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Permanents in linear optical networks

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arxiv quant-ph/0406127 v1 pith:FLDKCODN submitted 2004-06-18 quant-ph

Stefan Scheel This is my paper · ORCID
classification quant-ph
keywords matrixpermanentselementslinearnetworksopticalabstractacross
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We develop an abstract look at linear optical networks from the viewpoint of combinatorics and permanents. In particular we show that calculation of matrix elements of unitarily transformed photonic multi-mode states is intimately linked to the computation of permanents. An implication of this remarkable fact is that all calculations that are based on evaluating matrix elements are generically computationally hard. Moreover, quantum mechanics provides simpler derivations of certain matrix analysis results which we exemplify by showing that the permanent of any unitary matrix takes its values across the unit disk in the complex plane.

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Cited by 7 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Observation of associative-memory retrieval and spin-glass phases on a photonic quantum simulator

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    Experimental observation of memory retrieval, spin-glass blackout, and paramagnetic phases in a photonic quantum simulator of a four-body Hopfield associative memory model.

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    A multi-photon state behaves like a stochastic mixture of distinguishability patterns exactly when its interference parameters depend only on permutation cycle structure, enabling a compact partition representation an...

  3. Boson-fermion complementarity in a linear interferometer: An identity relating the determinant and permanent of a matrix

    quant-ph 2023-12 unverdicted novelty 7.0 of 10

    Derives a boson-fermion complementarity identity in linear interferometers that implies a previously unknown relation between |perm(A)|^2 and |det(A)|^2 for complex matrices, extending Muir's 19th-century identity.

  4. Universal linear manipulation via routing and projective measurements

    quant-ph 2026-08 accept novelty 6.0 of 10

    Routing a single measured output mode through N reconfigurations reproduces all output statistics of any N-mode unitary interferometer; multi-routing with m detectors extends this to m-photon boson sampling.

  5. Mitigating the barren plateau problem in linear optics

    quant-ph 2025-10 unverdicted novelty 6.0 of 10

    A dual-valued phase shifter in linear optics creates variational cost landscapes with fewer local minima and outperforms prior linear-optical variational algorithms by mitigating barren plateaus.

  6. An Algorithmic Upper Bound for Permanents via a Permanental Schur Inequality

    cs.DM 2025-09 reject novelty 6.0 of 10

    A new algorithmic upper bound for permanents via a permanental Schur inequality, but the proof has an off-by-one recurrence error and a flawed PSD lemma.

  7. Unveiling Hierarchical Invariants in Multiphoton Linear Optics

    quant-ph 2025-06 conditional novelty 4.0 of 10

    Purity in linear optics is shown to decompose into three conserved subspace contributions, experimentally confirmed for two-photon two-mode states.

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