Quantum Circuit Representation of Bosonic Matrix Functions

TL;DR

This work generalizes the Ising-spin correspondence from bipartite to arbitrary interaction graphs, showing that transition amplitudes of $ hat{H}^{N}$ can encode the hafnian and loop-hafnian of a real symmetric matrix $m{A}$, thereby unifying bosonic matrix-functions with quantum spin dynamics. By analyzing diagonal-free and general-diagonal cases, the authors derive explicit mappings: $raket{S| hat{H}^{N}|oldsymbol{0}}=N! ext{haf}(m{A}_{S})$ and $raket{S,S^{c}| hat{H}^{N}|oldsymbol{0},oldsymbol{0}}=N!(2(N-k)-1)!!( ext{prod}_{i otin S}A_{ii}) ext{haf}(m{A}_{S})$, which culminate in $raket{oldsymbol{ extphi}_{1}| hat{H}^{N}|oldsymbol{0},oldsymbol{0}}=rac{N!}{oldsymbol{ m L}_{N}} ext{lhaf}(m{A})$ for an appropriately prepared state $ig|oldsymbol{ extphi}_{1}ig angle$. The framework thus connects Gaussian boson-sampling quantities to spin dynamics and suggests efficient quantum-circuit implementations (XX gates, Hadamard tests, Dicke-state preparation) with potential complexity-theoretic hardness beyond bipartite graphs. This unification expands the applicability and hardness arguments of quantum spin models to more general architectures and matrix-function classes, paving the way for new quantum advantage demonstrations and algorithmic crossovers between bosonic networks and spin systems.

Abstract

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.

Figures (3)

• Figure 1: Graphical summary of the correspondences between matrix functions, graph structures, bosonic networks, and spin networks. Each matrix function counts perfect matchings of a particular class of graphs. The matrix function depends on the input states in bosonic networks and on the interaction structure of spin networks. Note that the interaction structure of spin models for each matrix function is very similar to the corresponding class of graphs. For the permanent and hafnian cases, the spin network follows the graph exactly. In contrast, for the loop-hafnian case, the spin network consists of two connected components: one encoding edges between distinct vertices and the other encoding self-loops.
• Figure 2: Diagram of our quantum spin model with $2N=6$. There are two connected components with the same number of spins generated by $\hat{H}_1$(red) and $\hat{H}_2$(blue), respectively.
• Figure 3: (a) the quantum circuit for the operator $\hat{V}$. (b) the quantum circuit for preparing $\ket{\phi_1}$.