Quantum Circuit Representation of Bosonic Matrix Functions

Bosonic counting problems can be framed as estimation tasks of matrix functions such as the permanent, hafnian, and loop-hafnian, depending on the underlying bosonic network. Remarkably, the same functions also arise in spin models, including the Ising and Heisenberg models, where distinct interaction structures correspond to different matrix functions. This correspondence has been used to establish the classical hardness of simulating interacting spin systems by relating their output distributions to #P-hard quantities. Previous works, however, have largely been restricted to bipartite spin interactions, where transition amplitudes, which provide the leading-order contribution to the output probabilities, are proportional to the permanent. In this work, we extend the Ising model construction to arbitrary interaction networks and show that transition amplitudes of the Ising Hamiltonian are proportional to the hafnian and the loop-hafnian. The loop-hafnian generalizes both the permanent and hafnian, but unlike these cases, loop-hafnian-based states require Dicke-like superpositions, making the design of corresponding quantum circuits non-trivial. Our results establish a unified framework linking bosonic networks of single photons and Gaussian states with quantum spin dynamics and matrix functions. This unification not only broadens the theoretical foundation of quantum circuit models but also highlights new, diverse, and classically intractable applications.