Boson Sampling is an important non-universal Quantum Computer which is estimated to achieve quantum supremacy in the near term future. Although for it to achieve Quantum Advantage over the classical computers, we need to scale the sampler to at least 50-100 input photons. This is difficult to implement currently as we do not have efficient deterministic single photon sources. The most widely used single photon sources are probabilistic and thus they reduce the efficiency exponentially, while scaling to larger inputs. Gaussian Boson Sampling (GBS) is the method suggested that uses the non-classical nature of light to its advantage while still being in the complexity class #P. We use N Single Mode squeezed states (SMSS) of light in an M (&gt;N) input mode sampler (M has to be N2 for the complexity to remain #P hard). This setup is used because there is no need to herald the input photons, rather they are all detected at the output. Thus it’s easier to use SMSS photons then. They are then evolved through an M-mode linear interferometer with all modes measured at the output by photon number resolving detectors.. A simplified example for 50 input SMSS with 100 mode interferometer is shown below. <img src="https://lh5.googleusercontent.com/gJATsinjPGdhpW9PVIRbnESPrLD4BWj-27Co00-78xPHo2i2uoozK1KaMo0RpybQ1SntjLt5EkOw9E0glKszDmi3zHAqZw5p8Q6VehSdjw90xt7Anrp5eTluJm1XHxAQtVnSN9Hw"/> To use the non-classical nature of light, several other variations were also suggested which calculated an instance of the sampling problem and thus reduced exponential repetitions to be performed. Scattershot boson sampling uses Two mode squeezed states of light and directly connects N Spontaneous Parametric Down Converted (SPDC) sources into a N mode interferometer. Here among the pair of photons generated at the SPDC source, one photon is detected at a detector thus heralding the other photon, before sending it to the interferometer. Gaussian Boson Sampling retains both the photons on the other hand, thus doubling the input bosons. The probability of measuring a specific output pattern in Gaussian Boson Sampling: P ∝| Haf ( TS ) |2 Where Haf is the Hafnian of the matrix TS and TS is the submatrix of T, describing the Unitary matrix of linear interferometer. TS only consists of rows and columns where the photons were detected. The complexity calculation of GBS protocol is done on the basis that Hafnian is a more general function than Permanent, hence deemed to be in the Complexity Class #P.Recently Prof. Zhong’s group in China has performed the GBS experiment with 50 SMSS photons and 100 input mode interferometer. The distribution obtained in 200s on the GBS device is estimated to take more than a billion years on supercomputers.Thus this speedup is beyond comparison and various applications of GBS have been recently theorized especially in the domain of Quantum Chemistry and Graph theory. References [1] Craig S. Hamilton, Gaussian Boson Sampling, Phys. Rev. Lett. 119, 170501 (2017)[2] HS Zhong, Quantum computational advantage using photons, DOI: 10.1126/science.abe8770 (2020)

Introduction to Gaussian Boson Sampling

Devanshu Garg