Connecting quantum circuit amplitudes and matrix permanents through polynomials

In this paper, we strengthen the connection between qubit-based quantum circuits and photonic quantum computation. Within the framework of circuit-based quantum computation, the sum-over-paths interpretation of quantum probability amplitudes leads to the emergence of sums of exponentiated polynomials. In contrast, the matrix permanent is a combinatorial object that plays a crucial role in photonic by describing the probability amplitudes of linear optical computations. To connect the two, we introduce a general method to encode an $\mathbb F_2$-valued polynomial with complex coefficients into a graph, such that the permanent of the resulting graph's adjacency matrix corresponds directly to the amplitude associated the polynomial in the sum-over-path framework. This connection allows one to express quantum amplitudes arising from qubit-based circuits as permanents, which can naturally be estimated on a photonic quantum device.