Connecting quantum circuit amplitudes and matrix permanents through polynomials

Hugo Thomas; Pierre-Emmanuel Emeriau; Rawad Mezher

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arXiv · 2408.08857

Connecting quantum circuit amplitudes and matrix permanents through polynomials

Hugo Thomas·Pierre-Emmanuel Emeriau·Rawad Mezher

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Abstract

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.

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Hugo Thomas, Pierre-Emmanuel Emeriau, Rawad Mezher. 2024-08-16. Connecting quantum circuit amplitudes and matrix permanents through polynomials. https://arxiv.org/abs/2408.08857

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