# Model of a three-qubit cluster in a thermal bath

E. Andre^{1}^{,2}, A.N. Tsirulev^{1}

^{1} Tver State University^{2} Faculty of Sciences, Agostinho Neto University

**DOI:** 10.26456/pcascnn/2023.15.223

* Original article*

**Abstract: ** This work studies a mathematical model of a quantum cluster consisting of three qubits and being in thermal equilibrium with the environment. The effective Hamiltonian is invariant under permutations of qubits and consists of two parts. The first part is similar to the Heisenberg XYZmodel with internal two-qubit interaction, while the second includes three-qubit interaction with the thermostat. Such a quantum system admits a fully analytical investigation and is considered in the context of mathematical modeling of quantum metamaterials, in which nanoclusters are elementary structural units with the strong internal interaction of qubits and the relatively weak coupling with the environment. For the Hamiltonian, we construct an orthonormal basis of eigenvectors, which includes the maximally entangled W-state. We also obtain the density operator of the cluster state in explicit form, and study the temperature dependences of the thermodynamic characteristics of the cluster: the partition function, entropy, and free energy. It is shown that the conditions of thermal equilibrium in this quantum system are satisfied at temperatures from 0,2 K to microkelvins, which correspond to the operating range of modern quantum logic elements and quantum simulators.

*Keywords: cluster of qubits, Hamiltonian, Pauli basis, operator exponential, density operator of a state, Gibbs-von Neumann state, partition function, entropy, free energy *

- Eduardo . Andre – 4th year postgraduate student, Applied Physics Department, Tver State University , Faculty of Sciences, Agostinho Neto University
- Alexander N. Tsirulev – Dr. Sc., Professor, Department of General Mathematics and Mathematical Physics, Tver State University

**Reference: **

**Andre, E.. ** Model of a three-qubit cluster in a thermal bath /
E.. Andre, A.N. Tsirulev //
Physical and chemical aspects of the study of clusters, nanostructures and nanomaterials. — 2023. — I. 15. — P. 223-230. DOI: 10.26456/pcascnn/2023.15.223. (In Russian).

**Full article (in Russian): **
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