Connecting quantum circuit amplitudes and matrix permanents through polynomials

TL;DR

This paper establishes a novel method linking quantum circuit amplitudes with matrix permanents by encoding polynomials into graphs, enabling estimation of quantum amplitudes on photonic devices.

Contribution

It introduces a general encoding technique that maps polynomials from quantum circuits to graph permanents, bridging qubit and photonic quantum computation frameworks.

Findings

Provides a method to encode polynomials into graphs

Shows how to express quantum amplitudes as graph permanents

Enables estimation of quantum amplitudes on photonic devices

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.