ising, and (

**d**) longitudinal Ising models.

**Figure 9.**(

**a**) Circuit implementation for the different interactions of the (

**a**) $\widehat{{\mathsf{\sigma}}_{1}^{x}}\widehat{{\mathsf{\sigma}}_{2}^{x}}$ term where the circuit consists of two CNOT gates and one U3 gate; (

**b**) $\widehat{{\sigma}_{1}^{y}}\widehat{{\sigma}_{2}^{y}}$ term where the circuit consists of two CNOT gates and one U3 gate, one $U{3}^{\u2020}$ gate and two NOT gates; and (

**c**) $\widehat{{\sigma}_{1}^{z}}\widehat{{\sigma}_{2}^{z}}$ term where the circuit consists of two CNOT gates and one Rz gate.

**Figure 10.**Properties of the quantum processor used: (

**a**) map view of the IBMQ Lima quantum processor, (

**b**) T1 (thermal relaxation time) of the qubits from qubit 0–4, (

**c**) T2 (dephasing time) of the qubits from qubit 0–4, and (

**d**) frequency of operation of the different number of qubits.

**Table 1.**Interaction Hamiltonians [52].

Interaction Model | Interaction Hamiltonian |
---|---|

Heisenberg Model | ${H}_{H}={H}_{0}+{\displaystyle {\displaystyle \sum}_{i=x,y,z}}{\displaystyle {\displaystyle \sum}_{l=1}^{N-1}}{J}_{l}^{i}{\sigma}_{l}^{i}{\sigma}_{l+1}^{i}$ |

XY Model | ${H}_{XY}={H}_{0}+{\displaystyle {\displaystyle \sum}_{i=x,y}}{\displaystyle {\displaystyle \sum}_{l=1}^{N-1}}{J}_{l}^{i}{\sigma}_{l}^{i}{\sigma}_{l+1}^{i}$ |

Transverse Ising Model | ${H}_{TIsing}={H}_{0}+{\displaystyle {\displaystyle \sum}_{l=1}^{N-1}}{J}_{l}{\sigma}_{l}^{x}{\sigma}_{l+1}^{x}$ |

Longitudinal Ising Model | ${H}_{LIsing}={H}_{0}+{\displaystyle {\displaystyle \sum}_{l=1}^{N-1}}{J}_{l}{\sigma}_{l}^{z}{\sigma}_{l+1}^{z}$ |

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**MDPI and ACS Style**

Chatterjee, S.; Behera, B.K.; Seo, F.J.
Pairing Hamiltonians of Nearest-Neighbor Interacting Superconducting Qubits on an IBM Quantum Computer. *Appl. Sci.* **2023**, *13*, 12075.
https://doi.org/10.3390/app132112075

**AMA Style**

Chatterjee S, Behera BK, Seo FJ.
Pairing Hamiltonians of Nearest-Neighbor Interacting Superconducting Qubits on an IBM Quantum Computer. *Applied Sciences*. 2023; 13(21):12075.
https://doi.org/10.3390/app132112075

**Chicago/Turabian Style**

Chatterjee, Shirshendu, Bikash K. Behera, and Felix J. Seo.
2023. "Pairing Hamiltonians of Nearest-Neighbor Interacting Superconducting Qubits on an IBM Quantum Computer" *Applied Sciences* 13, no. 21: 12075.
https://doi.org/10.3390/app132112075