Quantum-inspired permanent identities

  title={Quantum-inspired permanent identities},
  author={Ulysse Chabaud and Abhinav Deshpande and Saeed Adel Mehraban},
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… 



Quantum-inspired algorithm for estimating the permanent of positive semidefinite matrices

A quantum-inspired classical algorithm is constructed, by exploiting a connection between these mathematical structures and the boson sampling model, that approximates the matrix permanent from the corresponding sample mean and is shown to run in polynomial time for various sets of Hermitian positive semidefinite matrices.

Classical simulation of noninteracting-fermion quantum circuits

It is shown that a class of quantum computations that was recently shown to be efficiently simulatable on a classical computer by Valiant corresponds to a physical model of noninteracting fermions in one dimension.

A fast quantum algorithm for computing matrix permanent

The well-known Ryser’s formula is transformed into a single quantum overlap integral and a polynomial sum of quantum overlap integrals to estimate a matrix permanent with the multiplicative error and additive error protocols, respectively, and it is shown that the multiplier error estimation of a matrix Permanent would be possible for a special set of matrices.

A Quantum Optics Argument for the #P-hardness of a Class of Multidimensional Integrals

Matrix permanents arise naturally in the context of linear optical networks fed with nonclassical states of light. In this letter we tie the computational complexity of a class of multi-dimensional

Generalized concurrence in boson sampling

It is shown that the minimal classical runtime for all the known algorithms directly depends on CS, which is the summation over all the members of “the generalized Fock state concurrence” (a measure analogous to the generalized concurrences of entanglement and coherence).

LOv-Calculus: A Graphical Language for Linear Optical Quantum Circuits

A confluent and terminating rewrite system to rewrite any polarisation-preserving LO v -circuit into a unique triangular normal form, inspired by the universal decomposition of Reck et al. (1994) for linear optical quantum circuits.

Continuous-variable sampling from photon-added or photon-subtracted squeezed states

We introduce a new family of quantum circuits in Continuous Variables and we show that, relying on the widely accepted conjecture that the polynomial hierarchy of complexity classes does not

A linear-optical proof that the permanent is #P-hard

  • S. Aaronson
  • Mathematics
    Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
  • 2011
By using the model of linear-optical quantum computing and a universality theorem owing to Knill, Laflamme and Milburn, one can give a different and arguably more intuitive proof of Valiant's theorem that computing the permanent of an n×n matrix is #P-hard.

Boson sampling from a Gaussian state.

A quantum optical processor that can solve a randomized boson-sampling problem efficiently based on a Gaussian input state, a linear optical network, and nonadaptive photon counting measurements is described.