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Abstract Topological data analysis (TDA) methods can be useful for classification and clustering tasks in many different fields as they can provide two dimensional persistence diagrams that summarize important information about the shape of potentially complex and high dimensional data sets. The space of persistence diagrams can be endowed with various metrics, which admit a statistical structure and allow to use these summaries for machine learning algorithms, e.g. the Wasserstein distance. However, computing the distance between two persistence diagrams involves finding an optimal way to match the points of the two diagrams and may not always be an easy task for classical computers. Recently, quantum algorithms have shown the potential to speedup the process of obtaining the persistence information displayed on persistence diagrams by estimating the spectra of persistent Dirac operators. So, in this work we explore the potential of quantum computers to estimate the distance between persistence diagrams as the next step in the design of a fully quantum framework for TDA. In particular we propose variational quantum algorithms for the Wasserstein distance as well as the $d_p^c$distance. Our implementation is a weighted version of the quantum approximate optimization Algorithm that relies on control clauses to encode the constraints of the optimization problem. 

Abstract The quantum approximate optimization algorithm (QAOA) is an approach for near-term quantum computers to potentially demonstrate computational advantage in solving combinatorial optimization problems. However, the viability of the QAOA depends on how its performance and resource requirements scale with problem size and complexity for realistic hardware implementations. Here, we quantify scaling of the expected resource requirements by synthesizing optimized circuits for hardware architectures with varying levels of connectivity. Assuming noisy gate operations, we estimate the number of measurements needed to sample the output of the idealized QAOA circuit with high probability. We show the number of measurements, and hence total time to solution, grows exponentially in problem size and problem graph degree as well as depth of the QAOA ansatz, gate infidelities, and inverse hardware graph degree. These problems may be alleviated by increasing hardware connectivity or by recently proposed modifications to the QAOA that achieve higher performance with fewer circuit layers. 

We develop a global variable substitution method that reduces n-variable monomials in combinatorial optimization problems to equivalent instances with monomials in fewer variables. We apply this technique to 3-SAT and analyze the optimal quantum unitary circuit depth needed to solve the reduced problem using the quantum approximate optimization algorithm. For benchmark 3-SAT problems, we find that the upper bound of the unitary circuit depth is smaller when the problem is formulated as a product and uses the substitution method to decompose gates than when the problem is written in the linear formulation, which requires no decomposition.