Gaussian Boson Sampling (GBS) is one of the fundamental phenomena that demonstrate true Quantum Advantage — the point where it makes practical sense to use a quantum computer over a classical one to solve a specific problem.

Of course, Quantum Computers are probabilistic machines. So let us try to understand the concept of sampling from the perspective of Sir Francis Galton and his invention, The Galton Board.

The Galton Board 🧮

The Galton Board is a device designed to demonstrate the principles of Gaussian distribution. In a vertical board with interleaved rows of pegs, balls are dropped from the top, and bounce left or right as they hit the pegs. They collect into bins at the bottom, demonstrating the probability of balls collecting into different slots.

Most of the balls accumulate near the center compartments, remaining consistent even with repeated experiments. This forms a pattern that closely approximates a Bell Curve!

The principle behind the Galton Board is that each ball experiences a series of small, independent “accidents” (random bounces), causing slight deviations to the left or right. While a few balls may experience “lucky” streaks and land far from the center, the majority collect in the middle, reflecting the commonness of average outcomes.

The frequency of balls landing far from the center decreases rapidly, illustrating the core concept of probability in the Gaussian distribution.

As more balls are used, the bell curve pattern becomes more pronounced, but the fundamental distribution remains the same, with half the balls falling within a predictable distance from the center.

While it may seem simple, I think this quote by Sir Francis Galton in his book “Natural Inheritance” does a great job of describing the profoundness of the primary phenomena the Galton Board represents — “Order in Apparent Chaos”:

Boson Sampling 🎲

Alright, so we have a couple of balls bouncing here and there. What’s so special about that…😉

No seriously, the point is that the Galton Board is a sample of all the possible outcomes that could occur when dropping balls through it.

These samples allow us to make generalizations about the underlying probabilities of the Galton Board — the same concept applies when talking about Bosons.

Boson Sampling is the process of sampling from the distribution of boson patterns. This involves sending bosons (such as photons) through an Interferometer, a device that precisely measures the displacement of the boson patterns.

It turns out that this task is so complex that no classical computer can efficiently simulate it for large systems! That is why creating an Optical Quantum Computer where you can physically pass these bosons through an interferometer and sample them is so amazing.

But that’s not the only reason why Boson Sampling is so interesting. It is one of the most fundamental demonstrations of quantum advantage.

Showing that a quantum system can solve this problem efficiently is a direct challenge to the Extended Church-Turing thesis, which asserts that classical computers can efficiently simulate any physical process.

Now a regular Boson Sampling setup does suffer from many experimental issues. The biggest of which is optical noise from things like bending losses, surface scattering, and absorption.

That is where the Gaussian States come in…

Gaussian States 🛎

“Gaussian” in “Gaussian Boson Sampling” refers to the specific type of quantum states — Gaussian States — used as inputs for Boson Sampling. They are a special class of quantum states whose wavefunctions have a Gaussian Distribution shape.

They can be “squeezed” in a phase space’s position or momentum axis. If you don’t understand what that means, I recommend you check another one of my articles on Squeezed States for Biosensing Applications.

In simple terms, you can reduce the error of the Gaussian State when measuring a certain characteristic of the Boson. Mathematically, you are switching from calculating the permanent of the problem matrix to the hafnian instead.

By using Gaussian States as inputs to a Boson Sampling problem instead of regular Photons, you can get more accurate samples.

PS: The reason we use permanents and not determinants is that we are working with “Bosons”, not “Fermions”. Remember, the commutative properties of Bosons don’t involve a change of sign when particles are swapped!

Conclusion 💡

While this problem is certainly quite interesting, being one of the first to provide a foundation for quantum advantage, it doesn’t have any practical use cases in industry that solve real-world problems.

And so I think this is a great place to make a distinction between technologies that quantum advantage and solve real-world problems, versus those that don’t.

From a historical and foundational perspective, GBS proved invaluable to the development of Optical Quantum Computing, and QC as a whole.

At least now you can share this historical tidbit with your friends. Quantum history anyone?