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Abstract Ab initio calculations have an essential role in our fundamental understanding of quantum many-body systems across many subfields, from strongly correlated fermions^1–3to quantum chemistry^4–6and from atomic and molecular systems^7–9to nuclear physics^10–14. One of the primary challenges is to perform accurate calculations for systems where the interactions may be complicated and difficult for the chosen computational method to handle. Here we address the problem by introducing an approach called wavefunction matching. Wavefunction matching transforms the interaction between particles so that the wavefunctions up to some finite range match that of an easily computable interaction. This allows for calculations of systems that would otherwise be impossible owing to problems such as Monte Carlo sign cancellations. We apply the method to lattice Monte Carlo simulations^15,16of light nuclei, medium-mass nuclei, neutron matter and nuclear matter. We use high-fidelity chiral effective field theory interactions^17,18and find good agreement with empirical data. These results are accompanied by insights on the nuclear interactions that may help to resolve long-standing challenges in accurately reproducing nuclear binding energies, charge radii and nuclear-matter saturation in ab initio calculations^19,20. 

This thesis investigates quantum algorithms for eigenstate preparation, with a primary focus on solving eigenvalue problems such as the Schrodinger equation by utilizing near-term quantum computing devices. These problems are ubiquitous in several scientific fields, but more accurate solutions are specifically needed as a prerequisite for many quantum simulation tasks. To address this, we establish three methods in detail: quantum adiabatic evolution with optimal control, the Rodeo Algorithm, and the Variational Rodeo Algorithm.The first method explored is adiabatic evolution, a technique that prepares quantum states by simulating a quantum system that evolves slowly over time. The adiabatic theorem can be used to ensure that the system remains in an eigenstate throughout the process, but its implementation can often be infeasible on current quantum computing hardware. We employ a unique approach using optimal control to create custom gate operations for superconducting qubits and demonstrate the algorithm on a two-qubit IBM cloud quantum computing device. We then explore an alternative to adiabatic evolution, the Rodeo Algorithm, which offers a different approach to eigenstate preparation by using a controlled quantum evolution that selectively filters out undesired components in the wave function stored on a quantum register. We show results suggesting that this method can be effective in preparing eigenstates, but its practicality is predicated on the preparation of an initial state that has significant overlap with the desired eigenstate. To address this, we introduce the novel Variational Rodeo Algorithm, which replaces the initialization step with dynamic optimization of quantum circuit parameters to increase the success probability of the Rodeo Algorithm. The added flexibility compensates for instances in which the original algorithm can be unsuccessful, allowing for better scalability. This research seeks to contribute to a deeper understanding of how quantum algorithms can be employed to attain efficient and accurate solutions to eigenvalue problems. The overarching goal is to present ideas that can be used to improve understanding of nuclear physics by providing potential quantum and classical techniques that can aid in tasks such as the theoretical description of nuclear structures and the simulation of nuclear reactions. 

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.