Connection between BosonSampling with quantum and classical input states.

Area of Science:

• Quantum computing
• Computational complexity theory
• Linear optics

Background:

• BosonSampling is a key problem in quantum computing, involving sampling events from indistinguishable photons in optical networks.
• The computational hardness of BosonSampling is tied to the photon-number statistics of the input light.
• Fock-state BosonSampling (FBS) is computationally intractable, unlike BosonSampling with classical light inputs.

Purpose of the Study:

• To establish a mathematical connection between BosonSampling with quantum (Fock-state) and classical (thermal-light) inputs.
• To simplify the calculation of transition probabilities for Fock-state BosonSampling.
• To provide new perspectives on the computational hardness of BosonSampling.

Main Methods:

• Expressing the generating function of FBS transition probabilities using thermal-light input transition probabilities.
• Utilizing a closed-form expression for thermal-light transition probabilities.
• Analyzing the integral form of FBS transition probabilities involving Gaussian functions and Laguerre polynomials.

Main Results:

• A direct mathematical relationship was identified between quantum and classical light inputs for BosonSampling.
• The transition probabilities of FBS can be computed by evaluating a single matrix permanent.
• The transition probability of FBS was shown to be an integral of a Gaussian function multiplied by a Laguerre polynomial.

Conclusions:

• The study reveals a significant mathematical bridge between quantum and classical BosonSampling.
• This connection offers a more efficient method for calculating FBS transition probabilities.
• The findings contribute to a deeper understanding of the computational complexity of BosonSampling.