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.

💡 Research Summary

This paper establishes a unified theoretical bridge between bosonic counting problems—specifically the permanent, hafnian, and loop‑hafnian—and quantum spin dynamics in Ising models, and it translates this bridge into concrete quantum circuit constructions.
The authors begin by recalling that in standard boson sampling the output probabilities are proportional to the squared permanent of a sub‑unitary matrix, while Gaussian boson sampling (GBS) yields probabilities proportional to the squared hafnian or loop‑hafnian of a matrix determined by the interferometer, squeezing, and displacement parameters. Both permanent and hafnian are known to be #P‑hard to compute, and the loop‑hafnian, which generalises both, inherits the same hardness.
Previous work on quantum‑spin‑based sampling has been limited to bipartite interaction graphs, where the transition amplitude of the N‑th power of the Ising Hamiltonian between the all‑down and all‑up states is proportional to the permanent of a real matrix. The present work lifts this restriction and treats arbitrary interaction networks.
The core construction starts from a real symmetric 2N × 2N matrix A. When A has zero diagonal entries, the Hamiltonian reduces to
 Ĥ₁ = (1/2 2N) ∑{i≠j} A{ij} σₓ,i σₓ,j,
acting on 2N spin‑½ particles. The authors prove that the matrix element ⟨S|Ĥ₁^{k}|∅⟩, where |S⟩ contains exactly 2k up‑spins, equals k! · haf(A_S). Here haf denotes the hafnian of the principal submatrix A_S, i.e., the sum over all perfect matchings of the corresponding simple graph. Consequently, the N‑th power amplitude between the all‑down state |∅⟩ and the all‑up state |I⟩ is N! · haf(A).
When A possesses non‑zero diagonal entries, a second term
 Ĥ₂ = (1/2 2N) ∑{i≠j} A{ii} A_{jj} σₓ, \bar{i} σₓ, \bar{j}
must be added, where the barred indices refer to a second block of 2N spins. The full system now contains 4N spins. The authors show that the matrix element ⟨S, Sᶜ|Ĥ^{N}|∅,∅⟩ equals
 N! · (2(N−k)−1)!! · ∏{i∈Sᶜ} A{ii} · haf(A_S).
Summing over all subsets S of even size and normalising yields the transition amplitude between a specially prepared Dicke‑type superposition |ϕ₁⟩ and the vacuum |ϕ₀⟩ as N! · L_N · lhaf(A), where lhaf(A) is the loop‑hafnian of A and L_N is an explicit combinatorial factor. The loop‑hafnian is defined as
 lhaf(M) = Σ_{k=0}^{n} Σ_{S⊂