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Quantum topological data analysis with continuous variables

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  • I introduce a continuous-variable quantum topological data algorithm. The goal of the quantum algorithm is to calculate the Betti numbers in persistent homology which are the dimensions of the kernel of the combinatorial Laplacian. I accomplish this task with the use of qRAM to create an oracle which organizes sets of data. I then perform a continuous-variable phase estimation on a Dirac operator to get a probability distribution with eigenvalue peaks. The results also leverage an implementation of continuous-variable conditional swap gate.

    Mathematics Subject Classification: Primary: 62-07; Secondary: 81P68, 81P70.


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  • Figure 1.  The Betti numbers $ \beta_{0,1,2} $ for four example shapes (point, cirlce, spherical shell, and torus). They are the number of connected components, one-dimensional holes (also called tunnels or handles), and two-dimensional voids, respectively

    Figure 2.  (a) Given data represented by points. (b) For a given distance $ \varepsilon $, a circle is drawn around each point. (c) Between every two points with contacting circles a line is drawn. These connections are edges of $ n $-dimensional shapes (simplices), and the space of simplices in (c) is called a simplicial complex. For two different values of $ \varepsilon $, as in (b) i, ii, and (c) i, ii, one can get more or less connections between the data points resulting in different topologies. Therefore Betti numbers depend on the initial choice of $ \varepsilon $. It is useful to vary $ \varepsilon $ to find interesting structures

    Figure 3.  The $ k $-simplices for $ k = 0,1,2,3 $. These are a vertex, an edge, a triangle, and a tetrahedron, respectively

    Figure 4.  The action of the boundary operator is shown on a $ k = 2 $ simplex. A visual representation of a simplex being broken down into its boundary is depicted above. Its boundary consists of simplices of $ k-1 = 1 $. Below is the encoded representation of the boundary operator acting on the 2-simplex. In this encoding a 1 represents a vertex in the corresponding position in the string of bits. The boundary sum is represented by a clockwise rotation around the original simplex, and the negative sign in the result alternates as in Eq. (5)

    Figure 5.  Consider the $ k = 2 $ complex on the left, for a given value of $ \varepsilon $. In order to show that the striped area is a void, it itself must be boundary-less, and not a boundary for any part of the complex. Fulfillment of these two properties is equivalent to the combinatorial Laplacian (11) applied to the stripped area returning zero. Therefore this area would be part of the kernel of the combinatorial Laplacian for $ k = 2 $ contributing to the $ \beta_{2} $ Betti number

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