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One of the principal challenges in simulating fermions on a quantum computer is that qubits lack the anti-symmetry of fermions. The simplest solution, the Jordan-Wigner transformation, converts local interactions into non-local ones. I will describe a method based on Majorana fermions that preserves locality, and propose some improvements to it that reduce the CNOT gate cost and make the algorithm more suited to simulating nuclear matter. I will also suggest how a perturbation theory-based approach can be useful for studies in nuclear physics. Finally, I will discuss contributions I have made involving time fractals and quantum algorithms such as the rodeo algorithm, an eigenvalue estimation algorithm that can obtain precise results even on noisy quantum computers. 

The rodeo algorithm is an efficient algorithm for eigenstate preparation and eigenvalue estimation for any observable on a quantum computer. This makes it a promising tool for studying the spectrum and structure of atomic nuclei as well as other fields of quantum many-body physics. The only requirement is that the initial state has sufficient overlap probability with the desired eigenstate. While it is exponentially faster than well-known algorithms such as phase estimation and adiabatic evolution for eigenstate preparation, it has yet to be implemented on an actual quantum device. In this work, we apply the rodeo algorithm to determine the energy levels of a random one-qubit Hamiltonian, resulting in a relative error of 0.08% using mid-circuit measurements on the IBM Q device Casablanca. This surpasses the accuracy of directly-prepared eigenvector expectation values using the same quantum device. We take advantage of the high-accuracy energy determination and use the Hellmann-Feynman theorem to compute eigenvector expectation values for a different random one-qubit observable. For the Hellmann-Feynman calculations, we find a relative error of 0.7%. We conclude by discussing possible future applications of the rodeo algorithm for multi-qubit Hamiltonians.