Quantum-inspired permanent identities

Ulysse Chabaud1, Abhinav Deshpande1, and Saeed Mehraban2

1Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, CA 91125, USA
2Computer Science, Tufts University, Medford, MA 02155, USA

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The permanent is pivotal to both complexity theory and combinatorics. In quantum computing, the permanent appears in the expression of output amplitudes of linear optical computations, such as in the Boson Sampling model. Taking advantage of this connection, we give quantum-inspired proofs of many existing as well as new remarkable permanent identities. Most notably, we give a quantum-inspired proof of the MacMahon master theorem as well as proofs for new generalizations of this theorem. Previous proofs of this theorem used completely different ideas. Beyond their purely combinatorial applications, our results demonstrate the classical hardness of exact and approximate sampling of linear optical quantum computations with input cat states.

Some mathematical quantities are ubiquitous in mathematics, physics, and computer science. This is the case of a combinatorial object named the permanent.

By exploiting relations between the permanent and amplitudes of linear optical quantum circuits, we show that quantum-inspired techniques provide swift proofs of many important theorems about the permanent, such as the MacMahon Master Theorem.

Our quantum-inspired proofs provide new insight for the quantum scientist on combinatorial theorems and uncover new results in quantum complexity.

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[2] Jeffrey Marshall and Namit Anand, "Simulation of quantum optics by coherent state decomposition", Optica Quantum 1 2, 78 (2023).

[3] Youngrong Lim and Changhun Oh, "Approximating outcome probabilities of linear optical circuits", npj Quantum Information 9 1, 124 (2023).

[4] Erik Panzer and Karen Yeats, "Feynman symmetries of the Martin and $c_2$ invariants of regular graphs", arXiv:2304.05299, (2023).

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