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

@article{Chakhmakhchyan2017QuantuminspiredAF, title={Quantum-inspired algorithm for estimating the permanent of positive semidefinite matrices}, author={Levon Chakhmakhchyan and Nicolas J. Cerf and Ra{\'u}l Garc{\'i}a-Patr{\'o}n}, journal={Physical Review A}, year={2017}, volume={96}, pages={022329} }

We construct a quantum-inspired classical algorithm for computing the permanent of Hermitian positive semidefinite matrices, by exploiting a connection between these mathematical structures and the boson sampling model. Specifically, the permanent of a Hermitian positive semidefinite matrix can be expressed in terms of the expected value of a random variable, which stands for a specific photon-counting probability when measuring a linear-optically evolved random multimode coherent state. Our…

## Figures from this paper

## 20 Citations

### A fast quantum algorithm for computing matrix permanent

- Computer Science
- 2022

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.

### Inapproximability of Positive Semidefinite Permanents and Quantum State Tomography

- Computer ScienceArXiv
- 2021

It is shown that PSD permanents are NP-hard to approximate within a constant factor, and so admit no FPTAS (unless P=NP), and it is established that several natural tasks in quantum state tomography, even approximately, areNP-hard in the dimension of the Hilbert space.

### Quantum-inspired permanent identities

- Mathematics
- 2022

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…

### Simply Exponential Approximation of the Permanent of Positive Semidefinite Matrices

- Mathematics, Computer Science2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)
- 2017

A deterministic polynomial time cn approximation algorithm for the permanent of positive semidefinite matrices is designed and it is shown that the permanent is within a cn factor of the top eigenvalue of the Schur power matrix.

### Experimental linear optical computing of the matrix permanent

- MathematicsPhysical Review A
- 2019

Linear optical computing (LOC) with thermal light has recently gained attention because the problem is connected to the permanent of a Hermitian positive semidefinite matrix (HPSM), which is of…

### New Hardness Results for the Permanent Using Linear Optics

- Mathematics, Computer ScienceElectron. Colloquium Comput. Complex.
- 2016

A collection of new results about matrix permanents that are derived primarily via linear optical techniques are presented, which show that certain probabilities of boson sampling experiments with thermal states are hard to compute exactly despite the fact that they can be efficiently sampled by a classical computer.

### Connection between BosonSampling with quantum and classical input states

- Physics
- 2020

BosonSampling is a problem of sampling events according to the transition probabilities of indistinguishable photons in a linear optical network. Computational hardness of BosonSampling depends on…

### Connection between BosonSampling with quantum and classical input states.

- PhysicsOptics express
- 2020

It is shown that the generating function of a transition probability for Fock-state BosonSampling (FBS) can be expressed as a transition probabilities of thermal-light inputs, resulting in a fast oscillating integrand.

### Distinguishing noisy boson sampling from classical simulations

- Physics, Computer ScienceQuantum
- 2021

It is shown analytically and confirmed by numerical simulations that one can efficiently distinguish the output distribution of such a noisy boson sampling from the approximations accounting for low-order quantum multiboson interferences, what includes the mentioned classical algorithms.

### Quantum-inspired algorithms in practice

- Computer ScienceQuantum
- 2020

Overall, the results indicate that quantum-inspired algorithms can perform well in practice provided that stringent conditions are met: low rank, low condition number, and very large dimension of the input matrix.

## References

SHOWING 1-10 OF 23 REFERENCES

### Mathematical Foundations of Computer Science

- MathematicsLecture Notes in Computer Science
- 1974

Prove that if Y = Z and g = idY , then LidY (f) = f, for all f : X → Y, and that Lh◦g = Lh ◦ Lg, which is the equality of the following types: injective, surjective, and nonempty.

### Applied Numerical Linear Algebra

- Mathematics
- 1997

The symmetric Eigenproblem and singular value decomposition and the Iterative methods for linear systems Bibliography Index.

### Advanced Calculus

- EducationNature
- 1940

It is quite impossible to include in a single volume of reasonable size, an adequate and exhaustive discussion of the calculus in its more advanced stages, so it becomes necessary, in planning a thoroughly sound course in the subject, to consider several important aspects of the vast field confronting a modern writer.

### Phys

- Rev. Lett. 114, 060501
- 2015

### Quantum Inf

- Comput. 14, 541
- 2014

### Phys

- Rev. Lett. 10, 84 (1963); E. C. G. Sudarshan, ibid. 10, 277
- 1963

### Phys

- Rev. Lett. 116, 020401
- 2016

### Nature 409

- 46
- 2001

### JASA 58

- 13
- 1963