- A method for detecting the energy levels of an arbitrary spin system on a quantum computer is proposed. The method is based on investigating the evolution of the average value of the operator that commutes with the Hamiltonian of the system under study. The results of quantum calculations agree with theoretical ones. The method is generalized for the case when there is no operator that anticommutes with the Hamiltonian. It is based on the use of one additional spin (ancilla). The method has been demonstrated on IBM quantum computers. The results of quantum calculations agree with theoretical ones.
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A method for finding the geometric properties of graphs using quantum computing is proposed. The geometric characteristics of graphs depend on the properties of the corresponding quantum graphs. Specifically, it has been found that energy fluctuations in graph states and, consequently, the speed of quantum evolution, curvature, and torsion of states are related to the total number of edges, triangles, and squares in the corresponding graphs. This makes it possible to quantitatively determine the number of edges, triangles, and squares in a graph on a quantum device and achieve a quantum advantage in solving this problem with the development of a multi-qubit quantum computer.
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A method is proposed for finding the statistical sum of the Ising model on a graph using quantum computing. The Boltzmann factor is modeled on a quantum computer as the trace of a certain evolution operator with an effective Hamiltonian over the spins of the qubits corresponding to the graph edges. The low-temperature limit allows finding the ground state of the system. This allows using the proposed method to solve combinatorial optimization problems on a quantum computer. The method for finding the statistical sum and the ground state has been demonstrated for spin clusters on IBM quantum computers.
Quantum computers can be a powerful tool for solving such problems because they are capable of simultaneously processing many possible options thanks to the principle of superposition and quantum entanglement:
• Methods for finding energy levels of many-spin systems open up the possibility of achieving quantum advantage in solving eigenvalue problems.
• The method for finding the ground state allows quantum advantage to be achieved in finding the minimum or maximum energy of the Ising model with spatially anisotropic interaction using multicube quantum computers.
• The developed method for finding the statistical sum of spin systems allows studying their thermodynamic properties on a quantum computer.
• The method for finding the ground state of spin systems allows combinatorial optimization problems to be solved on a quantum computer. These problems are challenging for classical computers. Therefore, with the development of multi-qubit high-performance quantum processors, we expect to achieve quantum advantage in solving these problems.
Finding energy levels of physical quantum systems. Finding the statistical sum of spin systems, Fisher zeros, and Li-Yang zeros.
Finding the ground state of spin systems, which allows solving important applied combinatorial optimization problems.