aiqbench analysis
This paper is theoretical quantum computing work. It extends an Ising-model encoding so that hafnians and loop-hafnians, not just permanents, can be represented, and it shows the resulting dynamics can be simulated with an O(n²)-gate quantum circuit. For quantum readers, the practical question is whether this widens the family of problems that photonic and Ising-style hardware can target, and the paper's gate-count result is the concrete claim to watch. It is a mathematical framework, so it does not yet show an advantage on real devices.
Combinatorial Matrix Functions in Ising Models and Quantum Circuit Representation
arXiv:2602.01868v3 Announce Type: replace Abstract: Permanents, hafnians, and loop-hafnians are combinatorial matrix functions defined on the perfect matchings of graphs. These matrix functions arise in the quantum amplitudes for output photon configurations in linear optical networks, which has led both to sampling hardness results and to a variety of applications. The same matrix functions also appear in Ising models, since Ising models naturally encode graphs, and the matrix functions emerge from quantum interference. However, previous results have been restricted to balanced bipartite graphs, and hence to the permanent. In this paper, we extend the graph structure of the Ising model so that all three matrix functions can be obtained. In particular, self-loops can be encoded by introducing additional spins, and the loop-hafnian is obtained from a superposition of spin states, which we call the loop-hafnian state. We show how to recover the hafnian and permanent cases as the number of self-loops decreases. Moreover, we show that the Ising dynamics we consider can be simulated on a quantum circuit with \(O(n^2)\) gates for a \(2n \times 2n\) real symmetric matrix. Specifically, we propose a quantum circuit that maps the loop-hafnian state to a basis state with \(O(n^2)\) gates, and show that, by an adaptive measurement, the loop-hafnian states for all subsets of vertices can be measured simultaneously.
Abstract:Permanents, hafnians, and loop-hafnians are combinatorial matrix functions defined on the perfect matchings of graphs. These matrix functions arise in the quantum amplitudes for output photon configurations in linear optical networks, which has led both to sampling hardness results and to a variety of applications. The same matrix functions also appear in Ising models, since Ising models naturally encode graphs, and the matrix functions emerge from quantum interference. However, previous results have been restricted to balanced bipartite graphs, and hence to the permanent. In this paper, we extend the graph structure of the Ising model so that all three matrix functions can be obtained. In particular, self-loops can be encoded by introducing additional spins, and the loop-hafnian is obtained from a superposition of spin states, which we call the loop-hafnian state. We show how to recover the hafnian and permanent cases as the number of self-loops decreases. Moreover, we show that the Ising dynamics we consider can be simulated on a quantum circuit with \(O(n^2)\) gates for a \(2n \times 2n\) real symmetric matrix. Specifically, we propose a quantum circuit that maps the loop-hafnian state to a basis state with \(O(n^2)\) gates, and show that, by an adaptive measurement, the loop-hafnian states for all subsets of vertices can be measured simultaneously.
arXiv:2602.01868 [quant-ph]
(or arXiv:2602.01868v3 [quant-ph] for this version)
https://doi.org/10.48550/arXiv.2602.01868 Focus to learn more arXiv-issued DOI via DataCite
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